Imperatives and Deontic Logic

Imperatives and Deontic Logic

On the Semantic Foundations of Deontic Logic

Der Fakultat fur Sozialwissenschaften und Philosophieder Universitat Leipzig

eingereichte

D I S S E R T A T I O N

zur Erlangung des akademischen Grades

doctor philosophiae(Dr. phil.)

vorgelegt

von Herrn M. A. Jorg Hansen

g e b o r e n a m 09.09.1967in Cochem

Eingereicht am 24.01.2008Offentlich verteidigt am 18.11.2008Tag der Verleihung: 25.11.2008

Abstract

Deontic logic has so far been almost exclusively interpreted by a possibleworlds semantics and ideality or preference relations between the worlds. Indistinguishing itself from attempts at a logic of norms or logic of impera-tives, deontic logic has portrayed its formulas as ‘deontic propositions’, trueor false statements about what is obligatory, permitted or forbidden accord-ing to some unspecified normative system. Based on this idea, the ‘impera-tival tradition’ of deontic logic, an unorthodox sidearm of the mainstream,has interpreted the formulas of deontic logic not with respect to ideal or bestworlds, but to given sets of norms or imperatives. In a series of papers stand-ing in this tradition, I have shown that by using an explicitly represented setof imperatives and what they command as the logical semantics by whichthe truth of deontic formulas is defined, all the standard systems of monadicand dyadic deontic logic can be ‘reconstructed’, i.e. they are sound and com-plete with respect to such semantics. The basic concepts are motivated herefrom a position of ‘imperativological scepticism’, and the main results aresummarized.

Keywords:deontic logic, logic of imperatives, modal logic

Contents

Introduction 1

1 The “Logic of Imperatives” 31.1 Beginnings: Poincare’s Proposal . . . . . . . . . . . . . . . . . 31.2 Jørgensen’s Dilemma . . . . . . . . . . . . . . . . . . . . . . . 51.3 Dubislav’s Trick and Related Theories . . . . . . . . . . . . . 81.4 Explanations of Imperative Inferences . . . . . . . . . . . . . . 15

1.4.1 Logic of Satisfaction . . . . . . . . . . . . . . . . . . . 161.4.2 Logic of Existence . . . . . . . . . . . . . . . . . . . . . 171.4.3 Logic of Non-Factual Existence . . . . . . . . . . . . . 191.4.4 Formalistic Approaches . . . . . . . . . . . . . . . . . . 21

1.5 Ross’s Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . 251.6 Ordinary Language Arguments . . . . . . . . . . . . . . . . . 351.7 The Way to Go Forward . . . . . . . . . . . . . . . . . . . . . 46

2 Deontic Logic 502.1 Beginnings: Leibniz’s Discovery . . . . . . . . . . . . . . . . . 502.2 Standard Deontic Logic . . . . . . . . . . . . . . . . . . . . . . 532.3 Anderson’s Reduction . . . . . . . . . . . . . . . . . . . . . . 552.4 Possible Worlds Semantics . . . . . . . . . . . . . . . . . . . . 572.5 Dyadic Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . 602.6 Doubts and Questions . . . . . . . . . . . . . . . . . . . . . . 66

2.6.1 Doubts About Possible Worlds . . . . . . . . . . . . . . 662.6.2 The Question of Conflicts and Dilemmas . . . . . . . . 692.6.3 Interpretations of Deontic Logic . . . . . . . . . . . . . 72

2.7 A Fundamental Problem of Deontic Logic . . . . . . . . . . . 77

ii

3 Imperative Semantics for Deontic Logic 793.1 The Imperatival Tradition of Deontic Logic . . . . . . . . . . . 79

3.1.1 Anderson’s Reduction Revisited . . . . . . . . . . . . . 793.1.2 Kanger’s Welfare Programs . . . . . . . . . . . . . . . 813.1.3 Ziemba’s Deontic Syllogistics . . . . . . . . . . . . . . . 823.1.4 Stenius’ Logic of Normative Systems . . . . . . . . . . 833.1.5 Alchourron & Bulygin’s Expressive Conception . . . . 843.1.6 Van Fraassen’s Proposal . . . . . . . . . . . . . . . . . 853.1.7 Input/Output Logic . . . . . . . . . . . . . . . . . . . 87

3.2 Imperative Semantics for Deontic Logic . . . . . . . . . . . . . 893.3 The Reconstruction of Deontic Logic . . . . . . . . . . . . . . 94

3.3.1 Some Logics about Imperatives [2001] . . . . . . . . . . 943.3.2 Problems and Results for Logics about Imperatives [2004] 963.3.3 Conflicting Imperatives and Dyadic Deontic Logic [2005]1003.3.4 Deontic Logics for Prioritized Imperatives [2006] . . . . 1013.3.5 Prioritized Conditional Imperatives [2007] . . . . . . . 103

3.4 The Paradoxes of Deontic Logic Reconsidered . . . . . . . . . 1073.4.1 Ross’s Paradoxes . . . . . . . . . . . . . . . . . . . . . 1073.4.2 Contrary-to-duty Paradoxes . . . . . . . . . . . . . . . 1103.4.3 Paradoxes of Defeasible Reasoning . . . . . . . . . . . 113

4 Conclusion 117

Acknowledgements 120

Bibliography 121

Appendix: Research Papers 142Sets, Sentences and Some Logics about Imperatives[2001] . . . . . . 142Problems and Results for Logics about Imperatives [2004] . . . . . 164Conflicting Imperatives and Dyadic Deontic Logic [2005] . . . . . . 187Deontic Logics for Prioritized Imperatives [2006] . . . . . . . . . . . 215Prioritized Conditional Imperatives [2007] . . . . . . . . . . . . . . 249

Anhang 274Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274Bibliographische Beschreibung . . . . . . . . . . . . . . . . . . . . . 281Lebenslauf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

iii

iv

Publikationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Vortrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Versicherung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Introduction

At the biannual ∆EON-workshop on Deontic Logic in Computer Science,held at the University of Bologna, 8–10 January 1998, David Makinson drewattention to the distinction between norms, that cannot meaningfully betermed true or false, and deontic propositions, true or false statements aboutwhat is obligatory, permitted or forbidden according to the norms, and thatthis distinction has tended to get lost in modern deontic logic literature.He therefore called for a “reconstruction of deontic logic” as a logic con-cerned with norms, but in accord with the philosophical position that normsare devoid of truth values. Motivated by this call I have, in a series of re-search papers since then, provided a semantics for deontic logic that definesits operators by means of an explicitly given set of imperatives. It can bedemonstrated that all of the standard systems of monadic and dyadic deonticlogic can be ‘reconstructed’ with respect to such an imperative-based seman-tics, but also other systems are explored that consider specific issues, likethe possibility of conflicts and dilemmas, or the accommodation of priorityrelations between the imperatives.

Here, the topic is explained and the results are presented that have beenachieved so far. Chapter 1 addresses the question of whether there exists alogic of imperatives. If it does, then it would perhaps be easier to devise adeontic logic as a logic that reflects already given logical relations betweenimperatives, but the arguments make it clear that our start must be froma position of ‘imperativological scepticism’, and no direct logical relationsbetween imperatives can be assumed. Chapter 2 lays out the main systemsof deontic logic that are later to be ‘reconstructed’, and explains the back-ground of Makinson’s criticism directed at traditional deontic logic. Chapter3 first provides an overview over an already existing ‘imperatival tradition’of deontic logic that has, instead of using standard possible worlds seman-tics with ideality or preference relations, interpreted deontic operators with

1

INTRODUCTION 2

respect to given sets of imperatives or norms. Then the ‘imperative seman-tics’ used here to ‘reconstruct’ the systems of deontic logic is explained, andthe results of the previous research are summarized. Finally it is examinedin which way previous ‘paradoxes of deontic logic’ may be treated withinthe new semantical framework. The conclusion explains what the resultsmean for our understanding of deontic logic, and hints at possible furtherdevelopments.

Chapter 1

The “Logic of Imperatives”

1.1 Beginnings: Poincare’s Proposal

Can imperatives, i.e. sentences in the imperative mood, be part of logicalinferences? Henri Poincare considered this question in his 1913 essay “LaMorale et la Science” [172]. He begins by observing that if the premissesare all indicatives, then so will be the conclusion, hence for an imperativeconclusion at least one premiss in the imperative mood is required, and soscience alone cannot establish standards of morality. However, just as steamcan be put to use in different machinery, science may also be used for moralreasoning:

“It [the moral sentiment] will give us the major premiss of ourinference which, as it happens, is in the imperative mood. Atits side, science will put the minor premiss which will be in theindicative mood. From these a conclusion can be drawn that isin the imperative mood.”

Poincare seems to have in mind Aristotelian syllogisms of the following kind:

Hang all dwellers of the Nottingham Forest!All members of Robin’s band dwell in the Nottingham Forest.Therefore: Hang all members of Robin’s band!

Poincare then proceeds to give a second example of an inference with animperative conclusion:

3

LOGIC OF IMPERATIVES 4

“One can imagine inferences which are of the following type: dothis, but now if one does not do that, one cannot do this, so dothat. And such reasoning is not outside the field of science.”

The following is an example of the suggested inference:

Open the door!The door cannot be opened unless it is first unlocked.Therefore: Unlock the door!

So an order to do one thing includes orders to do all that is necessary to satisfythe primary command. Poincare’s proposals raise questions: by exchangingin his first example the syllogism barbara for camestres we obtain:

Hang all dwellers of the Nottingham Forest!No member of Robin’s band is hanged.Therefore: No dweller of the Nottingham Forest shall be amember of Robin’s band!

But it did not seem as if the speaker, e.g. the Sheriff of Nottinghamshire,was creating rules for band membership. The second type of inference isproblematic when there are no legal means to fulfill an imperative (cf. Foot[54] p. 384):

Sustain your aged parents!I can only sustain my aged parents if I rob somebody.Therefore: Rob somebody!

So if commanding means also commanding all necessary acts, then evenforbidden acts may be included. To improve matters, the second clause inPoincares scheme might be changed to ‘this can only legally be brought aboutby doing that’. But this introduces a normative element into a premiss thatPoincare assumed to be established by science alone.

While all this suggests that imperative inferences might require someadditions and modifications, the most difficult question has turned out tobe what makes them inferences. The problems attached to this question gounder the name of ‘Jørgensen’s Dilemma’.


1.2 Jørgensen’s Dilemma

Logic’s concern is with the soundness of arguments, or inferences. Theseconsist of sentences that represent the ‘premisses’ and usually one sentencethat forms the ‘conclusion’. The argument is then called ‘sound’, ‘valid’or ‘logical’, if it is not possible that all of its premisses are true but theconclusion false. The premisses are then said to ‘entail’ the conclusion whichthus ‘follows’ from them.1 Expressions in the imperative mood are not, inany usual sense, true or false. Therefore they are incapable of functioningas premisses or conclusions in logical inferences. However, people maintainthat the opposite is true and that there are inferences that have conclusionsin the imperative mood and premisses of which at least one is likewise inthe imperative mood (cf. Poincare’s examples above). This is a puzzlingsituation, which was first pointed out by Jørgen Jørgensen in [110]:

“So we have the following puzzle: According to a generally ac-cepted definition of logical inference only sentences which are ca-pable of being true or false can function as premisses or conclu-sions in an inference; nevertheless it seems evident that a con-clusion in the imperative mood may be drawn from two premisesone of which or both of which are in the imperative mood. Howis this puzzle to be dealt with?”

To find this puzzle, called ‘Jørgensen’s Dilemma’, perplexing, one mustaccept that imperatives cannot be meaningfully termed true or false. Thisseems to be the philosophical consensus, it can point to Aristotle’s definitionof an assertion as a grammatical entity that can be true or false, in distinctionto other grammatical entities like requests that are neither true or false (Deinterpretatione 17 a 4). Nevertheless, a way out of the dilemma may consistin giving up just this claim. Most prominently, Kalinowski [111], [114] hasargued that in the case of expressions of moral or legal norms, the attitude ofthe ‘ordinary’, non-philosophical person is to treat these as true or false. E.g.people say that it is true that another person’s right to live must be respected,or that slander is prohibited, and people would uphold these truths even ifparticular legislators did not enact such norms, or proclaimed otherwise.So Kalinowski concludes that legal or moral norms can be part of logicalinferences. I think that these considerations confuse truth with the notion of

1For such textbook definition cf. Mates [147] p.5, Lemmon [134] p.1, Hodges [98] p.55.


a legal or moral norm’s validity: the ‘external’ recognition of a norm as validin a certain society. For the present discussion it suffices that Kalinowskihimself restricts his view to legal and moral norms and does not claim that‘imperatives in the strict sense’ can be said to be true or false, and in factwrites that they are not true or false.2 But it is these that we are concernedwith.

On another view, any imperative can be equivalently replaced by a firstperson expressions like ‘I command you to ...’, ‘I order you to ...’, ‘I request ofyou that you ...’ or ‘I want you to ...’.3 But expressions like “I command youto ...” cannot only be used to command, but also to assert that I do in factcommand so-and-so. Therefore Sigwart [195] claims that each imperativeincludes the statement that the speaker wills the act which he commands.While he adds that nevertheless the import of what is said by the imperativeis not the communication of a truth but a “summons to do this, to leave thatundone”, Ledig [129] argues that because imperatives include such assertions,one must consequently apply the terms of truth and falsity to an imperative,and that it will be true unless it is e.g. stated for fun. Likewise, Kamp[118] points out that from the viewpoint of the addressee, it really makesno difference if an expressions like “I command you to ...” is understoodas a command or as an assertion, since if the utterance is appropriate (i.e.made in earnest in the appropriate conditions) then it must be true and sothe practical consequences are the same. Similarly, Opa lek & Wolenski [164]call a normative statement qua performative true if it is effective. However,it is hard to see how these considerations can solve Jørgensen’s dilemma. Ifany imperative that is meant seriously is true, then (i) any imperative eitherfollows from any other ‘true’ imperative or is not meant seriously, and (ii)all imperatives follow from one made for fun. If imperatives are ambiguousand include an assertion, then the fact that this assertion may be used in anindicative inference does not mean that the indicative conclusion is likewiseambiguous. A sign post bearing the words ‘Frankfurt/Main’ conveys theinformation that if I follow the indicated direction I will eventually arriveat Frankfurt. I may also infer that there exists a place called Frankfurt andthat a geographical entity called Main, namely a river, exists in its proximity.Knowing that sign posts tend not to include redundant information, I can

2Cf. Kalinowski [112] p. 36 and [114] p. 107.3In the case of English grammar, it has been claimed that the exclamation mark,

or rather the characteristic intonation that it replaces in written language, is only theremainder of such explicit performative lead-ins, cf. Harris [89] pp. 391–392.


also conclude that there must be another place that is also called Frankfurt(namely Frankfurt/Oder), and that this place does not lie near the Main,because otherwise the extra information would not have been discriminating.But the sign post does not point into the direction of this other city. Similarlythe information that is obtained by interpreting an imperative utterance maybe used to infer some other information. But this is not inferring imperatives.

The fact that imperatives are traditionally not considered to be true orfalse finds its explanation in the different intentions in which imperativesand indicatives are used. The main use of indicatives is to convey what thespeaker believes the world to be like. If it is so, then the sentence is called‘true’, if not, then it is called ‘false’ and the recipient might point out that Ishould perhaps change my beliefs. By use of an imperative I tell the addresseewhat I want to be done. If the addressee does what is demanded, the actionmay be qualified as ‘right’, or satisfactory with respect to the command, andif not, then the behavior of the addressee is in some sense ‘wrong’ and I willperhaps remind the agent of his or her obligation. So truth and falsity arethe qualities of descriptions when things are or are not as they have beendescribed, while ‘right’ or ‘wrong’ are the qualities of acts that are or are notin accordance with what has been prescribed. Descriptions and prescriptionshave a different ‘direction of fit’, and true/false are the terms used to expressthe match/mismatch on the language side in case of a descriptive use oflanguage, and right/wrong are the terms employed for the match/mismatchon the world side in case of a prescriptive use.4 Therefore it is a confusionof language, and indicates a misunderstanding of the intention in which thesentence has been uttered, if imperatives are termed true or false.

Accordingly, the most effort regarding Jørgensen’s dilemma has beenspent on developing alternative definitions for ‘imperative inferences’, ratherthan arguing for the application of the terms of truth and falsity to imper-atives – unless one is already convinced by the dilemma that such things asinferences with imperatives are at all impossible (e.g. Keene [120]).

4This explanation of why the terms of truth and falsity are not applicable to normativeuses of language originates with Anscombe [19] §32. Independent accounts can be foundin Kenny [122] p. 68 and Peczenik [166], [167] who speaks of the norm as a ‘qualifyingutterance’. The dual terms right/wrong are used as corresponding qualifications e.g. byEnglis [52] and Kelsen [121] p. 132.


1.3 Dubislav’s Trick and Related Theories

To deal with his own ‘dilemma’, Jørgensen [110] endorsed a proposal byWalter Dubislav [50] to transfer the ‘usual definitions’ of inferences betweenindicatives to imperatives ‘by analogy’. Dubislav gives the following example:

Though shalt not kill.Therefore: Cain shalt not kill Abel.

Here, he argues, the analogue of the following ‘ordinary’ inference is applied:

No human being kills any other human being.Cain and Abel are human beings.Therefore: Cain does not kill Abel.

Dubislav observes that to each imperative belongs a descriptive sentence thatdescribes the state of affairs that obtains if the subjects of the imperativerealize what the commanding authority demands. The formalisms of descrip-tive inferences are then transferred to imperatives by what he calls a ‘trick’(Kunstgriff ): imagine the state that the commanding authority desires real-ized, describe it, from this description infer some other descriptive sentence,which is then again interpreted as describing a state the authority wants tosee realized. He then proposes the following convention (DC) on the meaningof imperative inference (also see the next figure):

(DC) “An imperative F is called derivable from an imperativeE if the descriptive sentence belonging to F is derivable withthe usual methods from the descriptive sentence belonging to E,whereby identity of the commanding authority is assumed.”

The convention is illustrated by the next figure (where I write !A for animperative to which the descriptive sentence A ‘belongs’):5

Dubislav’s convention does not cover inferences with more than one im-perative premiss, though he mentions this possibility.6 For such inferences,(DC) can be modified as follows:

5Mally [145] p. 12 seems to have introduced the symbolism !A, which was then employedby Hofstadter & McKinsey [99] for the imperative that demands that A be the case.However, Mally intended !A to be interpreted theoretically, as an assertion or assumptionthat ‘A ought to be’, which we now call a deontic proposition and formalize by OA.

6Cf. Dubislav’s use of the plural when stating that “an inference from demand-sentenceswill now be formally facilitated by the following convention”, and Dubislav’s summary,in which he stresses that no demand-sentence can be derived from premisses that do notcontain at least one demand-sentence.


Figure 1.1: Dubislav’s convention (DC).

(DCM) An imperative F is called derivable from the imperativesE1, ..., En if the descriptive sentence belonging to F is derivablewith the usual methods from the descriptive sentences belongingto E1, ..., En, identity of the commanding authority presupposed.

Dubislav then proceeds to inferences in which the imperative premiss is ac-companied by another one in the indicative mood, and where the conclusionis again an imperative, for which he extends his convention:

(DEC) “An imperative F is called derivable in the extendedsense from an imperative E if the descriptive sentence belongingto F is at least jointly derivable from the descriptive sentencebelonging to E and true descriptive sentences that are consistentwith the first.”

This extended convention (DEC) may again be modified to facilitate in-ferences with more than one imperative premiss to produce the followingmodified extended convention

(DECM) An imperative F is called derivable in the extendedsense from imperatives E1, ..., En if the descriptive sentence be-longing to F is at least jointly derivable from the descriptive sen-tence belonging to E1, ..., En and true descriptive sentences thatare consistent with these.

Jørgensen [110] endorsed Dubislav’s proposal as one way to deal with hisdilemma and states that it seems clear to him that any imperative sentence


has an indicative parallel-sentence which describes the contents of the com-mand or wish. Jørgensen suggests that an imperative consists of an impera-tive factor and an indicative factor, where the first indicates that somethingis commanded, and the second what is commanded. The indicative factorcan then be separated from the imperative and formulated in indicative sen-tences describing the action, change or state of affairs which is commanded.Applying the rules to these latter sentences we can thus indirectly apply therules of logic to the imperative sentences to make their entailments explicit.Writing ` A for the assertion that the state of affairs A holds, this slightlydifferent analysis of imperative inferences is illustrated by the next figure:7

Figure 1.2: Jørgensen’s analysis of indirect imperative entailment.

Jørgensen’s concept was in turn further refined by R. M. Hare [86], accord-ing to whom an imperative sentence and an indicative sentence ‘correspond’if they have the same ‘descriptor’, but different ‘dictors’, where what is de-scribed by the descriptor is what would be the case if the sentence is true orthe command obeyed, and the dictor is what does the saying or command-ing.8 This is still not much different from Jørgensen’s analysis, but Hare findsit is misleading to speak of an ‘indirect’, ‘parallel’ or ‘analogous’ applicationof logic. Instead, in Hare’s view imperatives are logical in the same way asindicatives; he argues that “most inferences are inferences from descriptor todescriptor and we could add whichever set of dictors we pleased”. Since most

7This formalization is used in Reichenbach [180] §57.8This distinction was anticipated by Ledig [130], who wrote that norms and descrip-

tions have an isolable imaginary content (isolierbarer Vorstellungsinhalt). Hare’s [87] laterterminology of a ‘neustic’ and ‘phrastic’ mirrors his earlier distinction.


logical reasoning is done with descriptors only - this Hare calls the ‘principleof the dictive indifference of logic’, there is no special need for a logic ofimperatives. Rather, all logic is recast as a logic of descriptors, where if thedescriptors of the premisses describe a state of affairs, then the descriptor ofthe conclusion describes, at least partially, the same state of affairs. Hare’sview of logic is pictured in the next figure:

Figure 1.3: Hare’s descriptor logic.

It is immediate that neither Jorgensen’s nor Hare’s account make a ma-terial difference for what imperative inferences should be accepted. Hare’sconcept of a ‘dictor’ that operates on a ‘descriptor’ poses problems: gram-matically, it is hard to see how dictors can be removed from sentences, orexchanged in them, so that the remainder or the new composite is a mean-ingful expression.9 So there may be reasons not to follow Hare’s analysis,but – closer to Dubislav’s original concept – speak of a thematically parallelsentence in the indicative mood. However, this way or another, the idea of adescriptive sentence that parallels an imperative or of an imperative’s indica-tive factor has become the most successful part of the Dubislav-Jørgensen-Hare analysis. Ross [188] calls this element the ‘theme of demand’, a statethe realization of which is requested by the demand, and proposed that to

9Opa lek [162] points out that even if the imperative is rephrased as ‘I command that...’ or ‘it is obligatory that ...’, the ‘...’-part is a Latin ut-expression that only due toa peculiarity of English grammar may be confused with an indicative sentence, also cf.Opa lek [163] ch. 2, Kalinowski [113] and Rodig [186] for similar criticism. On the otherextreme, Leonard [135] has argued that it is the descriptor that is called ‘true’ and ‘false’,and so imperatives share these properties with descriptive sentences.


any imperative corresponds an ordinary indicative sentence which containsa description of the imperative’s theme of demand. Frey [58] p. 440 usesthe term Erfullungsaussage for the parallel sentence that indicates if the im-perative is satisfied or violated, which Rescher [184] p. 52 calls a ‘commandtermination statement’ and Keene [120] the ‘actualization’ of the imperative.Geach [60] states that for every imperative there is a future-tense statementwhose ‘coming true’ is identical with the fulfilment of the imperative, Sosa[200], [202] speaks of the ‘propositional core’ of an imperative and Hanson [83]of a state s the commanding agent ‘envisages’, of which then a description Sis used. Von Wright (e.g. [255] p. 269) calls the state the norm ‘pronounces’as obligatory or allowed the ‘content of a norm’. The seemingly universalconsensus is explained by the pragmatic function of imperatives, that is, theregulation of human behavior: if there were no imperative-correlated indica-tive sentences, it could not be understood what ought to be, and neitherwould it be possible to determine whether the norm is satisfied or violated.10

This is why even ‘anti-reductionist’ authors that oppose the idea that im-peratives or imperative reasoning can be reduced to indicatives or indicativelogic, agree on the following principle:11

Principle (W) (Weinberger’s Principle).To each imperative there corresponds a descriptive sentence that is true if theimperative is satisfied and false if it is not-satisfied (violated).

It is clear that an acceptance of (W) does not force us to also accept theDubislav-Jørgensen-Hare account of imperative inference, which neverthelesshas been accepted by a number of authors.12 (W) can then be used to

10This explanation and the formulation of the principle below is most clearly expressedin Weinberger [234] p. 172. Weinberger uses the term ‘coordination’ instead of ‘corre-spondence’, but this suggests an onto or even one-to-one mapping.

11Besides Weinberger cf. Hamblin [74] pp. 151-152: “Take the exact words of the imper-ative, and transform them into indicative mood (...) Now the worlds which extensionallysatisfy the imperative are just those of which the description is true.”, and Moutafakis’theorem T3 ([155] p. 155), which expresses the equivalence of the statements that animperative is satisfied and that a description of the prescribed action as performed is true.

12These include Simon [196], who converts commands to declarative mode “by removingthe imperative operators from them”, obtaining a theory in which all recipients obey thecommands, and then applies the ‘ordinary laws of logic’ to derive new relations that maybe converted back into commands. According to Niiniluoto [156], an “imperative !p entailsimperative !q if p entails q”. Very close to (DCM) is von Wright’s account in [246] pp. 71,164, where he defines the content of a prescription as ‘the prescribed thing’, and definesthat a command is entailed by a second command or by a set of commands if the content


show that seemingly differing explanations of imperative inferences are in factequivalent to this account. Thus Rescher [184] defines command inferencesin terms of satisfaction in the following way:

A command inference is valid if there is no possible world inwhich the premisses are all satisfied and the conclusion fails tobe satisfied.

which, using (W), is equivalent to

A command inference is valid if there is no possible world in whichthe descriptive sentences corresponding to the premisses are alltrue and the descriptive sentence corresponding to the conclusionis false.

Using the textbook definition of an argument, this is equivalent to

A command inference is valid if the descriptive sentences corre-sponding to the premisses entail the descriptive sentence corre-sponding to the conclusion.

which in turn is the modified Dubislav convention (DCM).13

Using the idea that norms qualify the states of affairs that satisfies orviolates them as ‘right’ and ‘wrong’, one can define:14

An imperative !A entails an imperative !B if and only if (iff)every state of affairs that is qualified as wrong by !B is qualifiedas wrong by !A.

of a command is a consequence of the conjunction of the content of a command with thecontents of none or one or several other commands.

13Sosa’s [200], [202] definition of a ‘directive argument’ is very similar to Rescher’s, butadditionally demands that the imperatives that function as premisses are jointly satisfiablein order to cope with normative conflicts. Sosa adds a second condition, demanding thatif the imperative conclusion is violated, at least one imperative premiss must be violated,in order to also cope with conditional imperatives. This equals Keuth’s [123] condition B1

that it must be logically impossible to violate the conclusion without violating a premiss.Obviously, the second condition makes no difference for unconditional imperatives.

14The definition is similar to the one used by Peczenik [166], [167] for forbidding normsand the quality ‘forbidden’. Kamp [117] uses an analogous definition for permissions,where one permission entails another if the second makes only such courses of actionspermissible that were already so before.


which can then be translated into

An imperative !A entails an imperative !B iff every state of affairsthat violates !B also violates !A.

which using (W) is equivalent to

An imperative !A entails an imperative !B iff every state of affairsin which B is false also makes A false.

which using classical logic is equivalent to

An imperative !A entails an imperative !B iff every state of affairsin which ¬B is true also makes ¬A true.

which using tertium non datur and modus tollens is equivalent to

An imperative !A entails an imperative !B iff every state of affairsin which A is true also makes B true.

which by definition of entailment equals

An imperative !A entails an imperative !B iff A entails B.

and this is again Dubislav’s convention (DC).Lemmon [133] defines an entailment relation for imperatives via a defi-

nition of inconsistency of a set of imperatives and indicatives, where such aset is called inconsistent if it cannot be the case that all indicatives are trueand all imperatives obeyed.15 Lemmon’s entailment relation is then definedas follows:

An imperative !A is entailed by a set indicatives and imperativesif this set together with !¬A is inconsistent.

15Lemmon expresses reservations regarding his definition, but only because he thinksthat it does not sufficiently restrict imperative conclusions to statements about futureactions. A definition similar to Lemmon’s seems to be intended by Philipps [171] p. 364who defines: ‘to do p is forbidden!’ is true iff the indicative ‘someone does p’ is incompatiblewith the class of valid prescriptions, where ‘compatible’ means that if the indicative is true,then at least one prescription is violated.


where !¬A is the imperative that is satisfied if and only if A is false. Using(W) we obtain:

An imperative !A is entailed by a set indicatives and impera-tives if the set of indicatives together with the set of descriptivesentences corresponding to the imperatives together with ¬A isinconsistent.

which, using classical logic, is equivalent to

An imperative !A is entailed by a set indicatives and imperatives ifthe set of indicatives together with the set of descriptive sentencescorresponding to the imperatives entails A.

and this is Dubislav’s modified extended convention (DECM). This showsthat, with (W), Dubislav’s proposal is equivalent, or at least very similar, toa number of other proposals how imperative inferences are possible in theface of Jørgensen’s dilemma.

1.4 Explanations of Imperative Inferences

Dubislav’s ‘trick’ provides a formal method that explains how inferencesbetween imperatives can be defined without having to assign them truthvalues. What has not been made clear is what is achieved by the method, i.e.why we should think that this is what it means to ‘infer’ an imperative fromsome other imperative or a set of imperatives and indicatives, or formally,what is means that some imperative inference scheme

(ImpInf) !A∴ !B

is valid. Dubislav’s trick can easily be applied to e.g. sentences of the form‘I doubt that ...’. Then from ‘I doubt that he is staying at his sister’s placein San Francisco’ follows ‘I doubt that he is staying in San Francisco’, which,though we can derive ‘he is staying in San Francisco’ from ‘he is staying atthis sister’s place in San Francisco’, seems wrong: I might not doubt that heis staying in San Francisco, but doubt very much that he is staying with hissister. So why should Dubislav’s trick work for imperatives if it would notfor other expressions?


1.4.1 Logic of Satisfaction

On one interpretation, which has been called the ‘logic of satisfaction’, thescheme (ImpInf) is understood as stating that if the imperative sentence !Ais satisfied, then it must be that some other imperative sentence !B is alsosatisfied. This interpretation is usually attributed to Hofstadter & McKinsey[99], whose formalization of the scheme (ImpInf) would be !A >!B, whichis derivable in their axiom system whenever A → B is classically derivable.It is immediate that (ImpInf) must be valid on this interpretation wheneverthe ‘ordinary’ argument

A∴ B

is valid for the descriptive sentences A and B: If A classically implies B, thenit must be that if !A is satisfied, then A is true and also B is true and hence!B is satisfied. Thus Dubislav’s ‘trick’ receives its semantic justification. Butif (ImpInf) is interpreted in this way, then it seems one should also acceptthe following scheme:

!A

∴ A

According to our (informal) convention, !A represents an imperative sentencethat is satisfied iff A is true. So it must be that if !A is satisfied, then A istrue and so the above scheme is valid. But on the look of it, this schemeseems to state that from an imperative that demands A it can be inferredthat A is the case, which is nonsense. And this misunderstanding revealsthat when we spoke of the possibility of an inference in which the premissesand the conclusion are imperatives, it seems that we talk about inferring animperative from some other imperatives, and not about reasoning whetheror not the imperatives in question are satisfied. So though the inferencesof a ‘logic of satisfaction’ are valid in the interpretation in which they wereintended, thus interpreted inferences seem not to be what we want from alogic of imperatives. For these reasons, Ross [188] p. 61 and also Hare [88]doubted that a logic of satisfaction is what one has in mind in the case ofpractical inferences.16

16Kanger [119] p. 49 and Føllesdal & Hilpinen [53] p. 7 criticize Hofstadter & McKinseyfor making !A and A ‘equivalent’, which is somewhat unfair since the intended interpre-tation of their formulas (in terms of satisfaction) is not presented.


1.4.2 Logic of Existence

When we speak of inferring one imperative from some other imperative, thiscould mean that the existence of an imperative is logically deduced from theexistence of some other imperative.

What is meant by saying that an imperative ‘exists’? First, it could be theexistence of an utterance of some sentence in the imperative mood by somecommanding agent towards some commanded subject.17 Second, one mightdemand that the utterance, as a performative use of the imperative sentence,was effective and established an imperativum, i. e. a ‘command’, ‘demand’,‘request’ or the like. For this it may be required that the commanding agenthad the will to command (and did not use the words for fun) as well as someauthority (power to punish or reward) over the addressee.18 Third, for anorder by legal authorities in this capacity to come into ‘legal existence’ itmay be required that the authority was competent to utter it according tothe legal rules of some normative system that confers such competence, andsimilar for bodies that are constituted not by law but by other rules like afirm or Robin’s band.19

Yet however much the concept of existence is thus refined, it seems torequire the presence of actual facts: a (still alive?) speaker, a linguisticentity like an utterance and circumstances of speaking, a certain attitude ofthe speaker towards the act of speaking, a backing of the speaker by force oran authority conferred by existing and/or valid rules, etc. But it is difficultto see how logic can stipulate such an existence. This is illustrated in thefollowing example by Aleksander Peczenik:20

17This existence is what Frey [58], along with an additional property of ‘justification’,infers in imperative inferences: “If the imperatives that appear in the premisses exist andare justified, then also the imperatives derived from these exist and are justified” (p. 465).Frey’s ‘justification’ means that what is demanded is ‘good’ regarding some aim of thecommanding agent, called ‘axiological validity’ in Ziembinski [260].

18Cf. von Wright [246] p. 120-126. This is Ziembinski’s [260] ‘thetic validity’. Lemmon[133] seems to have this notion of ‘validity’ or ‘existence’ in mind when he demands thatthe entailment of imperatives must be defined in terms of what imperatives are in forceat a given time.

19Bulygin [39] uses the term ‘systemic validity’. According to Weinberger [228], [231] p.259, this validity takes the place of truth as the ‘hereditary trait’ (Erbeigenschaft) that istransferred from the premisses to the normative conclusion in inferences with normativesentences (Normsatze).

20Quotation from a letter by A. Peczenik to R. Walter, printed in [218] p. 395


“The premiss ‘love your neighbour’ may be regarded as describingthe fact that the authority – Jesus – has in fact said ‘love yourneighbour.’ The imperative existed because it was uttered byJesus. But the conclusion, for example, ‘love Mr. X’ does notdescribe anything which in fact has been said by Jesus.”

Here, the intended argument from ‘love your neighbour’ to ‘love Mr. X’is not accepted because the commanding agent ‘did not actually say’ whatappears in the conclusion, and so unlike the premiss the conclusion did never‘exist’ as a fact. An imperative sentence that has not been expressed was notreceived and cannot be understood by its addressee as a command or legalorder. Thus the required ‘existence’ of the imperative, or ‘validity’ of thecommand seem to be the analogues not of truth of a descriptive sentence, butof ‘stated’ and ‘asserted’. Yet indicative logic does not force anyone to stateor assert anything, even if some other descriptive sentence was used in a waythat expresses one’s commitment to it. It only explains what people meanwhen they use an indicative sentence in order to assert some fact, by sayingwhat other sentences must be true if the stated sentence is true.21 Becausethe imperative sentence in the conclusion may not exist as an utterance,Hamblin [74] p.89 warns against speaking of inferences between imperatives.Because the ‘telling part’ (or attitude) of the speaker cannot be inferred, thepossibility of command inferences was denied by Sellars [193] p. 239-240, andfor the same reason, Lemmon’s [133] attempt to define imperative inferencesvia the notion of an imperative’s being ‘in force’ was rejected by Sosa [202]p. 61 who argues that such notions involve attitudes by the authority or itssubject that cannot be inferred. Von Kutschera [128] argues that a (used)imperative is an action, actions do not follow from actions, so there is nologic of imperatives. In Alchourron & Bulygin’s [5] ‘expressive conceptionof norms’, the existence of a norm is seen as dependent on empirical factsand the possibility of a logic of norms is consequently denied as there areno logical relations between facts. For similar reasons, Philipp [168], [169]denied both the possibility of a logic of imperatives and of norms.

21Kenny [122] first pointed out that ‘valid’ and ‘invalid’, interpreted as meaning ‘com-manded’ and ‘not commanded’, are not the analogues of ‘true’ and ‘false’, but of ‘stated’and ‘not stated’.


1.4.3 Logic of Non-Factual Existence

To get around this difficulty one may consider to interpret ‘existence’ notwith respect to natural facts, but with respect to some ideal ‘world of ought’or an assumed ‘normative system’ that is closed under consequences, wherethe closure operation may be understood e.g. in the sense of derivability byuse of Dubislav’s convention. So if the agent’s use of the imperative moodhas resulted in the existence of a command in the ‘world of ought’, or, dueto the agent’s legal competence as e.g. a police officer, created an orderthat now belongs to the normative system, then all the ‘consequences’ of thecommand that can be derived by an appropriate method exist in this world orsystem as well.22 What thus has ideal existence is not the imperative sentenceas a spatial and temporal phenomenon or as a grammatically correct ormeaningful combination of words. Rather, it is what the use of an imperativesentence expresses or accomplishes – a command, request etc. Then it mustbe that not only commands, requests etc. ‘exist’ in this sense that in facthave been expressed by a performative use of a sentence in the imperativemood, but also some that only can be expressed. For if all that ‘exists’in the ‘normative system’ already exists as a result of a pragmatic use oflanguage, then there would be no need to let the ‘normative system’ e.g.be closed under a consequence operation. That what we can express byusing language (commands, requests, assertions etc.) has some existence,

22The ‘world of ought’ terminology originates with Walter [217], who is however followingKelsen [121] p.195 in that an individual norm does not exist before the general normwas applied by a judge, so orders that can ‘only be deduced’ do not exist in Walter’s‘world of ought’. The idea to explain logical relations between ‘norm sentences’ (likeimperative sentences) in terms of their existence in a ‘system of norms’ that is closedunder consequences is that of Stenius [205]. In Opalek & Wolenski [165], norms arenon-linguistic entities expressed by (descriptively interpreted) deontic statements, andnormative systems consist not only of norms that have been expressed by a normativeauthority, but also of the consequences of these ‘basic obligations’. In Alchourron &Bulygin’s ‘hyletic’ variant of a conception of norms [8], ‘implicitly promulgated’ norms have‘existence’ in a logically closed normative system, and descriptively interpreted (deontic)norm propositions are then “propositions about the existence of norms (in that system)”.Hollander [100] promotes the idea of a ‘deontically perfect world’ where norms exist thatobey logical principles, like that conflicts are excluded. Kelsen [121] pp. 187–188 rejectsthe idea of an ‘ideal existence’ of norms because there is no ‘ideal’ act of will that createsthem, and rejects the whole idea of a logic of norms.


possibly independent from any pragmatic origin23, in some ideal ‘world ofought’, is a difficult concept that possibly creates more problems than itsolves.24 But it is even difficult to see that it solves the problem of entailmentbetween imperatives. For to say that some ideal object created by use of animperative implies the existence of some other ideal object in some ‘world ofought’ or normative system, is not to say that an imperative implies anotherimperative. The existence of a forest might imply the existence of a tree, butto say that ‘the forest implies the tree’ is making a categorical mistake. Thefirst is an indicative argument, which can be formalized in the usual way:

(1) ∃x : Forest(x)∴ ∃y : Tree(y)

The argument is analytical when the words ‘forest’ and ‘tree’ have theirusual meaning and for all that understand this meaning and thus know thatthere cannot be forests without trees. By starting to talk about the (ideal)existence of commands it seems that we silently changed (ImpInf) into

(2) the command given by α to x by the use of the imperative sentence!ATherefore: the command given by α to x by the use of the imper-ative sentence !B

which appears confused. This is because the argument form is not used asit is usually used, and now we do not know what to make of it. We are usedto filling in the blanks of the argument form

(3)∴

with sentences. Whether also imperative sentences can be meaningfully usedto fill in the blanks is the open question. However, there is no pre-establishedusage of filling in the blanks with names of objects, as in

(4) a∴ b

23Cf. Stenius [205] according to whom all normative systems include a norm thatdemands a tautology.

24Note that the topic of this discussion has not suddenly become the ontological sta-tus of notoriously difficult concepts of practical philosophy and jurisprudence, like moralobligations, natural law, human rights, laws of custom etc. Our concern are still ordinarysentences in the imperative mood, addressed e.g. to a husband, secretary, student, childor dog (cf. Ziemba [259] p. 386).


where a means a forest and b means a tree. At most, this is a mistaken wayto try to express (1). Similarly, the scheme (2) must be corrected into (5):

(5) There exists a command given by α to x by the use of the imper-ative sentence !A.Therefore: There exists a command given by α to x by the use ofthe imperative sentence !B.

Now we have arrived at something that resembles an argument – though wehave not yet shown when such a scheme constitutes valid arguments. Butthis argument is one that uses only descriptive sentences that can be true orfalse. It is not a case of (ImpInf), i.e. not an argument where imperativesentences function as premisses or conclusion.25 So by appealing to the notionof existence we obtain at most an inference relation between sentences thatdescribe the existence of certain commands, but not a logic of commands.

1.4.4 Formalistic Approaches

Maybe we should have gone looking for the explanation of an entailmentrelation between imperatives not in some relation between entities in a ‘worldof ought’, but in the meaning of the sentence that the speaker uses. Someonewho says that ‘either Barack Obama or Hillary Clinton will be the 44thpresident of the U.S., but it won’t be Obama’, thereby expresses the beliefthat Hillary Clinton will be the 44th U.S. president. The speaker did notsay explicitly that it will be Clinton (these were not the words used), butthe speaker can be said to have implicitly, or tacitly, expressed this opinion.Therefore if a person asserts something, she may be taken to ‘implicitly’assert something else. Likewise, when a person commands something, shemay also be ‘commanding something by implication’. Consider the followingexample employed by Hare [88]:

Go via Coldstream or Berwick!Don’t go via Coldstream!Therefore: Go via Berwick!.

25That the ‘world of ought’ approach thus only provides arguments with descriptivesentences is accepted by Walter [217], for he turns to identify imperatives with descriptionsof the ‘world-of-ought’-existence of a command that is created by the use of an imperative– consequently such sentences can be true or false and therefore part of logical inferences.Thus imperative logic is reduced to indicative logic, where the difficult part is now theverification of some descriptive sentences.


Here, an officer who must go from London to Edinburgh is ordered ‘go viaColdstream or Berwick,’ and – a bit later – is given the order ’don’t go viaColdstream’. Both commands have been taken to imply that the officer is(now) ordered to go via Berwick, and that the inference is therefore valid.For the question what commands should be considered to have been ‘im-plicitly commanded’ by explicitly given commands, one may then point toDubislav’s conventions or similar rules – e.g. (DCM) clearly accounts forthe above inference.26 To search for an explanation of imperative inferencesby interpreting explicitly given commands seems a much better idea thanto find it in otherworldly relations. Unfortunately, it is also a circular idea:according to it one can infer a command from another command if by givingthe latter command one implicitly commands the former. Or: we may useDubislav’s convention to infer imperatives from other imperatives because itprovides the imperative sentences that are implicitly commanded when thefirst imperative sentences are used for commanding.

An interesting response to this reproach of circularity is to say that bygiving rules for inferring one imperative from another, one did all that isrequired to explain the meaning of imperative inferences. It is this viewthat seems to lie at the root of Dubislav’s proposal to “formally facilitateinferences” from demand-sentences through a ‘convention’ (Ubereinkunft) or‘trick’ (Kunstgriff ). In fact, in his main work Die Definition [49], Dubislavgives a similarly ‘formalistic’ characterization of propositional logic (and laterpredicate logic). There, Dubislav starts with a ‘pure, game-like calculus’ thatis played with ‘pieces’ and signs (‘¬’, ‘∨’, brackets) by first arranging someinto initial positions and then replacing pieces and re-arranging pieces intonew game positions according to the rules of the game. This game thenbecomes the calculus of propositional logic by interpreting its elements as

26The term of ‘commanding something by implication’ was introduced by Geach [61].Alchourron [3] speaks of the consequences of what is prescribed as ‘indirectly prescribed’,Alchourron & Bulygin [5] write that e.g. if teacher commands that all pupils should leavethe class-room, he also implicitly commands that John (who is one of the pupils) shouldleave the class-room, even if the teacher is not aware of the fact that John is there, andin [8] they view the ‘deductive consequences’ of norms as ‘implicitly promulgated’, wherethe deduction process is equivalent to the modified Dubislav convention (DCM). Hare [88]and Rescher [184] both propose to define command inferences in terms of ‘implicitly givencommands’ – analogously, Rescher’s ‘assertion logic’ [185] is concerned with assertions thata speaker is ‘implicitly committed to’ in virtue of overtly made assertions. It was shownabove that Rescher’s explanation of imperative inferences is equivalent to the modifiedextended convention (DECM).


indicated: the ‘pieces’ as propositions, the signs as ‘not’, ‘or’, and brack-ets, the ‘initial positions’ as axiomatic basis, the game rules as the usualrules of substitution and modus ponens, and the achievable game positionsas derivable formulas. This characterization of propositional logic is meantby Dubislav as an exposition of Boole’s [34] idea that the “validity of theprocesses of analysis does not depend upon the interpretation of the sym-bols which are employed, but solely upon the laws of their combination”. InDubislav’s view, which he calls ‘the formalistic theory’, this description oflogic functions as a mould for all scientific theory: a theory is constitutedby a pure calculus (of formulas and rules), combined with a fixed interpre-tation. Observational sentences are captured in formulas that can be usedalongside axioms or derivable formulas of the system to derive other formulaswithin the calculus. Then the assignment of these derived, or better: ‘calcu-lated’ formulas is reversed, i.e. they are translated back into observationalsentences. If these are regularly true, then the observational sentences are‘explained’ by the theory. If a calculated observational sentence turns outto be false, then the theory is erroneous. Thus it also becomes possible todecide between competing, non-isomorphic theories.

The usefulness of the Dubislav’s formalistic approach for the problemof imperative logic is immediate. In fact, Dubislav’s own proposal in [50]satisfies all requirements in [49] for being a theory of imperative inference:there are entities that may function as premisses and conclusions, namelyimperative sentences. There is an interpretation that assigns each impera-tive sentence a formula, namely that of the indicative ‘parallel sentence’ inthe calculus of ‘ordinary logic’. There is a calculus, namely ‘ordinary logic’,that tells us what formulas can be derived from the formulas assigned to theimperative sentences that function as premisses. And finally, this assignmentis reversible to provide derived imperative sentences. Other authors – takingtheir cues from Tarski’s [210] syntactical definition of consequence relationsand deductive systems,27 Tarski’s [211] definition of truth,28 Gentzen’s [63]

27Cf. Alchourron & Bulygin [8] who employ a formal consequence relation to explainwhat norms are ‘implicitly promulgated’ by a set of norms.

28Both Rodig [187] and Yoshino [257] appeal to Tarski and argue that meaningful op-erations with prescriptions are made possible by supposing that normative attributes like‘obligatory’ or ‘punishable’ may be applied to actions. Rodig draws attention to the prob-lem of objective verifiability and therefore truth of such statements. But he circumventsthe problem by assuming that meta-language truth conditions can be given, which is suf-ficient to handle normative attributes as normal predicates in the object language. Rodig


idea that to define a symbol is to give rules for its introduction and elimina-tion,29 and Wittgenstein’s dictum that the meaning of a word is its use ([239]§43) – have similarly argued that instead of searching in vain for analoguesof truth values, it suffices for an explanation of imperative inferences to giveformal rules for obtaining imperatives from other imperatives.

If this ‘formalistic’ approach to the logic of imperatives is accepted, we arestill not finished yet. If the assignment of formulas, calculations and back-translations of derived formulas are to be more than a game, there must besome way to judge the adequateness of the theory, and be it only to decidebetween competing proposals.30 In analogy to Dubislav’s general approach,where a theory is only an explanation of phenomena if its calculated ob-servational sentences are regularly true, one should require of any proposed‘logic of imperatives’ that the imperative it ‘derives’ from other sentences arenormally – not ‘true’ of course, but accepted as ‘implicit’ in other sentencesthat are used as premisses. This resembles what is called the ‘soundness’ of acalculus: if the calculus allows ‘false’ (unacceptable) conclusions to be drawnfrom ‘true’ (accepted) premisses, then it must be discarded as ‘unsound’.31

I now turn to the question of adequacy in this sense.

and Yoshino then use these predicates to formalize e.g. a norm that says that helpingin an emergency situation is obligatory as ∀acts: In emergency(act) ∧ Helping(act) →Obligatory(act). The puzzling thing is that if this really is a prescription (norm), i.e. itmakes so far unregulated acts of helping in cases of emergency obligatory, then for nosuch act the ‘truth’ of the part Obligatory(act) can be established before the ‘truth’ of thewhole is established. This at least differs from Tarski’s compositional truth definition.

29Cf. Alchourron & Martino [10] who provide a calculus with an ‘introduction rule’for a prescriptively interpreted O-operator, where their rule corresponds to the modifiedDubislav convention (DCM) plus a requirement of joint satisfiability.

30It seems consensus that there must be some ‘test’ of adequacy. Weinberger [224] writesthat one must test a rule for the logical manipulation of norm sentences for its adequacyfor the area of normative thought, and Sosa [201] speaks of a ‘control of commonsense’that is necessary because otherwise there would simply be no end to the possible “logics”.

31The other possibility, that the calculus does not provide all the inferences from pre-misses that are acceptable (usually called ‘completeness’), is less harmful and can be dealtwith by e.g. refining it. For a similar definition of adequacy cf. Chellas [44] p. 4, wherehowever the terminology is vice versa.


1.5 Ross’s Paradoxes

Shortly after Jørgensen’s dilemma and Dubislav’s workaround for a logic ofimperatives had been described, Alf Ross re-considered inference schemes in‘the most simple form’, where a ‘new’ imperative is inferred from one imper-ative premiss, i.e. where the scheme used is that of Dubislav’s convention(DC). The following is an instance of such a scheme:

!A∴ !(A ∨B)

Here, !A means (as now usual) an imperative sentence that is satisfied if andonly if the descriptive sentence A is true, and !(A∨B) is an imperative thatis satisfied if and only if either A or B are true. It is immediate that thesecond imperative can be inferred from the first sentence !A by Dubislav’sconvention. Fine, said Ross, let !A be interpreted as the imperative ‘post theletter’, so we can infer from the imperative ‘post the letter’ the imperative‘post the letter or burn it’ !(A ∨B). So

(1) Post the letter!Therefore: Post the letter or burn it!

is a valid imperative inference. Ross himself points out that his paradox isnot paradoxical if this ‘validity’ of an imperative inference is understood inthe sense of a logic of satisfaction. If the letter is posted and the imperative!A satisfied, then the imperative !(A ∨ B) will likewise be satisfied – thisis no more paradoxical than that A ∨ B can be inferred from A. But ifthe meaning of ‘imperative inference’ refers to anything like the ‘validity’ or‘existence’ of an imperative, then Ross claims that his inference is not onlynot immediately felt to be evident, but rather evidently false.

Why does Ross’s example of an imperative inference seem paradoxical?In particular, regarding the ‘formalistic theory’ of imperative inference givenin the last section, why should it be paradoxical to say that if one uses theimperative !A for commanding, then one ‘implicitly’ also commands !(A∨B)?One explanation has been that that by using a disjunctive imperative, i.e.an imperative sentence that like !(A∨B) is satisfied if some state of affairs orsome other state of affairs holds, the authority has left it to the subject howto satisfy her command. Suppose Romeo hands a letter to Mercutio withthe words ‘Post the letter or burn it, but relieve me from deciding its fate


and mine’, would his friend not be free to do as he pleases? Analyzing this‘freedom’, it has been argued that giving a command entails an ‘imperativepermission’ or implicitly authorizes to carry out the actions required to satisfythe command.32 So the imperative ‘post the letter or burn it’ would containthe permission ‘I hereby permit you to post the letter or burn it’. Nowexplicit disjunctive permissions are often understood in a ‘strong’ sense thatgrants both disjuncts: when someone says ‘help yourself to a cup of coffee ora cup of tea’, then the guest is permitted to help herself to coffee and alsopermitted to help herself to tea (though possibly not both). So one obtainsthe following chain:

(2) Post the letter!Therefore: Post the letter or burn it!Therefore: You may post the letter or burn the letter!Therefore: You may burn the letter!

But it seems counterintuitive to say that by ordering a letter posted onepermitted it to be burned.33 To avoid this result, one may argue that it isnot the first step in (2) that is problematic, but the second, i.e. we shouldnot be allowed to infer a strong permission from an imperative. Yet nothingseems wrong with the following piece of Mercutio’s reasoning:

(3) Romeo asked me to post the letter or burn it.Therefore: I may post the letter or burn the letter, as I wish.Therefore: I may burn the letter.

The reason why the inference from the first line of (3) to the second line seemsnot objectionable, while the similar inference from the second line in (2) toits third line appears somehow wrong, may lie in the fact that the imperativeto ‘post the letter or burn it’ that is used in the reasoning is only implicit,i.e. derived, while Mercutio’s reasoning was about an imperative that wasexplicitly used by Romeo. So one could modify one’s view on the secondstep in (2) by saying that one is only allowed to infer a strong permissionto do what is commanded if this command is not itself derived. I return to

32Cf. Chellas [44] p. 19 for the term ‘imperative permission’ and Keene [120] for the‘implicit authorization’.

33The idea to explain the counterintuitive nature of Ross’s paradox using the also, oreven more, counterintuitive inference to ‘you may post the letter or burn it’ was vonWright’s in [249] pp. 21–22, also cf. von Wright [256] pp. 121–122 and Hintikka [96].


such a distinction between ‘explicitly given’ premisses and ‘implicitly given’imperatives in a moment. But consider the example from the last section,where an officer was commanded to go via Coldstream or Berwick, and (alittle later) told not to go via Coldstream, where both commands were viewedas implying the command to go via Berwick. The proposed modificationwould still allow us to make the following inference:

(4) I was commanded to go via Coldstream or Berwick.Therefore: I may go via Coldstream or Berwick, as I wish.Therefore: I may go via Coldstream.

So it seems the authority contradicted herself when ordering (a little later)not to go via Coldstream, i.e. first a choice between the two routes wasgranted, and later this choice was retracted, or rather: the second commandmodified the original command.34 Whenever a command contradicts, cancelsor modifies another command, the conflict may be absorbed e.g. by applica-tion of the rule of lex posteriori, which says that as a rule authorities shouldbe considered competent to modify their own orders. But the puzzling thingis that the example was originally presented as a smooth application of im-perative logic as facilitated by Dubislav’s convention (DCM). Nothing madeit appear as if there is some contradiction or modification involved and thatmore is used or required than just a flat application of the rules.35

An answer to these problems could be to give up the idea of strong im-perative permission altogether: without it, the agent cannot reason thatburning the letter is permitted. But while strong permission might be seenas problematic,36 it is not clear why it should be altogether discarded. Inparticular, nothing seemed wrong with assuming strong permission in thecase of Romeo’s request (3). But whatever view is taken on strong permis-sion, there is another point that makes Ross’s paradox seem counterintuitive

34According to Hare [88] it is just a conversational implicature that gets canceled. Butit seems that by saying “go via Berwick or Coldstream” the authority really leaves it tothe agent which route she wants to take – and later retracts this choice –, while someonewho says e.g. “the tickets are upstairs or in the car”, and later adds “they are not in thecar” only made it seem as if the tickets could be in either location. If the order was onlygiven “further orders pending’, as Hare also argues, then the first order was not complete,because it left the agent unable to determine how to fulfill it. It is as if the authority hadsaid in the middle of a sentence: “hang on, I’m not finished yet, I’ll be right back.”

35This was the point in Williams’s [238] criticism of Hare’s [88] scheme.36Cf. Stenius [206]: “Free choice permission is too strong a concept to be useful.”


without appealing to some ‘implied’ permission: Imagine that, having beengiven the command ‘Go via Coldstream or Berwick’, the agent finds the roadColdstream blocked. Then the following reasoning seems logical:

(5) I was commanded to go via Coldstream or Berwick.I cannot go via Coldstream.Therefore: I should go via Berwick.

It seems the kind of deliberation that one would expect of reasonable agents.Likewise, Mercutio, having been asked by Romeo to ‘post the letter or burnit’, might be found reasoning in the following way:

(6) Romeo asked me to post the letter or burn it.For fear of Tybalt’s revenge, I cannot bring myself to postthe letter.Therefore: I should burn the letter.

One might dispute whether Mercutio’s fear is really on a par with a roadblocked e.g. by a landslide. But if we suppose it is, then Mercutio’s reasoningseems as impeccable as that of the officer. Now return to Ross’s paradox:here the agent was ordered to ‘post the letter’. Implicit in this imperative, sowe are told by Dubislav’s convention (DC), is the imperative ‘post the letteror burn it’. Imagine that the agent is not able to post the letter for somecause (the postal workers are on strike and the mail bins have been lockedup). So the agent could reason in the following way:

(7) I was (implicitly) ordered to post the letter or burn it.I cannot post the letter.Therefore: I should burn the letter.

But this reasoning is absurd. Just because the agent cannot fulfill her obliga-tion to post the letter, this does not mean that she is obliged to do somethingthat was never mentioned, and in fact could be anything: the words ‘burnthe letter’ could be replaced e.g. by ‘go to the zoo’, ‘kill a passer-by’ or ‘loveyour neighbor’ and the inference would be just as valid – if it is valid.37

Now the agent, in reasoning in the above settings, used indicative state-ments about natural facts – like that something cannot be done – to reason

37This is Weinberger’s [225], [226] explanation of why Ross’s paradox poses a problem.


about the imperatives ‘go via Berwick or via Coldstream’, ‘post the letter’or ‘post the letter or burn it’. But inferences that mix imperatives and in-dicatives are notoriously troublesome and should perhaps be avoided. AsMacKay [139] points out, both of the following ‘inferences’

Go fly a kite!You are going to drop dead.Therefore: Drop dead!

You are going to fly a kite.Drop dead!Therefore: Go fly a kite!

are validated by Dubislav’s extended convention (DEC), where both infer-ences seem plainly invalid, and so perhaps (DEC) should not be accepted.Yet consider again the case of the officer. Imagine that it was not the originalauthority that issued the command not to go via Coldstream, but someoneelse, like the officer’s husband (who in the past had some bad experience onthis road). Since there is some discretion in the authority’s order, there isno reason why the officer should not give in to her husband’s request, and sothe following reasoning of the officer seems correct:

(8) I was commanded to go via Coldstream or Berwick.My husband asked me not to go via Coldstream.Therefore: I should go via Berwick.

Similarly, we can imagine Mercutio to reason in the following fashion:

(9) Romeo asked me to post the letter or burn it.Tybalt threatened me not to post any of Romeo’s letters.Therefore: I should burn the letter.

But then the following reasoning of the agent to whom Ross’s imperative‘post the letter’ was addressed must be likewise correct:

(10) I was (implicitly) ordered to post the letter or burn it.I have been asked not to post the letter.Therefore: I should burn the letter.

There is some discretion in the (implicit) order to ‘post the letter or burn it’,


so why should the agent not take an additional request into account? Theonly difference between the (9) and (10) is that in (9) the reasoning appealsto an explicitly used imperative, whereas in (10) it starts by considering anorder that was only ‘implicit’ in the use of some imperative. So maybe whatwas wrong was that derived imperatives were used, without paying enoughattention to the fact that the derived imperatives are only ‘part of a sys-tem’, that the ‘explicitly’ used imperatives have not ceased to exist, thatimperatives that are only derived do not ‘exist’ on quite the same level asexplicit imperatives, or that the agent is somehow expected to make use ofthe logically strongest information that is available.38 So we are back at theproposal that a difference must be made between ‘explicitly used’ impera-tives, and imperatives that ‘only derive’ from explicit imperatives. But torequire that reasoning with imperatives starts with ‘explicit’ imperatives, andmust not start with imperatives that are only inferred, reveals an unusual,non-classical meaning of ‘imperative inference’. For classically, logical infer-ences may very well be conducted by proving first that some assumptionshave some desired conclusion, and then show that the assumptions followfrom an accepted set of premisses. This is facilitated by the transitivity ofclassical consequence: if A ∈ Cn(B) and B ∈ Cn(C) then A ∈ Cn(C) (‘con-sequences of the consequences are also consequences’), or the monotonicityrule: if A ∈ Cn(X) then A ∈ Cn(X ∪ Y ) (what follows from some axiomsalso follows from a larger set of axioms).

Ross’s paradox seems to demonstrate that given the imperative inferencesprovided e.g. by Dubislav’s convention, it becomes necessary to distinguishbetween the imperatives that are explicitly given and the imperatives that areinferred: agents can use the former for their reasoning, but not always the lat-ter, or not the latter by themselves, which makes reasoning with imperativessomehow non-classical. And so there may yet be another way to get aroundthe difficulties: perhaps Ross’s example is not really a case of an imperative

38Rodig ([187] p. 184–185) points out that by deriving the norm to ‘post the letter orburn it’, the original order to ‘post the letter’ does not ‘cease to exist’, and that it is theconjunction of both norms that must be satisfied. That the entailed norms do not ‘exist’in quite the same way as explicit norms is the idea of von Wright e.g. in [255] and [256] p.114 and p. 122. According to Stenius [206], the use of ‘post the letter or burn it’ carriesthe tacit information that a stronger regulation like ‘post the letter’ does not ‘belong tothe codex’. For the idea that using a weaker sentence ‘post the letter or burn it’ violatesa conversational presupposition cf. Hintikka [96]. Also cf. Hamblin [74] p. 88: “ ‘implicitimperatives’ may be different from the real thing, and we should be wary of loading themup with the full range of imperative properties.”


inference. Perhaps (1) is simply invalid. It is obvious that the scheme is anapplication of Dubislav’s convention, so (DC) must be modified. One wayto do that is to let the logic that is used for the right hand side inference infigure 1.1 not classical logic (propositional or predicate logic), but some otherlogic that does not allow one to infer A ∨B from A. Let Dubislav’s conven-tion therefore be reinterpreted in terms of a very strong ‘relevant deduction’developed by Weingartner & Schurz [237] and Weingartner [236], which wastailored specifically to eliminate Ross’s paradox.39 Unlike other relevant log-ics, their ‘R-consequence’ not only blocks the inference of A∨B from A, butalso that of A∨B from (A∨B)∧A or from (A∨B)∧¬B. Dubislav’s schemeis then changed accordingly (cf. the next figure). Weingartner & Schurz’s

Figure 1.4: Dubislav’s convention with relevance.

proposal produces a very strange consequence relation: not only monotonic-ity fails, but also reflexivity, i.e. X ⊆ Cn(X) is not valid. But as we cannotderive A∨B from A any more, we also cannot derive !(A∨B) from !A, andso Ross’s paradox is solved.

However, in [188] Ross also presented another paradox, a variant of hisfirst paradox, that remains valid even on such a ‘relevant’ reinterpretation of(DC): the inference from !(A ∧ B) to !A, where A ∧ B means the sentencethat is the conjunction of A and B. I consider the paradox in the form of‘Weinberger’s paradox’ or the ‘window paradox’:40

39 R-consequence is defined by Weingartner & Schurz’s [237], [236] in the following way:a propositional formula A is a R-consequence of a set of formulas X iff (i) X classicallyimplies A and (ii) it is not possible to uniformly replace a proposition letter at at leastone of its occurrences in A by a random proposition letter without making the classicalinference invalid.

40The origin of the example is unclear. The name ‘Paradox of the Window’ is used e.g.by Stranzinger [207] and Weinberger [232].


(11) Close the window and play the piano!Therefore: Close the window!

(12) Close the window and play the piano!Therefore: Play the piano!

Suppose that α wants x to practice the piano, but neighbors have alreadycomplained about the disturbance and even called the police on a previousoccasion. So α does not want x to play the piano while the window is open.Closing the window will reduce the noise so much that the neighbors are leftwith nothing to complain about. Suppose then that α sends x to play thepiano, using the words ‘close the window and play the piano’. A little bitlater, the following discussion ensues between α and β:

α: I told her to play the piano, but I didn’t hear her doing it allafternoon.

β: Well, at least she closed the window.α: Why should she do that?

Here, the positive view on x’s behavior by β is not accepted by α. Closingthe window by itself is meaningless. It might even be unwanted in general– it blocks out fresh air – if it weren’t for the sake of piano practice. Butbacked with the inference (11), β can continue in the following way:

β: You ordered her to close the window, that’s what she did, soshe did something right, didn’t she?

Now consider the following, alternative dialogue:

α: She practised the Khachaturian with the window wide open.What shall we tell the police this time?

β: It was you that told her to play the piano.α: But I didn’t. She was also to close the window.

β’s reproach for x’s playing the piano is not accepted by α, because playingthe piano without closing the window first was not what α had asked x todo. However, backed with the inference (12), β could reply in this way:

β: You ordered her to play the piano, that’s what she did, sodon’t try to wiggle out of your responsibilities.


In these dialogues, α’s position seems natural, while β’s reaction is strangeand uncomprehensible. But given the inference schemes (11) and (12), β isright: from α’s command, the imperatives ‘close the window’ and ‘play thepiano’ can be inferred – so we are told by Dubislav’s convention (DC), andDubislav’s convention restricted by relevant implication. Moreover, thesederived imperatives are used by β as imperatives are meant to be used,namely compared with reality, and reality accordingly qualified as ‘right’ or‘wrong’. So x did something right by closing the window, as x satisfied an(implicit) imperative, and similarly when playing the piano without closingthe window. But intuitively, closing the window by itself produces nothinggood, and playing the piano with the window open seems a clear violationof obligations and not satisfactory in any way.41

The ‘window paradox’ seems to arise whenever the states of affairs men-tioned in the imperative are only conjunctively desired by an authority. Thatfor this reason we cannot detach conjuncts in wishes, i.e. we cannot concludefrom ‘she wishes for a and b’ that ‘she wishes for a’ was pointed out by Menger[150] for the case of complementary goods, e.g. when a is ‘a cigarette’ andb is ‘a match’, for one may not wish either one of the goods by itself. Ross[188] points out that the same difficulty arises for imperatives, e.g. when theimperative is to ‘write a letter and post it’. Other examples have includedthe imperatives ‘take the parachute and jump’, ‘pay the bill and file it’ or‘fill up the boiler with water and heat it’.42 Goble [65] showed that even aseemingly innocuous obligation to ‘sing and dance at Gene’s party’ may beplanted in a setting that makes it impossible to speak of fulfilling the obli-gation when only one act, singing or dancing, is performed. To determinewhether an imperative is ‘separable’ or ‘inseparable’, i.e. whether doing Aalone produces something ‘right’ with respect to an imperative !(A ∧ B) ornot, it is necessary to examine the intentions and wishes of the authority

41It gives the paradox a further twist if we imagine that playing the piano with thewindow open is explicitly forbidden. For by Dubislav’s convention (DC), the imperative!¬(A ∧ ¬B) (‘don’t play the piano while the window is open’), is derivable from theimperative !(A ∧ B) (‘close the window and play the piano’). But it seems that theadditional prohibition is best formalized as a conditional imperative (in Hofstadter &McKinsey’s [99] formalism: ¬B ⇒!¬A). Conditional imperatives pose other problemsoutside the current topic. In any case, one would still have to say that playing the pianowith the window not closed was satisfactory with regard to some (derived) imperative.

42Cf. Hare [88], Weinberger [223] and [235]. These difficulties led Weinberger to rejectthe validity of an inference from !(A ∧B) to !A in his publications since [223].


that used the imperative, it is not a matter of logic.43

To solve these difficulties, Kenny [122] proposes a logic of ‘satisfactori-ness’. This logic uses a set of propositions to represent the wishes of theauthority under considerations. A fiat (an impersonal imperative like ‘letthere be light’) is called satisfactory if and only if whenever the fiat is satis-fied then every proposition in the set of wishes true. Finally an inference ofone fiat from another fiat is defined as follows:

!B may be inferred from !A in the logic of satisfactorinessif and only if

if !A is satisfactory then !B is satisfactory.

It is clear that the troublesome inferences (1), (11) and (12) are invalidatedby this logic of satisfactoriness: when posting the letter is satisfactory forthe wishes of the authority, then burning the letter need not be so. Likewise,if closing the window and playing the piano is satisfactory with respect toall wishes, then playing the piano alone does not guarantee that the wishesof the authority are also satisfied. But Kenny’s approach gives rise to otherparadoxes: in the logic of satisfactoriness we can e.g. derive:44

(13) Open the door!Therefore: Open the door and wear a tie today!

The inference is clearly absurd and so Kenny’s logic does not help us to solvethe paradoxes.

In Ross’s first paradox, the imperative to ‘post the letter or burn it’ was‘inferred’ from the imperative to ‘post the letter’, thus forcing one to acknowl-edge that some (though only inferred) imperative is satisfied by burning theletter. In the ‘window paradox’ we could ‘infer’ the imperative ‘play the pi-ano’ from the imperative ‘close the window and play the piano’, thus forcingus to acknowledge that an (inferred) imperative is satisfied when the pianois played with the window wide open. In both cases, we would much rathersay that no imperative was satisfied by burning the letter that was meant tobe posted, and by playing the piano with the window open when it shouldhave been closed. This, I think, is the main cause why Ross’s paradox andthe window paradox give rise to counterintuitive feelings, or are ‘paradoxi-

43The terminology here is that of Hamblin [74] p. 184.44A similar counterexample was given by Gombay [71], also cf. Sosa [201].


cal’. So we should not be allowed to infer such imperatives. So Dubislav’sconvention is not an apt theory to explain how an imperative may be derivedfrom another one. And so we are back at square one: all theories, includingthe formalistic approach, have so far failed to explain what it means to inferan imperative from some other imperative in spite of Jørgensen’s Dilemma.

1.6 Ordinary Language Arguments

Maybe it is not really the case that all options have run out to redefineDubislav’s scheme in a way so that it avoids Ross’s paradoxes. Maybe wehave to replace the classical logic that appears in his scheme by yet anotherlogic, or develop such a logic.45 But it is hard to see what kind of logic thiscould be, since most logics, including other non-monotonic logics, will permitus to either infer !(A∨B) from !A or !A from !(A∧B), and so at least one ofthe two paradoxes will arise. So I think, after all these troublesome attemptsto define a ‘logic of imperatives’, it is worthwhile to take another look atPoincare’s proposal that originally started the controversy.

Poincare’s only explicit example of an inference with an imperative con-clusion has the following form:

(1) Do this!This cannot be done without that.Therefore: Do that!

The following is an instance of this scheme:

(2) Drive me to the airport!To get to the airport, one must drive in a northerly direction.Therefore: Drive me in a northerly direction!

In which setting could these sentences be used? Suppose I have entered ataxi and used the above sentences. But some confusion could arise. Thedriver could reply: “So what do you want me to do, drive you to the airportor just drive north?” The driver needs a direction. Ordering her to go to theairport alone is sufficient for this, and the behavior expected of a passenger

45Cf. Keene [120]: “What we wanted here is a logic of actions, in which a well-definedconcept of inclusion plays a leading role.”


entering a taxi. Using two imperatives where each contains an instruction ofwhere to go is unexpected and confusing.

So suppose I have just used the sentence ‘drive me to the airport’. A littlelater I realize that we seem not to be going north, and I say to my partner:

“Is she hijacking us? I ordered her to go to the airport, andthe airport lies to the north. So she ought to be driving us in anortherly direction. But she is not.”

This reasoning seems flawless. Yet it only involved sentences about impera-tives, and did not involve sentences in the imperative mood, and so it cannotbe an example of an imperative inference. But maybe this would be a goodtime to say to the driver:

(3) I ordered you to go to the airport.To get to the airport, one must drive in a northerly direction.Therefore: Drive me in a northerly direction!

But here the two sentences that function as premisses are both descriptive.Since Poincare explained that an imperative cannot be derived from indica-tive premisses alone (and there is no reason not to follow him), this cannotbe an imperative inference, and there must be something more involved thanthe drawing of a logical conclusion. One such other function of the ‘there-fore’ appearing at the front of the last sentence of (3) is not to reason, butto motivate, as in:

(4) The car is broken.Therefore: Take the bus into town!

Here the speaker motivates the imperative to take the bus by explaining thatdriving into town is impossible, since the car is broken. So similarly, whatseems to happen in (3) is that I motivate my (new) imperative ‘drive me ina northerly direction’ by an already given command and an assumed fact.

Consider again the proposed inference (2). Just like indicative inferencesare explained by the fact that someone who accepts (or: assents to) thepremisses must also accept the conclusion, Hare [87] has argued that animperative inference is one where someone who assents to all imperativepremisses must also assent to the imperative conclusion:

“A sentence p entails a sentence q if and only if the fact that a


person assents to p but dissents from q is a sufficient criterion forsaying that he has misunderstood one or other of the sentences.(...) A person who assented to this command [‘Take all the boxesto the station’], and also to the statement ‘This is one of theboxes’ and yet refused to assent to the command ‘Take this tothe station’ could only do so if he had misunderstood one of thesethree sentences.”

But what does it mean that a person ‘assents’ to a command? SupposeJohn’s mother tells him ‘John, clear the table and do the washing up’, andJohn’s little sister echoes: ‘John, do the washing up’. If John ‘assents’ tohis mother’s order, does he also have to ‘assent’ to an order by his sister,whom he might not accept as an authority? Perhaps the analysis assumesidentity in the person who uses the commands. Suppose then it was notJohn’s mother but some officer who used the imperative, and John is notobliged qua son, but as this officer’s orderly. The second command is alsoused by the officer, maybe a little later. But suppose that John is onlyobliged to the officer if the commanding is done in a certain fashion, e.g.when the officer is standing up, or when the officer is not drunk, and thatwhen the second imperative was used the officer was, as a matter of fact,not standing up or already had more than her fill. Or suppose that John isnot an orderly, but some djinni, and the officer is the person who rubbed thelamp, but that, when the first imperative was used, this already was the lastof the three wishes that had been granted. Does John, in these cases, haveto ‘assent’ to the second command? It seems that such an interpretation of‘assent’ would have to get involved into reasoning about whether the act ofusing an imperative ‘really creates’ a command. But it did not seem as ifsuch reasoning is involved in Hare’s proposal.

So let the word ‘assent’ be understood in its weakest possible interpreta-tion. A person could hardly be said to assent to a command given to her ifshe did not satisfy or to try to satisfy it. Returning to the situation whereI have asked the taxi driver to take me to the airport, when the taxi driverassents to this request, she will start driving me to where she thinks the air-port lies, i.e. start to satisfy, or try to satisfy, my request. If the taxi driveragrees that the airport lies in a northerly direction, she will, in obeying myrequest, eventually drive in what she thinks is a northerly direction. So shemight be said to additionally satisfy, or try to satisfy, a request to drive mein a northerly direction, had such a request been made. But did I request


the taxi driver to drive me north? I might be absolutely sure that the air-port is to the north, but I would still blame the taxi driver for not goingwhere I requested if, opposite to what I believed, the airport is in fact tothe south-west of my starting point and the driver still went north. So I didnot utter such a request, and would not even imply such a request, lest I becharged by the driver for going there instead of the airport. All we can sayis that the taxi driver would also be satisfying, and so seemingly assentingto, a purely hypothetical request to drive me in a northerly direction, if shesatisfies the request to drive me to the airport and the airport does in factlie in a northerly direction. But this is again not a logic that infers one im-perative from some set of other imperatives and/or indicatives, but the logicof satisfaction as explained in sec. 1.4.1.46

It would be nice to have ‘real life’ examples, cases of ‘ordinary’ reasoningwith imperative premisses and an imperative conclusion, i.e. instances of

(ImpInf) !A∴ !B

where !A and !B are sentences in the imperative mood, and where the useof the inference – not the imperatives – is either accepted in some ordinarydiscourse, or opposed (and the person who uses it blamed for being ‘unrea-sonable’ or ‘illogical’).

Use of indicative arguments in everyday discourse often occurs in singularsentences, like

(5) Unemployment is rising, so there are not enough jobs created.(6) She has got an ‘A’ in English, so she achieved top-marks in at least

one subject area.(7) I have read all of Vladimir Nabokov’s novels, so I have read Pnin.

Here two descriptive sentences are linked with the adverb ‘so’ (similar adverbswould be ‘therefore’ or ‘hence’). (5) seems analytical if one understands‘enough’ to be elliptical for ‘enough to make up job-losses elsewhere’. (6)is analytical if one knows that ‘A’ is a top-grade and that English is oneof several high-school subjects. (7) is made into a logical argument by theassumed background knowledge that Pnin is a novel by Nabokov. It isoften not easy to distinguish such indicative arguments from sentences thatpresent reasons, motives or are otherwise explanatory, for these also use the

46This resembles the criticism by Keene [120] of Hare’s proposal.


form of descriptive sentences that are concatenated by an adverb like ‘so’ or‘therefore’, as in the following examples:

(8) I couldn’t get the car started, therefore I took the bus.(9) I wanted to make friends with her, therefore I asked her if she

would go shopping with me.(10) There were holes in the roof, so birds had come in and were roosting

in the rafters.

(8) explains why today the speaker used the bus. Since the bus need nothave been the only means to get into town, or the speaker may have stayedat home, the hearer cannot just conclude the second part from the first. (9)presents the psychological motive why the speaker asked the other person togo shopping with her. Other people might have been motivated differentlyby the desire to make friends with that person. In (10), a natural event isexplained by a certain state of affairs. Again, this is not a logical argument:the birds could also have not flown in, or flown in but not nested in theceiling. Now the adverbs ‘so’ and‘therefore’ can also be used to meaningfullylink imperatives. Consider the following examples:

(11) Stop the rise of unemployment, so see to it that more jobs arecreated!

(12) Make your guests comfortable, so introduce your guests to eachother!

(13) Don’t let vermin into you house, therefore patch up the roof!(14) Read all of Nabokov’s novels, so read Pnin!

(11) might be encountered in some political debate. At first it appears tobe a good argument, but then doubts arise: is the speaker really appealingto logic, or is she just complementing her first imperative by a second, morespecific one, as when we say: “Go there! Go there now!”? And one could alsostop the rise of unemployment by e.g. prohibiting companies to dismiss theirworkers, or making it more difficult for them (maybe 5 was not so analyticalafter all). Then (11) would seem to be rather a case of a motivating useof ‘so’: the imperative to see to it that more jobs are created is motivatedby the primary aim to stop unemployment. Likewise, in (12), the adviceto introduce guests to each other is rationalized by the more general aim tomake guests comfortable. It is hard to see what could be analytical here: toease tensions, the host may equally encourage the guests to guess each others


names, or serve them plenty of alcohol, or maybe the guests are easygoingand do not really require any effort on the host’s part to make themselvesat home. Similarly, in (13) the more readily accepted advice to keep verminout of the house is used as a rationale to make the addressee accept thedrudgery of having to patch up the roof. The most promising candidatefor an appeal to analyticity seems to be (14), i.e. that the imperative toread all of Nabokov’s novels includes the imperative to read Pnin, given thebackground knowledge that Pnin is a novel by Nabokov. Note that whenmaking the background knowledge explicit, it becomes a case of Dubislav’sextended convention (DEC). Such a sentence may be used e.g. by a teacherof a literature course when addressing her students. But again we cannotrule out that this is just a case of complementing an imperative by a second,more specific one, as we sometimes do to get things done.

Adherents of Dubislav’s convention (DC) must also accept the followingargument:

(15) Aim for an ‘A’ in English, so aim for top-marks in at least onesubject area!

But it seems dubious what reason the speaker could have for adding the ‘so’part. Just aiming for top-marks in some subject area is clearly not what thespeaker wants the addressee to do. More meaningful would be the converse,

(15a) Aim for top marks in at least one subject area, so aim for an ‘A’ inEnglish!

where the advice to aim for ‘A’ in English is rationalized by the wish to havethe student achieve top-marks somewhere. But since the student could notknow from the first imperative that it was the subject of English that thespeaker wanted her to achieve top marks in, this would – like (12) and (13)– rather be a ‘motivating so’, and not a use of ‘so’ that appeals to a logicalcapability.

Matters are further complicated by the fact that expressions of the fol-lowing kind can also be meaningfully employed:

(16) The car isn’t working properly, so take the bus!(17) I forgot my keys, therefore leave your key under the mat!(18) Gill is your best friend, so invite her to your party!


In all three sentences, the first part is descriptive and the second is in theimperative mood. We have already noted in the case of (3) that such argu-ments exist, but for anyone who agrees to Poincare’s thesis that imperativeconclusions do not follow from an indicative premisses it is clear that (16)– (18) cannot represent valid arguments. (16) seems again a case where the‘so’ is used to motivate the advice that is expressed by the imperative. The‘so’ does not express a logical relation, for sometimes it is better to use a carthat stutters than a coach that won’t take one back. In (17) the indicativegives a reason why the speaker wants her request to be followed. Accord-ing to Hamblin [74], such reason-providing indicatives are often attached toadvice-expressing imperatives, yet here the imperative might also be an order(e.g. of a parent). For the same reason the speaker might have ordered theagent to hand over her key, and not to leave it under the mat, and so whatis expressed is again not a logical relation. (18) seems also like presentinga motive for inviting Gill to the party (she is the addressee’s best friend),but here things might be a bit more complicated – the expression could beelliptical for:

(18.a) Invite your best friends to the party, Gill is your best friend, soinvite her to your party!

This is very similar to what Dubislav considered a valid argument, namelyhis inference from ‘thou shalt not kill’ to ‘Cain shall not kill Abel’. But then,(18) might also be elliptical for

(18.b) Gill is your best friend, one invites one’s best friends to one’s parties,so invite her to your party!

where the second part (which is not in the imperative mood) appeals to theexistence of a rule that the speaker might consider binding, or binding for theaddressee. Then this is rather a case of reason-giving, and not of a logicalinference: the speaker motivates her imperative by asking the speaker toconform to some preexisting rule.

To tell the uses of ‘therefore’s’ and ‘so’s’ that are motivating, reason-giving or explanatory in a non-logical sense, apart from those that separatethe premisses from the conclusion in an argument that is intended to be alogical one, we can use the following trick: instead of ‘therefore’ or ‘so’, use aclause like “... It follows logically from this that ...” to separate the sentences.


The new phrase makes the appeal to a logical capability explicit. Where theoriginal adverbs ‘so’ and ‘therefore’ were used to indicate a (claimed) logicalinference, the new formulations

(5.a) Unemployment rates are rising. It follows logically from this thatnot enough jobs are created.

(6.a) She has got an ‘A’ in English. It follows logically from this that sheachieved top-marks in at least one subject area.

(7.a) I have read all of Vladimir Nabokov’s novels. It follows logicallyfrom this that yes, I have read Pnin.

appear only to be changes in expression. The speaker, just as before, appealsto a shared understanding of words, concepts and background knowledge, tomake the second sentence seem to be expressing nothing new, but only a logi-cal consequence from what has been said before. Note that it does not matterwhether the arguments are, in fact, analytical. People sometimes think theyuse valid arguments when they are not. But the rephrased sentences makeit clear that the speaker intends the sentences to be just that. And the newformulations seem not to change the meaning of the original sentences when-ever a ‘logical’ use of the adverbs ‘so’ and ‘therefore’ was really intended. Bycontrast, when the first part was used to give some background information,a reason, explanation or motive, the rephrased expressions appear odd:

(8.a) I couldn’t get the car started. It follows logically from this that Itook the bus.

(9.a) I wanted to make friends with her. It follows logically from thisthat I asked her if she would go shopping with me.

(10.a) There were holes in the roof. It follows logically from this that birdshad come in and were roosting in the rafters.

The phrase ‘it follows logically from this’ makes again an appeal to someshared understanding of used words, concepts and background. But here,this background knowledged obviously does not allow one to ‘conclude’ thesecond sentence from the first. The listener could not have known from thefirst sentences in these examples that the speaker took the bus, asked someoneto go out shopping or has birds nesting in the roof of her house. So claiming,as the rephrased sentences do, that the second part can be concluded from thefirst, makes the sentences seem irritating, weird and false, while the earlier


sentences appeared quite harmless.Now consider what happens if such a method is used on imperatives. So

far, (14) seemed the best candidate for a sentence that ‘appeals to logic’, soI will concentrate on this example. First note that

(14.a) Read all of Nabokov’s novels. It follows logically from this that readPnin!

is not grammatical. Now if the grammar is difficult, this may already hintat little usage of such statements, but I think that instead of the ‘that’ wecan easily use e.g. a colon, corresponding to a pause in oral language, as inthe following expression:

(14.b) Read all of Nabokov’s novels. It follows logically from this: readPnin!

But here, the part that follows the colon seems strangely detached. Is this acommand, i.e. is the speaker, using the expression following the colon, stillgiving a command? Or is the emphasis on the part before the colon, andso the purpose of the second sentence is merely to tell (truly or falsely) thatsome consequence relation holds? The impression that this is a strange useof words increases if we add the subject of the request:

(14.c) John, read all of Nabokov’s novels. It follows logically fromthis: John, read Pnin!

Here, the phrase ‘it follows logically from this’ makes it appear as if thespeaker was not giving commands to John at all. It seems what the speakerreally does is talking about logical relations between sentences – maybe it isa logician presenting an example of an imperative inference. So perhaps weshould try out another phrase:

(14.d) John, read all of Nabokov’s novels. We can conclude from this:John, read Pnin!

Yet this expression also has a false ring: who is doing the commanding of the‘conclusion’ – the speaker? Or the ‘we’ that is to do the concluding? Do thespeaker and the listeners all join into giving John the command? Apparentlyit was wrong to use the first person plural, and so we might want to change


the sentence into:

(14.e) John, read all of Nabokov’s novels. I conclude from this: John, readPnin!

But this seems to be the worst alternative so far. Is the speaker concludingthe last sentence? Or is the speaker commanding it? And if so, then whyis she saying that she is concluding it? The performative acts of concludingand commanding seem to collide, whereas the acts of stating and concludingseemed to go hand in hand. But we have yet another phrase to try out:

(14.f) Read all of Nabokov’s novels. So you can conclude for yourself:read Pnin!

Though this is perhaps a less common phrase to signal logical arguments, thenew sentence seems to be the most successful so far. But it appears necessarythat the ‘you’ is the person to whom both commands are addressed. So letus make the addressees explicit. Of the following sentences

(14.g) John, read all of Nabokov’s novels. So John, you can conclude foryourself: read Pnin!

(14.h) John, read all of Nabokov’s novels. So Mary, you can conclude foryourself: read Pnin!

(14.i) John, read all of Nabokov’s novels. So Mary, you can conclude foryourself: John, read Pnin!

only the first seems somehow acceptable. In (14.h) it appears as if Mary isasked to read the book, but this can hardly be ‘concluded’ from a commandnot directed at Mary. (14.i) makes it seem as if Mary is asked to give acommand to John (and not just to draw a conclusion). Moreover, if theaddressee is expressly included in the inferred command, then also (14.f),which seemed so promising at first, looks strange:

(14.j) John, read all of Nabokov’s novels. So you can conclude for yourself:John, read Pnin!

It seems that in (14.f) and (14.g) the speaker has not just asked the addresseeof the first command to ‘draw a conclusion’, but in this process to ‘givehimself’ the command expressed by the second sentence, i.e. to ‘tell himself to


read Pnin’. When the addressee is made explicit in the ‘inferred’ command,it looks as if the addressee is additionally asked to use his own first namewhen telling himself to read Pnin – which is a weird thing to ask of anybody.And this points at another problem of (14.f) and (14.g): if the person whocommands ‘read all of Nabokov’s novels’ (the teacher) and the person whocommands ‘read Pnin’ (John himself) are not identical, how can the secondimperative be inferred from the first?

By contrast, all of the above phrases can be employed for ‘deontic sen-tences’ (non-imperative sentences that do not prescribe, but describe whatought to be done) without difficulty:

(19) You ought to read all of Nabokov’s novels, therefore you ought toread Pnin.

(19.a) John ought to read all of Nabokov’s novels, therefore John ought toread Pnin.

(19.b) John ought to read all of Nabokov’s novels. It follows logically fromthis that John ought to read Pnin.

(19.c) John ought to read all of Nabokov’s novels. We can conclude fromthis that John ought to read Pnin.

(19.d) John ought to read all of Nabokov’s novels. I conclude from thisthat John ought to read Pnin.

(19.e) John ought to read all of Nabokov’s novels. You can conclude foryourself that John ought to read Pnin.

(19.f) John ought to read all of Nabokov’s novels. Mary, you can concludefor yourself that John ought to read Pnin.

All these sentences seem grammatical, meaningful and not confusing. Wemight even view the inferences they express as sound, but this is not thequestion here. Yet as we have seen, all attempts to use the phrases thatlink these sentences, normally used to indicate logical arguments in indica-tive discourses, to link imperatives to indicate ‘imperative inferences’, resultin expressions that seem somehow confused and wrong. When used to linkimperatives, they mix up the roles of commanding, command-receiving, anddrawing conclusions. And since the method to use such clauses to distin-guish appeals to logic from e.g. motivating uses of ‘therefore’s and ‘so’s, failsto produce sentences that do not appear strange or confused in the case ofimperatives, perhaps it did so because these adverbs really are not used toindicate a claimed analyticity when linking imperatives. A motivating use


of the adverb ‘so’ suffices to explain why the sentence (14) seemed meaning-ful: the teacher, perhaps asked by John whether he also has to read Pnin,motivates the more specific imperative to read this book by prefixing to itthe general requirement to read all of Nabokov’s novels, thus making it clearthat Pnin is in fact one of the books that John has to read. (14.f) appearscomparatively less strange than the other reformulations because to ask Johnto ‘give himself’ the imperative to read Pnin may be a (roundabout) wayto make sure he actually reads it. To understand (18) we do not need todetermine whether the speaker refers to an explicit command to ‘invite one’sfriends’, or a social custom to do so, because what is in any case implicit in(18) is an appeal to a preexisting obligation to motivate the agent to do whatthe speaker wants her to do. It also explains why (15) seemed so strangelypointless: the reason for using the less specific imperative to achieve sometop marks is not sufficiently explained by prefixing to it a more specific im-perative to achieve top marks in English.47 And so it seems that all of theimperative arguments (11)–(18) are really cases of reason-giving and mo-tivation, and the ‘so’s and ‘therefore’s’ used in these expressions that likePoincare’s ‘donc’, or the ‘also’s, ‘daher ’s and ‘deshalb’s of German language,may be used to connect both indicative and imperative sentences, provideonly reasons, explanations or motives in the case of imperatives, and do notindicate claims of analyticity.

So I want to dare the hypothesis that there are no examples of imperativeinferences, i.e. logical conclusions in the imperative mood, drawn from atleast one premiss in the imperative mood, to be found in ordinary languagearguments. They only appear in the writings of some philosophers.

1.7 The Way to Go Forward

Other authors have noted before the conspicuous absence of imperative ar-guments from natural language. Wedeking [221] argued that there are nocases in which we actually use commands in arguments, and that words like‘therefore’ before imperative sentences are employed not to mark inferences,but for the purpose of reason-giving, of motivating the subject. Harrison[90] argued from a point of grammar and semantics that there is no logic

47Note that the same strangeness does not necessarily arise for deontic logic. The deanof one faculty may say to another: “Our students are obliged to have an ‘A’ in English,so yes, ours are – like yours – obliged to achieve top marks in at least one subject are.”


of imperatives; the difference between his position and mine (that there areno ‘imperative inferences’ in ordinary language, and so a logic of impera-tives has no point) seems very subtle.48 Even more conspicuously, there havebeen little challenges to these arguments.49 On the contrary, there is a wholetradition of ‘normological’ or ‘imperativological skepticism’, of authors whohave denied the existence of a logic of norms or imperatives.50

If there are, as a matter of fact, in ordinary language, no argument formsthat resemble ‘imperative inferences’, then there also is no place for a formaltheory for such a logic. Presenting formalizations of such a logic would bewriting about what Dubislav [49] called an Unding or chimaera: a non-thingthat exists only as a concept, but no real object falls under the concept.

So did Poincare commit a mistake? Did he confuse an important insightby Hume [107] on the use of ‘is’ and ‘ought’ – that facts cannot be used to

48Harrison writes (pp. 110/111, 124/125): “The expression ‘I conclude: Shut the door’does not make sense, nor does the expression ‘I conclude that don’t’. One can say: ‘Soshut the door’ and ‘Therefore shut the door’ and ‘Shut the door because ...’ but thefunction of the words ‘so’, ‘therefore’ and ‘because’ is not in this context to indicate that‘Don’t shut the door’ is a conclusion. They have some other function, which philosophershave confused with that of indicating that what follows them is a conclusion. (...) Thereason, therefore, why ‘therefore’ and ‘so’ can precede ‘post the letter’, but ‘I conclude’ cannot, is that ‘I conclude’ can precede only propositions (and only they can be conclusions)and indicate that reasons have been given for the proposition, but ‘therefore’ and ‘so’ canprecede either propositions or imperatives. When ‘so’ and ‘therefore’ precede imperatives,however, they are not reasons for the imperative, as the unwary might suppose, but forthe action enjoined, advised, recommended or directed by it.” Harrison concludes (p.81): “There is no such thing as imperative logic (...) There are indeed logical relationsbetween one imperative and another, but this simply supports a logic in which the premisesand conclusions are indicative statements about imperatives” (p. 81, my emphasis). Thecloseness of Harrison’s account to mine was pointed out to me by Lou Goble.

49Castaneda [43], replying to Wedeking, grants that differences in the meaning of infer-ential words in indicative and imperative ‘inferences’ may exist, but argues that they donot prohibit a concept of imperative inferences in parallel to indicative ones. This seems tomiss the point, it echoes Dubislav’s convention without explaining what such formalismsare to formalize. I owe the reference to Wedeking and Castaneda to Peter Vranas.

50Such authors include G. H. von Wright who writes in [256] p. 109: “And now Itoo, after a long and winding itinerary have come to the same view: logical relations,e.g. of contradiction and entailment, cannot exist between (genuine) norms.”. Abovewe have already noted that Hamblin [74] p.89, Sellars [193] p. 239-240, von Kutschera[128] and Philipp [168], [169] have expressed scepticism or denied the possibility of alogic of imperatives altogether. For imperatives also cf. Moritz [153], Williams [238],Keene [120], Opa lek & Wolenski [164]. The term is coined from Weinberger’s [230] term‘normological skepticism’ which denies logical relations not only between imperatives, butany prescriptive language. The main proponent of normological scepticism is Kelsen [121].


argue that they must be so or that other facts should be made similar to them– with a statement about grammar? Curiously, in his essay, Poincare [172]never claimed to have discovered the logic of imperatives of which he washeralded as the pioneer. His main argument is that findings of science caninfluence moral reasoning. He just presumes that, like scientific argumentsconsist of sentences in the indicative mood, moral reasoning is conductedusing sentences in the imperative mood. It is true that facts can influencethe reasoning of agents about their obligations: Hare’s officer, who uponbeing commanded to go to Edinburgh via Coldstream or Berwick finds theroad via Coldstream blocked, acts quite reasonably by concluding that shenow ought to go via Berwick. But this is a reasoning about what obligationsshe has, it is a deontic argument, and not a case of ‘inferring’ imperatives. SoPoincare’s main argument is correct, but the assumed parallelism betweensentences in indicative and imperative mood, that they can both feature inlogical arguments, does not exist. Our language does not work that way.

There are several ways to go forward from a position of ‘imperativologicalskepticism’. First, one might continue the ‘logic of imperatives’ as a logic ofsatisfaction. The logic of satisfaction states which imperatives must also besatisfied if some other imperatives are satisfied, and it may also be used tostate which imperatives will be violated by satisfying other imperatives. Wecan use the notion of satisfaction to distinguish imperatives that might beseen as redundant in a set of imperatives in the sense that these will also besatisfied if some other, different imperatives are satisfied, or identify subsetsof imperatives that cannot be all satisfied and so conflict. By providing theseconcepts, the logic of satisfaction, though it may appear trivial, remains ameaningful and correct way to talk about imperatives.51

Second, imperatives normally express the wish or desire on the part ofthe person or authority using the imperative that what is commanded issatisfied. But it seems unreasonable to wish for A to be realized, but alsofor ¬A to be realized, and in this sense two wishes may exclude another. Ifimperatives express wishes of one particular person, we can then point outto her what wishes may be unreasonable. Likewise it might be desirable toview the norms of a particular society as if they all were the wishes of oneperson, the ‘law giver’, and logic may then give advice as to which normsmust be revised so that the system is ‘reasonable’. This is the position of G.

51Cf. C. G. Hempel’s [91] remark with regard to Ross’s Paradox that a logic of satis-faction should not be so easily rejected.


H. von Wright in his late work on normative logic, cf. e.g. [255], [256].52

Finally, there is deontic logic. Deontic logic uses the modal expressions ‘itis obligatory that’, ‘it is permitted that’, ‘it is prohibited that’ etc. to describewhat ought to be, is permitted or is prohibited according to given imperativesor norms. Deontic logic has been disparagingly called a “kind of ersatz truth”,that merely mirrors logical relations that already exist between imperativesor norms, and so we should rather look for this logic than studying a deonticlogic that only reflects it and so must result in a “dull isomorphism”.53 But ithas been the ‘logic of imperatives’ that has kept escaping us, while sentencesthat use deontic expressions can easily be used to form valid arguments. Somaybe it is not a logic of imperatives that is the ‘proper’ subject of studyand makes deontic logic just an ersatz theory, but it is the other way round,54

and the idea of a logic of imperatives has been a fata morgana, leading us toever more futile attempts to explain inference relations between imperatives,to find analogues of truth values, or new logics to explain Dubislav’s scheme,whereas any plausibility of this idea was just a reflection of the real, but quitedistinct possibility of a logic about imperatives, namely of deontic logic.55 Ifmy hypothesis holds, then it is the only logic regarding normative conceptssuch as obligation that we should be concerned with. So it is to this logicthat I now turn.

52The idea that commands can be identified with wishes, which in the above senserelate to each other, goes back to Bentham [29] pp. 95–97. Note that this does not forceone to acknowledge that there is a logic of commands. There is a difference between atheoretical inconsistency and a practical inconsistency, or: “what is inconsistent, and whatis inconsistent to say, are two entirely different things” (Harrison [90] p. 95).

53This is Hare’s view in [88] p. 325; also cf. Alchourron [2] pp. 264–266; Kalinowski[112] p. 134; Weinberger [230] p. 58, [232]; Wagner & Haag [216] p. 102. The idea thatdeontic logic reflects the logical properties of norms is that of von Wright in [246] p. 134.

54Cf. Alchourron & Bulygin [7] p. 463: “This logic of norms is, so to say, a reflectionof the logic of normative propositions. It is because we regard as inconsistent a system inwhich it is true thatOxp andOx¬p, that we say that the norms !p and !¬p are incompatible.So it is the logic of norm propositions which yields the foundations for the logic of norms.”.

55Thus Hare’s [87] proposed inference of “Take this box to the station” from the im-perative “Take all the boxes to the station” and the statement “This is one of the boxes”reflects a deontic deliberation of the agent: “I have been ordered to take all the boxes tothe station, this is one of the boxes, so I must take this box to the station.” Castaneda [43]argues in favor of imperative inferences that “we can certainly say: ‘John, go home’ followsfrom, and was inferred by Smith from, ‘John, go home and study’.” While logicians usesentences of the form: ‘...’ follows from ‘...’, with all kinds of expressions filling the gaps,it seems to me that the underlying natural inference is again deontic: Smith concludesfrom John’s having been ordered to go home and study that John ought to go home.

Chapter 2

Deontic Logic

2.1 Beginnings: Leibniz’s Discovery

Deontic logic is a branch of modal logic that is concerned with the logic of nat-ural language expressions such as “it is obligatory that ...”, “it is permittedthat ...”, “it is forbidden that ...”. It seems that Gottfried Wilhelm Leibniz(1646–1716) was the first to explain in detail that these ‘legal modalities’appear to be related in a similar way to each other as the ‘alethic modalities’that describe something as necessary, possible and impossible. Leibniz, in anunpublished manuscript dating from 1671 for his book Elementa Juris Nat-uralis, proposes that what is obligatory (debitum) is what is necessary for agood man to do, what is permitted (licitum) is what is possible for a goodman to do, and what is forbidden (injustum, illicitum) is what is impossiblefor a good man to do. Leibniz lets a fourth legal modality, that of ‘indiffer-ence’, correspond to ‘contingencies’ – as contingent is what is possible butnot necessary, so what is indifferent is what is permitted but not obligatory.Thus Leibniz seeks to “transfer all the complications, transpositions and op-positions of the modalities from Aristotle’s De Interpretatione to the iurismodalia” ([131] pp. 480-481).1

While it would be unfair to say that this analogy between legal conceptsand their ‘natural’ or ‘alethic’ counterparts became forgotten among logical

1As Knuuttila [124] explained, the idea that a good man cannot violate the law can betraced through the middle ages back to Aristotle’s Nicomachean Ethics EN 1100b 33-35,where he states that “no happy man can become miserable, for he will never do the actsthat are hateful and mean”.

50

DEONTIC LOGIC 51

philosophers – it did not2 – the analogy was first formally explored in G. H.von Wright’s 1951 “Essay on Modal Logic” [243]. The ‘legal’, or ‘deontic’modalities are studied here in comparison to other modal categories such asthe alethic, quantificational and epistemic, as illustrated by the table below:

Modal Categories

alethic epistemic deontic existentialnecessary verified obligatory universalpossible unfalsified permitted existingcontingent undecided indifferent partialimpossible falsified forbidden empty

So among the above modal categories, the deontic concept of ‘obligatory’corresponds to the alethic concept of ‘necessary’ or the quantificational con-cept of ‘universal’, while the deontic concept of ‘permitted’ corresponds tothe alethic concept of ‘possible’ or the quantificational concept of ‘existing’.3

For the alethic modalities, consider the modal ‘square of oppositions’ fromAristotle’s De Interpretatione chapters 12, 13:

It is necessary that A It is necessary that not A=It is not possible that not A contraries =It is not possible that A=It is impossible that not A =It is impossible that A

q ↖ ↗ qimplies oppositions implies⇓ ↙ ↘ ⇓

It is possible that A It is possible that not A=It is not necessary that not A subcontraries =It is not necessary that A

=It is not impossible that A =It is not impossible that not A

2Watts [220] mentions the analogy several times, cf. part II ch. 2 sect. iv. As Hruschka[106] points out, Gottfried Achenwall [1] §§12, 44 uses language very similar to Leibnizwhen distinguishing between the actio moraliter possibilis, actio moraliter impossibilis andthe actio moraliter necessaria, identifying the necessitas moralis with obligatio. Bentham[29] p. 97 constructs a deontic square similar to the one below, and shortly before theadvent of axiomatized deontic logic, Grelling [73] suggests the construction of a formalsyntax of ought-sentences after the image of modal logic.

3Cf. von Wright [242] p. 2. The changes to von Wright’s original table are due toAnderson and Moore [18] who added the quantificational concept of ‘partial’ (for somebut not for all), and the epistemic concept of ‘unfalsified’ (verified or undecided).

DEONTIC LOGIC 52

In its corners we find three equivalent statements, expressed in terms of thethree alethic modalities ‘necessary’, ‘possible’, ‘impossible’. Horizontally,the statements in the left and right upper corner are contraries, meaningthat not one statement from the left and one from the right corners canbe true together. The statements in the left and right lower corner presentsubcontraries, meaning that not one statement from the left and one from theright corners can be false together. Vertically we find the sub-alternatives,meaning that any statement from the right (left) upper corner implies anystatement from the right (left) lower corner. The statements that standdiagonally to each other are oppositions, meaning that each statement fromone corner implies that any statement from the diagonally opposite corner isfalse. Given the similarity between the two, the alethic and deontic modalcategories, this square can then be translated to a ‘deontic square’ as follows(cf. Prior [178] p. 220):

It is obligatory that A It is obligatory that not A=It is not permitted that not A contraries =It is not permitted that A

=It is forbidden that not A =It is forbidden that Aq ↖ ↗ q

implies oppositions implies⇓ ↙ ↘ ⇓

It is permitted that A It is permitted that not A=It is not obligatory that not A subcontraries =It is not obligatory that A

=It is not forbidden that A =It is not forbidden that not A

The expressions ‘it is obligatory that A’, ‘it is not permitted that not Aand ‘it is forbidden that not A are taken to be equivalents. Each corner hasthree such equivalents, expressed in terms of the three modalities ‘obligatory’,‘permitted’ and ‘forbidden’. If it is obligatory that A, then A should not beforbidden at the same time, and so the statements in the right and left uppercorners are contraries to each other. Furthermore, if ‘it is obligatory that A’implies that A is not forbidden, this means that A is permitted, and so any ofthe statements in the upper left corner implies all statements in the lower leftcorner, and any of the statements in the upper left corner stand in oppositionto any of the statements in the diagonally opposite lower right corner, andso on for all the corners. Thus the relations from Aristotelian modal logicare transferred to the deontic modalities, just as Leibniz suggested.

DEONTIC LOGIC 53

2.2 Standard Deontic Logic

Not counting an earlier attempt by Ernst Mally [145],4 the first successfulaxiomatization of the relations between deontic expressions was presentedby Georg Henrik von Wright in his 1951 article ‘Deontic Logic’ [242]. VonWright employs three modal operators ‘O’, ‘P ’ and ‘F ’, where OA meansthat it is obligatory that A, PA means that it is permitted that A and FAmeans that it is forbidden that A. The three operators are cross-defined byletting ‘O’ be primitive, and defining PA as ¬O¬A and FA as O¬A, soPA means that it is not obligatory that not A, and FA means that it isobligatory that not A.5 His axiomatization, called the ‘classical system ofdeontic logic’ [251], can be characterized by the following axioms and rules:

(Ext) If A↔ B is a theorem, then OA↔ OB is a theorem.(M) O(A ∧B)→ (OA ∧OB)(C) (OA ∧OB)→ O(A ∧B)(D) OA→ PA

Here, the principle of extensionality (Ext) expresses that if A and B areequivalent, then so are the sentences OA and OB. The principle of monotony(M) expresses that if it is obligatory that both A and B, then it is alsoobligatory that A and obligatory that B, and the principle of cumulativity(C) expresses vice versa that if it is obligatory that A and obligatory thatB, then it is also obligatory that both A and B. Together, (M) and (C) areequivalent to von Wright’s principle of distribution (PA∨PB)↔ P (A∨B),meaning that if a disjunction of acts is permitted, then one of the disjunctsis permitted, and vice versa. (D) is the characteristic ‘deontic’ axiom thatexpresses that if it is obligatory that A then it is permitted that A; it is theequivalent of von Wright’s principle of permission PA∨P¬A that expressesthat any act is either permitted or its negation permitted, and it is also – dueto the definition of PA as ¬O¬A equivalent to ¬(OA ∧ ¬O¬A). It is alsocrucial for obtaining the modal relations expressed by the deontic square:6

4Mally’s axiomatization culminated in the result that ‘it is obligatory that A’ impliesthat ‘A is the case’, and vice versa, or, in Mally’s words, the Aquivalenz von tatsachlichGefordertem und Tatsachlichem (the equivalence between the factually obligatory and thefacts) cf. [145] p. 269. Mally’s error lies in his missing distinction between logical andmaterial implication, cf. Wolenski [241], Goble & Lokhorst [70].

5Von Wright originally used ‘P ’ as primitive and equivalently defined OA as ¬P¬A,and FA as ¬PA, but the use of ‘O’ as primitive became standard.

6Since (D) is missing from Oskar Becker’s independent publication [28], von Wright is

DEONTIC LOGIC 54

OA, (D) O¬A,¬P¬A, F¬A ¬PA, FA

q ↖ ↗ q(D) × (D)⇓ ↙ ↘ ⇓PA, P¬A,

¬O¬A, ¬FA (D) ¬OA, ¬F¬A

If one adds to von Wright’s classical system the additional axiom (N)7

(N) O(A ∨ ¬A)

one obtains the equivalent of the so called ‘standard deontic logic’ systemSDL, which is more rigorously formalized as follows:8

Definition 2.1 (The Language of Deontic Logic) The alphabet of thelanguage of deontic logic LDL is like that of propositional logic LPL exceptthat it additionally contains the deontic operator symbol ‘O’. LDL is definedby the same rules as the set LPL plus the following: if A ∈ LDL, then alsoOA ∈ LDL. PA abbreviates ¬O¬A, and FA abbreviates O¬A.

Definition 2.2 (The System SDL) SDL is the smallest set that containsall LDL-instances of tautologies and the axiom schemes (M), (C), (D) and(N), and is closed under the rules of (Ext) and modus ponens.

rightly acclaimed to be the founder of modern deontic logic.7(N) makes all tautologies obligatory. While von Wright [242] p. 11 rejected it, its

philosophical import seems rather small: if something is obligatory, e.g. if OB is true,then the truth of O(A ∨ ¬A) can be derived by means of (C) and (Ext).

8The deontic systems studied here will be based on propositional logic PL. The alpha-bet of its language LPL consists of a set of proposition letters Prop = {p1, p2, ...}, Booleanconnectives ‘¬’, ‘∧’, ‘∨’, ‘→’ and brackets ‘(’, ‘)’. LPL is then the smallest set such that(a) any proposition letter p1, p2, ... is in the set, (b) if A,B are in the set, then so are¬A, (A∧B), (A∨B), (A→ B). Semantically, valuation functions v : Prop→ {1, 0} definethe truth of a sentence A ∈ LPL (written v |=PL A) as follows: (a) v |=PL pi if and onlyif (iff) v(pi) = 1, (b) v |=PL ¬A iff it is not the case that v |=PL A, (c) v |=PL A ∧ B iffv |=PL A and v |=PL B, (d) v |=PL A ∨ B iff v |=PL A or v |=PL B, (e) v |=PL A → B iffv |=PL ¬A or v |=PL B. If all valuations that make true all sentences of a set of sentencesΓ also make true A, then Γ implies A (written Γ � A). If A is true under all valuations v,A is called a tautology, if it is false under all valuations, it is called a contradiction. Fromthe set PL of all tautologies, the notions of provability, consistency and derivability (Iwrite Γ `PL A) are defined as usual. > denotes an arbitrary tautology, and ⊥ an arbitrarycontradiction.

DEONTIC LOGIC 55

Disregarding varying axiomatizations, SDL was first introduced by Prior[178] ch.1 §6, in the nomenclatura of modal logic it is the normal modalsystem KD, i.e. the modal system K to which (D) is added. It differs fromthe alethic system KT only in so far as that (D) replaces the axiom

(T) OA→ A

typical for such systems: for alethic modalities, (T) states that if it is neces-sary that A then A (is true), which should not hold in the deontic interpre-tation ‘if it is obligatory that A then A (is true)’. It should be noted thatthe move from von Wright’s first proposal to standard deontic logic SDL alsobrought about a change in the meaning of the expression A in the scope of thedeontic operators. Von Wright [242], [243], similar to Becker [28], intendedA to be the name of an act: PA is interpreted as the proposition that the actnamed by A is permitted. By contrast, Prior lets A designate a propositionthat asserts that an act of the sort A is done. Deontic logic literature hasfavored Prior’s interpretation, and OA is standardly read as ‘it is obligatorythat the situation described by the descriptive sentence A is realized’.9

2.3 Anderson’s Reduction

Independently from each other, Stig Kanger [119] and Alan Ross Anderson[12], [14] showed in 1956/7 that the analogy between the alethic and deonticmodal concepts may not only be used to transfer the rules of the former tothe latter, but to reduce deontic logic to alethic modal logic altogether. Tothis effect, a ‘prohairetic constant’ Q (Kanger, Q denoting ‘what moralityproscribes’) or S (Anderson, S meaning a ‘sanction’ or ‘a bad state of affairs’)is added to any modal system from K upwards, and the deontic O-operatoris then defined in the following way, respectively:

(Def-OQ) OA =def 2(Q→ A)(Def-OS) OA =def 2(¬A→ S)

So in Kanger’s interpretation, that A is obligatory means that ‘what moralityprescribes’ necessitates A, and in Anderson’s interpretation it means that notdoing A necessitates the sanction (or the ‘bad thing’).10 If the alethic modal

9Cf. Allen [11], Føllesdal & Hilpinen [53] p. 14.10By closer inspection, Kanger’s variant of this reduction of the deontic modalities to

alethic modal logic may well be attributed to Leibniz [131], when he wrote that what isobligatory is what is necessarily done by a good man (who by nature does what morality

DEONTIC LOGIC 56

system is K, and the following respective axioms are added:11

(DQ) �Q(DS) �¬S

then the deontic fragment of this system coincides with SDL. More formally:

Definition 2.3 (The Language of the Modal System KQ+ [KS+])The alphabet of the language of the language LKQ+ [LKS+ ] is like that ofpropositional logic LPL except that it additionally contains the alethic oper-ator symbol 2 and the prohairetic constant Q [S]. LKQ+ [LKS+ ] is definedby the same rules as the set LPL plus the following two: the constant Q [S]is in the set; if A is in the set, then so is 2A. �A abbreviates ¬2¬A.

Definition 2.4 (The Modal System KQ+ [KS+])The axiomatic system KQ+ [KS+] is the smallest set of that contains allLKQ+ [LKS+ ]-instances of tautologies and the axiom-scheme

(K) 2(A→ B)→ (2A→ 2B)

as well as (DQ) [(DS)] and is closed under the rules of modus ponens and

(RN) If A is in the set, then so is 2A.

Theorem (Translation Theorem for SDL [Aqvist [22] pp. 683-688]).Let φ : LDL → LKQ+ [LKS+ ] be a translation from the language of deonticlogic to the language of alethic modal logic with the constant Q [S] such that

(a) φ(pi) = pi, (b) φ(¬A) = ¬φ(A),(c) φ(A ∧B) = φ(A) ∧ φ(B), (d) φ(A ∨B) = φ(A) ∨ φ(B),(e) φ(A→ B) = φ(A)→ φ(B), (f) φ(OA) = 2(Q→ φ(A)) [2(¬φ(A)→ S)].

Let the deontic fragment DF (KQ+) [DF (KS+)] contain all A ∈ KQ+[KS+]with A = φ(B) for some B ∈ LDL. Then DF (KQ+)[DF (KS+)] = φ(SDL).

The reduction of deontic to alethic modal logic is interesting in its ownright, it can even point to an history of its own. Kanger’s variant, thatsomething is obligatory if it is necessary for an ideal state, stands right in

prescribes). Knuuttila [124] p. 236 points out that a reverse reduction of the alethicmodalities to deontic modalities can be found in the Dialectica of Peter Abelard (1079–1142) who identified necessity with what nature demands, possibility with what natureallows, and impossibility with what nature forbids.

11Anderson showed that even this addition can be avoided if one chooses the modalsystem KT and replaces S by the formula B∧�¬B. For Kanger’s constant Q, the formulaB → 2B constitutes a similar replacement.

DEONTIC LOGIC 57

the footsteps of Leibniz [131], according to whom what is obligatory is whatis necessarily done by a good man;12 Husserl, Menger and Bohnert have alsopresented related ideas.13 Anderson’s method of defining the obligatory viaa sanction or ‘bad state’ has been called ‘the most natural interpretation ofdeontic sentences’ (Hilpinen [92]) and was later taken up by dynamic deonticlogic.14 Nevertheless, Anderson’s and Kanger’s reductions can also be seenas a first shift from an informal understanding of deontic expressions asbelonging to a discourse about moral or legal norms to an interpretation viaprohairetic terms like the ‘ideal’ or ‘bad’.

2.4 Possible Worlds Semantics

Axiom systems characterize the logic in question by means of sets of theo-rems. E.g. we know that

OA ∧ PB → P (A ∧B),

meaning that if something is obligatory and something else is permitted,then we may realize both conjunctively, is a theorem of standard deonticlogic SDL, because its theoremhood can be established by means of thissystem’s axioms and the rules, namely (K) and modus ponens. If A is atheorem of e.g. SDL, we write `SDL A and may call A provable in SDL. Asusual, from this notion a variety of others derive: a set of sentences Γ ⊆ LDL

is called SDL-inconsistent if and only if there are A1,..., An in Γ, n ≥ 1, suchthat `SDL ¬(A1 ∧ ... ∧ An), and Γ is SDL-consistent otherwise. A sentence

12Very similarly, Anderson’s sanction has been interpreted to represent “simply failureto be perfect” (Prior [179] p. 146.)

13Cf. Husserl ([108] p. 41/42, my translation): “The following forms are equivalent:‘A ought not to be B’, and ‘an A which is not B, is generally a bad A’, or ‘only an A,which is not B, is a good A’ ”, “ ‘an A may be B’ = ‘an A, which is B, needs, for thisreason, not to be a bad A’ ”; Menger [150]: “The statement ‘I command p’ (...) maybe interpreted as ‘unless p, something unpleasant will happen’ ”; Bohnert [33] reducescommands to motivational declarative sentences like P ∨ B(x), where B is the behaviordesired of x and P a penalty, or E → B(x), where E is some desired end. Both, Mengerand Bohnert, are aware that their definitions, phrased in terms of ordinary disjunctionand material implication, have unwanted consequences (like that all factual truths arecommanded), to avoid them Menger applies a three-valued logic, Bohnert demands thatB(x) should be ‘causally necessary’ for the end E.

14By singling out an ‘undesirable state of affairs’, an action αmay be defined as forbiddenif it necessarily results in this state, as obligatory if the action not-α is forbidden, and aspermitted if α is not forbidden (cf. Meyer [152]).

DEONTIC LOGIC 58

A ∈ LDL is SDL-derivable from a set Γ ⊆ LDL (we write Γ `SDL A) iffΓ ∪ {¬A} is SDL-inconsistent, etc.

However, axiom systems, and particularly those of deontic logic, haveoften been devised ad hoc, philosophers taking their cue from intuitions asthey come along. Was it a good idea to add (N) to von Wright’s system?If one thinks that the object of human wishes and commands are neithernecessities nor impossibilities (cf. Menger [150] p. 97) should one not retract(N) and instead add ¬O(A ∨ ¬A) and ¬O(A ∧ ¬A) to the set of axioms,thus making sure that neither tautologies nor contradictions can ever beobligatory? The first addition would yield ¬OB for anyB, and so incidentallyfree us of all obligations: responsible is the seemingly innocuous (M), so wewould have to retract (M), but what replace it with? By contrast, the secondaddition would be harmless, in fact it is a theorem of SDL.

A much better way to go forward in developing a logical system is bymeans of a logical semantics. A logical semantics describes mathematicalstructures or ‘models’ that enable us to evaluate each sentence of the for-mal language as ‘true’ or ‘false’ with regard to the model. The sentences ofthe language that are valid according to a semantics (true in all its models)may correspond to all, and only, the theorems of an axiomatically describedlogical system; this is then called sound, and complete, with respect to the se-mantics.15 The logical-semantical interpretation of the logical symbols (con-nectives, operators, predicates etc.) of the formal language may then becompared to intuitions about the use of the ordinary language expressionsthey symbolize, where the aim is to let the properties of the mathemati-cal models mirror the aspects and regularities of ordinary language usage asclosely as possible, while on the other hand the logical semantics might helpus clarify our intuitions about that usage.16

Since 1957, deontic logic has been interpreted in terms of a possible worldssemantics (cf. Hintikka [97], Kanger [119]).17 Its models M are triples〈W,R, V 〉, where W is a nonempty set of ‘possible worlds’, R ⊆ W ×W isa binary relation between worlds, where for any w, v ∈ W , wRv means thatv is a ‘deontic alternative’ to w, and v : Prop ×W → {1, 0} is a valuationfunction that associates a truth value with each proposition letter in any

15Axiomatic systems may thus correspond to a range of different semantics.16For the uses of a logical semantics and its limits cf. Stekeler-Weithofer [204] p. 141.17With Kripke [127], interpretations in terms of possible worlds became standard for

systems of modal logic, and the account I present here is Kripke-semantics as reapplied todeontic logic by e.g. Hanson [82], also cf. Aqvist [22] pp. 665-669.

DEONTIC LOGIC 59

world w ∈ W . Truth of LDL-sentences in any model M and world w ∈ Wis then defined recursively:

(a) M , w |= pi iff v(pi, w) = 1,(b) M , w |= ¬A iff M , w 2 A,(c) M , w |= A ∧B iff M , w |= A and M , w |= B,(d) M , w |= A ∨B iff M , w |= A or M , w |= B,(e) M , w |= A→ B iff M , w |= ¬A or M , w |= B, and(f) M , w |= OA iff ∀v ∈ W : if wRv then M , v |= A.

So the truth of sentences of propositional logic at a world w is defined asusual, while a sentence of the form OA is true at a world w if and only if Ais true at all deontic alternatives to w. The deontic alternatives may also beviewed as the ‘ideal worlds’ that are accessible from w, so what is obligatoryis what is the case in all ideal worlds accessible from the actual world w.(The semantics are illustrated below by a model M = 〈W,R, V 〉 such thatM , w1 |= O(p1 ∨ p2); the deontic alternatives to w1 are here the worlds w2

and w3.) A LDL-sentence A is called ‘valid’ iff it is true at any world w in

Figure 2.1: Possible worlds semantics for monadic deontic logic

any model M (we write |= A). If we add the restriction that for all modelsM , R is serial, i.e. for any w ∈ W there exists at least one v ∈ W that is adeontic alternative to w, then the set of valid LDL-sentences coincides withSDL, i.e. SDL is sound and complete with respect to this semantics.

Turning back to the axiomatization of deontic logic, we can now see howthe axiom (N) can be defended from the viewpoint of possible worlds seman-tics: something is obligatory if it is true in all accessible ideal worlds. But

DEONTIC LOGIC 60

at any world, for all models M , A ∨ ¬A, being a tautology, is true, and soO(A ∨ ¬A) is true. It is claimed that this shows the role of possible worldssemantics in the clarification of our intuitions concerning the behavior of de-ontic operators, whereas a purely axiomatic approach is not conclusive (cf.Wolenski [240]).

2.5 Dyadic Deontic Logic

Early in the development of deontic logic, doubts arose as to the adequacy ofstandard deontic logic SDL for the formalization of conditional obligations,and in particular that of contrary-to-duty obligations. It is possible that Ais forbidden, and additionally that if A – contrary-to-duty – obtains, then itis obligatory that some act of reparation B is done. But how to express thissecond consequence relation? It cannot be formalized as O(A → B), for asPrior [177] pointed out, we have

(PP) FA→ O(A→ B),

as a theorem of deontic logic. Possible worlds semantics reveals it as anoffspring of one of the paradoxes of material implication ¬A → (A → B):FA means that at all accessible ideal worlds ¬A is true, and so also A →B is true there, hence O(A → B) is also true, for any B. Prior thoughtthis theorem, later called ‘Prior’s Paradox’, troublesome, since it can beunderstood as stating that the doing of what is forbidden commits us to thedoing of anything whatsoever, e.g. the forbidden act of stealing commits usto committing adultery.

One may think that the situation can easily be remedied by using theformalization A→ OB, instead of O(A→ B), for conditional obligations, orat least by doing so when contrary-to-duty obligations are involved. However,Chisholm [46] showed that whatever combination of these formalizations ischosen for the following sentences

(1) It ought to be that a certain man go to the assistance of his neighbors.(2) It ought to be that if he does go he tell them he is coming.(3) If he does not go, then he ought not to tell them he is coming.(4) He does not go.

then any of the four sets {OA,O(A → B),¬A → O¬B,¬A}, {OA,O(A →B), O(¬A→ ¬B),¬A}, {OA,A→ OB,O(¬A→ ¬B),¬A}, and {OA,A→OB,¬A → O¬B,¬A} will either be inconsistent in SDL, or one of its four

DEONTIC LOGIC 61

sentences will derive one of the others. But intuitively, the normative situa-tion is not inconsistent and the sentences are independent from each other,so their formalization should not make them otherwise. Thus this examplewas termed ‘Chisholm’s Paradox’.

In reaction to Prior’s paradox, G. H. von Wright in 1956 [244] introduced astroke ‘/’ in the scope of the operator O, where the ‘dyadic’ formula O(A/B)means that A is obligatory under the condition B. The old, ‘monadic’ O-operator is then defined as OA = O(A/>), i.e. as an obligation that holdsin arbitrary, or default conditions. More formally, the new language is asfollows:18

Definition 2.5 (The Language of Dyadic Deontic Logic)The alphabet of the language of dyadic deontic logic LDDL is like that of LDL,except that it additionally contains the stroke ‘/’. LDDL is then the smallestset such that(a) if A,B are in LPL, then O(A/B) is in the set,(b) if A,B are in the set, then so are ¬A, (A∧B), (A∨B) and (A→ B).

Again, P (A/B) abbreviates ¬O(¬A/B) and F (A/B) abbreviates O(¬A/B).OA, PA and FA abbreviate O(A/>), P (A/>) and F (A/>) respectively.

With the introduction of the stroke, soon numerous new axiomatic sys-tems of dyadic deontic deontic logic emerged. Systematically, the followingdyadic analogues of the axiom schemes of SDL appear to be good candidatesfor an axiom system of dyadic deontic logic:

(Ext) If A↔ B is a tautology, then O(A/C)↔ O(B/C) is a theorem.

(ExtC) If C ↔ D is a tautology, then O(A/C)↔ O(A/D) is a theorem.(DM) O(A ∧B/C)→ (O(A/C) ∧O(B/C))

(DC) (O(A/C) ∧O(B/C))→ O(A ∧B/C)

(DD) O(A/C)→ P (A/C)

(DN) O(A ∨ ¬A/C)

These axioms allow reasoning with antecedents like C or D only in the veryweak form of (ExtC). But sometimes we want to use information from the an-tecedent for our reasoning with consequents. Consider the following exampleby Powers [174]:

18Note that the language LDDL is more restricted than LDL and e.g. does not allowiterated or nested O-operators as in OOA or O(OA → A), or ‘mixed’ propositional anddeontic formulas as in B ∧ OA. I simplify matters, as an extended language would e.g.require operators and axioms for universal necessity in order to make the system completewith respect to the kind of semantics given below.

DEONTIC LOGIC 62

John must either not impregnate Suzy Mae or marry her. SuzyMae is pregnant by John. So John must marry Suzy Mae.

A rule that permits such inferences is achieved by strengthening (Ext) to aprinciple of ‘circumstantial extensionality’ :

(CExt) If C → (A↔ B) is a tautology, then O(A/C)↔ O(B/C) is atheorem.

Using this rule, we obtain the theorem O(¬A ∨ B/A) ↔ O(B/A), i.e. ina situation where A is true (Suzy is pregnant by John) the obligation torealize ¬A ∨B (Either Suzy is not pregnant by John or John marries Suzy)is equivalent to an obligation to realize B (John marries Suzy). If (Ext) isreplaced by (CExt), it is clear that either (D) or (N) must be weakened to:

(DD-R) Unless C is a contradiction, O(A/C)→ P (A/C) is a theorem.

(DN-R) Unless C is a contradiction, O(>/C) is a theorem.

Otherwise for impossible conditions ⊥ the theorem O(⊥/⊥) is derivable from(DN) and (CExt), and also the contradictory theorem ¬O(⊥/⊥) from (DD),(DC) and (CExt), i.e. the set of axioms would be inconsistent.

Two other early dyadic deontic axioms or theorems were the principles of‘conditionality’ and of ‘rational monotony’:19

(Cond) O(A/C ∧D)→ O(D → A/C)

(RMon) P (D/C)→ (O(A/C)→ O(A/C ∧D))

(Cond) expresses that if A is obligatory in the situation C-and-D, then itis obligatory in the situation C that if one realizes D then one also realizesA. (RMon) expresses that if one realizes something permitted, then one’sobligations do not change. Together, these axioms, with the choice betweenunrestricted (DD) and restricted (DN-R), or restricted (DD-R) and unre-stricted (DN), describe the two most successful systems in dyadic deonticlogic:20

19(Cond) is Rescher’s [182] axiom A4, it also derives from the left-to-right direction ofvon Wright’s [244] axiom A2: P (A∧B/C)↔ P (A/C)∧P (B/C∧A). (RMon) is derivablefrom its right-to-left direction and is also Rescher’s [182] theorem T4.16. Given (CExt),the conjunction of (Cond) and (RMon) is equivalent to A2.

20As for the not so successful approaches, note first that these should not be theorems:Up O(A/C)→ O(A/C ∨D)Down O(A/C)→ O(A/C ∧D)

With Up or Down, any contrary-to-duty obligation, contrary to intuitions, creates a con-flict of duties. E.g. consider a Chisholmian set {OA,O(B/A), O(¬B/¬A)}. With Up wederive OB and O¬B from O(B/A) and O(¬B/¬A), so there is a conflict. From OA and

DEONTIC LOGIC 63

Definition 2.6 (Systems DDLH, DDLL)DDLH and DDLL are the smallest sets that contain all LPL-instances oftautologies and the axiom schemes (DM), (DC), (Cond), (RMon), whereDDLH additionally contains all LPL-instances of (DD-R) and (DN), andDDLL additionally contains all LPL-instances of (DD) and (DN-R). Bothsets are closed under (CExt), (ExtC) and modus ponens.

DDLH is Hansson’s [84] system DSDL3 as axiomatized by Spohn [203], and,neglecting differences of language, DDLL is the synoptical Lewis-type systemDFL of Aqvist in [23]. The following are typical theorems of the two systems:

(Ref) O(A/A) [ only DDLH ](Pres) O(⊥/C)→ (O(A/D)→ O(A ∧ ¬C/D))(P -Pres) P (⊥/C)→ (O(A/D)→ O(A ∧ ¬C/D))(CCMon) O(A ∧D/C)→ O(A/C ∧D) [ = (P -Cond) ](Cut) O(D/C)→ (O(A/C ∧D)→ O(A/C)) [ = (P -RMon) ](Or) O(A/C) ∧O(A/D)→ O(A/C ∨D)(DR) O(A/C ∨D)→ (O(A/C) ∨O(A/D))(FH) P (C/D)→ (O(A/C ∨D)→ O(A/C))(Trans) P (A/A ∨B) ∧ P (B/B ∨ C)→ P (A/A ∨ C)(P -Loop) P (A2/A1) ∧ ... ∧ P (An/An−1) ∧ P (A1/An)→ P (An/A1)(Loop) O(A2/A1) ∧ ... ∧O(An/An−1) ∧O(A1/An)→ O(An/A1)

The names are from analogues in the study of nonmonotonic reasoning(cf. Makinson [140]), namely reflexivity, preservation, conjunctive cautiousmonotony, disjunctive reasoning and transitivity. (Ref) is typical of Hansson-type logics. (CCMon) is Rescher’s [182] th. 4.4, (Or) is the right-to-leftversion of von Wright’s [247] (B3), (DR) is Hansson’s [84] th. 13, and (FH)is Føllesdal & Hilpinen’s th. 77. (Trans) is transitivity of weak preferencedefined as A 4 B =def P (A/A ∨ B) in Lewis [136] p. 54. (P-Loop) wasintroduced by Spohn [203] to define the relevant equivalence classes in hiscompleteness proof of Hansson’s DSDL3, and its O-form was rediscovered byKraus, Lehmann & Magidor [126].

The reason that Hansson’s system DDLH and Lewis-type system like

O(B/A) we derive OB with (Cond), (DC) and (DM), and with Down we then obtainO(B/¬A), which conflicts with O(¬B/¬A). Nevertheless, axiom systems by von Wright[247], [248] and A. R. Anderson [15] derive Up, and Rescher’s system [182], that alsoprovides (Cond), (DC) and (DM), derives Down.

DEONTIC LOGIC 64

DDLL were more successful than others was not so much because they in-cluded some widely recognized theorems, but because they are also soundand (weakly) complete with respect to a preference semantics that modelscontrary-to-duty situations in an intuitively plausible way:21 Let � (‘at leastas good as’) be a reflexive, transitive and connected preference relation onB (the set of all Boolean valuations v : Prop → {1, 0}) that additionallysatisfies the ‘Limit Assumption’:

(LA) For all A ∈ LPL, if ‖A‖ 6= ∅ then best(‖A‖) 6= ∅Here ‖A‖ = {v ∈ B | v |= A} is the subset of Boolean valuations or ‘worlds’that make A true (the ‘A-worlds’), and for any X ⊆ B, best(X) = {v ∈X | ∀v′ ∈ X : v � v′}) is the set of ‘best’ worlds in X. The Limit Assump-tion tells us that for all PL-sentences that are true at some ‘world’, there areworlds at which they are true that are at least as good as any other worldat which the sentence might be true. Finally, alongside the usual truth def-initions for Boolean operators the following truth definitions for the deonticO-operator are used, for DDLH and DDLL respectively:

DDLH : �|= O(A/C) iff best(‖C‖) ⊆ ‖A‖DDLL: �|= O(A/C) iff ∃v ∈ ‖C ∧ A‖) : ∀v′ ∈ ‖C ∧ ¬A‖ : v′ � v

So for DDLH , O(A/C) is defined true iff the best C-worlds are A-worlds, andfor DDLH , O(A/C) is defined true iff there is a (C ∧ A)-world such that no(C ∧ ¬A)-world is at least as good.

Just as the introduction of the stroke ‘/’ into the scopes of the deonticoperators was motivated by the search for a representation of contrary-to-duty imperatives, so was the development of preference semantics for dyadicdeontic logic. This was expressed particularly clearly by Bengt Hansson:22

“The problem (...) is what happens if somebody neverthelessperforms a forbidden act. Ideal worlds are excluded. But it maybe the case that among the still achievable words some are betterthan others. There should then be an obligation to make the bestout of the sad circumstances.”[84]

21A proof of (weak) completeness for DDLH was first presented by Spohn [203], it canbe straightforwardly adapted to DDLL. As the semantical details vary between Hanssonand Lewis, I use a description that fits both systems.

22Also cf. Mott [154]: “The lesson to be learned from Chisholm’s paradox has been theeminently convincing, indeed obvious, one: that what we ought to do is not determinedby what is the case in some perfect world, but by what is the case in the best world we can‘get to’ from this world. What we ought to do depends upon how we are circumstanced.”

DEONTIC LOGIC 65

Hansson’s and Lewis’ semantics capture this idea in the form of a (possiblyinfinitely descending) chain of sets of ideal, subideal etc. worlds: the so-calledsystem of spheres (cf. fig. 1). Ideal worlds are those that are best overall:those in best(B). The second-best worlds are those that are best in the setof all worlds without the best worlds, i.e. those in best(B\best(B)), and soon. For any possible situation C, the best C-worlds are those that are in thesmallest C-permitting sphere, i.e. the smallest sphere that has a non-emptyintersection with the set of C-worlds ‖C‖. The Limit Assumption guaranteesthat for all such C there is such a smallest C-permitting sphere. As to how

Figure 2.2: System of spheres for dyadic deontic logic

contrary-to-duty situations are modeled, consider again Chisholm’s Paradox.Things are best when the man goes to his neighbors’ assistance and tells thathe is coming, so ‖A ∧ B‖ are the best worlds. It seems second-best if theman doesn’t call but still goes, and so the best ¬(A ∧ B)-worlds are theworlds in ‖A ∧ ¬B‖. Then come the worlds in ‖¬A ∧ ¬B‖ where the mandoes not go but also does not tell that he is coming, and the worst arethe worlds in ‖¬A ∧ C‖ where he tells his neighbors that he would go, butdoesn’t. This exhausts the logical possibilities. As may be checked, the thusconstructed model makes true all of {OA,O(B/A), O(¬B/¬A)}. If, uponlearning that ¬A is true, i.e. the man does not go, we restrict the model23

to the worlds in ‖¬A‖, then only the obligation O¬C not to call remains,

23Though such a restriction is not an option for the semantics as outlined, it is possiblein Lewis [137] semantics where the set of worlds may be a subset of B, e.g. one where itis necessary that ¬A. Then we have (O(¬B/¬A) ∧2¬A)→ O¬B.

DEONTIC LOGIC 66

which is how preference semantics solves Chisholm’s Paradox. Other so-called paradoxes receive similar solutions in dyadic deontic logic, like the‘Paradox of the Good Samaritan’ [179] and that of the ‘Gentle Murderer’[56]: in monadic deontic logic it follows from the fact that it is forbiddento assault a man that it is also forbidden to help the assaulted man, andan obligation not to commit murder, but otherwise murder gently, is notexpressible for the reasons given for Prior’s Paradox. In dyadic deontic logicwe can state without contradiction that assault and murder are forbidden,but that in the ‘bad’ situation that either happens secondary obligations tohelp or ‘do it gently’ obtain.

2.6 Doubts and Questions

Since its emergence as a subject of philosophical logic in 1951, deontic logichas undergone multiple periods of refinement. While it was very much anaxiomatic theory in the 1950s, study in the 1960s turned to an applicationof possible worlds semantics, and the study of the late 1960s and early 1970sis characterized by the use of preference semantics to interpret systems ofdyadic deontic logic. The late 1970s and the 1980s have reinterpreted thepossible worlds models for monadic and dyadic deontic logic in terms oftemporal tree structures and added temporal operators and/or temporallyindexed O-operators (cf. e.g. van Eck [51], Aqvist [20]). The 1990s have inturn used these tree systems to interpret action-based deontic operators andcombined the theory of deontic logic with the notion of STIT (‘seeing-to-it-that’, cf. Horty [103]). At the core of all these later developments still liesthe the theory of deontic logic as sketched in the previous section. But therehave also been doubts and questions as to the adequacy of this theory.

2.6.1 Doubts About Possible Worlds

The axiomatic basis of deontic logic has always been feeble at best. It hasbeen claimed that deontic logic has not a single axiom that is not incompati-ble with some reasonable moral position and so they all violate the principleof logical neutrality (cf. Sayre-McCord [191]). Worse still, the main seman-tical tool of traditional deontic logic, possible worlds semantics, is stretchedin its task to clarify our intuitions about possible theorems of deontic logic,as it is itself the target of philosophical criticism. Consider the definition

DEONTIC LOGIC 67

‘OA is true at the actual world iff A is true at every deonticalternative’.

First, to establish the truth of OA, what does it mean that there exists analternative world in which A is true? Are possible worlds real or imaginedentities? In an ontological sense, the only possible world is our own, theactual world, with its future developments hidden in it like the ear in a grainof corn (Kalinowski [115]). So second, given the past history of our world, itcannot be ‘deontically perfect’, and so “we cannot conceive of a Trans WorldAirline that would take us to a deontically perfect world” (Geach [62]). Andthird, even if we restrict our view to possible developments ‘from now on’, aspreference semantics for dyadic deontic logic does, given the provenly falliblenature of human beings, these developments will never be deontically perfect:each development is one where someone sins and so someone’s sinning isobligatory, which is hardly intuitive (cf. Geach [62] p. 10). Fourth, even ifall ideal alternatives to our world are e.g. such that nobody smokes, it maybe that our subideal legislators, as a matter of fact, do not forbid smoking:what justification have we then to call smoking forbidden? maybe it ought tobe forbidden, but this is not the same (cf. Wansing [219]).24 Fifth, considerthe case of the habitual milk-drinker (Solt [199]): suppose that you have thehabit of drinking a glass of milk every day, a habit so strong that at everydeontic alternative each of your alteregos also drinks a cup of milk everyday: does that suffice for the statement ‘it is obligatory that you drink acup of milk every day’ to be true? Solt argues that in order to distinguishbetween norms and laws of nature, habit or logic, the biconditional of thetruth definition should be reduced to the part ‘if OA is true at the actualworld then it is true at every deontic alternative’. But sixth, as Kalinowski[115] points out, this direction of the iff-clause commits us to the view thatnorms only prescribe what is good, for otherwise what they demand couldnot be true in an ideal world, a view that is hardly justified.25 Kalinowski’s

24Wansing [219] therefore calls possible worlds semantics “inappropriate as an informalsemantical foundation of deontic logic”.

25Goble [65] pp. 193/4 points out a related problem: suppose Smith robs a bank, oughthe to be punished? what if, unpunished, he would go on “to perform wonderful works ofphilanthropy with his loot”, making the world a much better place? Goble argues thatwe ought not to consider best logically possible worlds, nor even only actually availablealternatives, but only those worlds that result through minimal change from the actualworld. I think things are much simpler if we have a way to represent norms in deontic logic:Smith ought to be punished because there is a norm that says he ought to be punished.

DEONTIC LOGIC 68

subsequent advice not to evaluate the truth of deontic propositions in termsof the good, better or ideal, but to return to reasoning with existing norms,echoes a description of ordinary deontic reasoning by Ross [189] p. 17:

“When a plain man fulfils a promise because he thinks he oughtto do so, it seems clear that he does so with no thought of its totalconsequences, still less with any opinion that these are likely tobe the best possible. He thinks in fact much more of the pastthan of the future. What makes him think it right to act in acertain way is the fact that he has promised to do so – that and,usually, nothing more. That his act will produce the best possibleconsequences is not his reason for calling it right.”

Some of the criticism of deontic logic’s possible worlds semantics seemstoo harsh – e.g. when we aim to prove mathematical results about an ax-iomatically specified logic, its meaning is just that of a formal tool. Still, itseems that possible worlds semantics has, at least for a number of authors,lost its power of also mirroring and guiding our intuitions. And Ross’s viewthat ordinary deontic reasoning starts with norms or laws, and not with pro-hairetic notions like ‘ideal’ or ‘best’, that these notions are secondary at best,is even apparent in an author who expressed that

“deontic logic, to put it in a nutshell, is the study of logical rela-tions in deontically perfect worlds”

insofar as the same author – who in [252] turned away from possible worldssemantics altogether, it is G. H. von Wright [254] – explains that ideal worldsare what is ‘envisaged’ by a system of norms, and that “any normative codemay be said to contain an implicit description of a state of affairs, viz. theideal state envisaged by the norm” (ibid.). In particular, von Wright refersto the idea of a Kantian ‘Kingdom of Ends’, used by Hintikka [95] p. 73 tomotivate the idea of a deontic alternative:

“The ‘Kingdom of Ends’ is the world such as it would if all andsundry rational beings always honored all their obligations (du-ties). In this respect, a Kantian ‘Kingdom of Ends’ is like adeontic alternative to the actual world.”

Not because it makes the world a better place.

DEONTIC LOGIC 69

But this seems like a bad defense of possible worlds semantics. If the motiva-tion for deontic possible worlds semantics starts with given norms or duties,and defines the ideal worlds as the ones where all is as is envisaged by thesenorms, then why not appeal to these reasons directly and try, as Kalinowskiadvises, to reconstruct deontic logic on the basis of norms?

2.6.2 The Question of Conflicts and Dilemmas

A second, more practical problem is that existing norms may conflict, or, inunforeseen circumstances, result in dilemmas. For moral norms the questionof the existence of true conflicts has been the subject of intense philosophicaldebate since E. J. Lemmon [132] started it; in positive law they are, accord-ing to Alchourron [2] and Alchourron & Bulygin [6], ‘frequently found’ and‘no rarity’, while computer scientists seem to have come to view conflictsbetween specifications as inevitable. But standard deontic logic excludes thepossibility that OA and O¬A are both true by its axiom (D), which is equiv-alent to ¬(OA ∧ O¬A). Possible worlds semantics makes it impossible thatall deontic alternatives make true A and also ¬A – unless there are no deonticalternatives, which is excluded by the seriality restraint that corresponds toaxiom (D). But to drop seriality seems not a proper way to model conflicts:in a conflict, there are not too little, but rather too many ‘ideal’ ways to act.Similarly, for the axiomatic approach, simply removing (D) from the set ofaxioms will not be sufficient: unless one accepts that a case of conflict makesanything obligatory, one would also have to remove or replace (M) or (C),which make it possible to derive from OA and O¬A that OB for any B (socalled ‘deontic explosion’).

For dyadic deontic logic there is not only the problem of conflicts, but alsothat of dilemmas: for any two apparently harmless statements of obligationOA and OB, the situation ¬(A∧B) creates a dilemma, i.e. a situation wheresomething that is obligatory can no longer be realized (unless there is therather artificial assumption that either A implies B or B implies A). Just likestandard deontic logic cannot accommodate conflicts, the traditional dyadicdeontic systems DDLH or DDLL do not accept dilemmas: given that ¬(A∧B)is consistent, O(A/¬(A∧B)) derives ¬O(¬A/¬(A∧B)) with (DD) or (DD-R), whereas with (CExt) we derive O(¬A/¬(A ∧B)) from O(B/¬(A ∧B)).In dyadic deontic semantics, to model any two sentences OA and OB, the‘system of spheres’ must specify which state of affairs A or B is to be preferredin the section ||¬(A∧B)|| of the spheres, though nothing might be specified

DEONTIC LOGIC 70

about such a situation by the sentences that one aims to model – considerthe above modeling of Chisholm’s dilemma: we assumed that not going andnot telling is better than not going and telling, though the original sentenceswere silent about this question.

An easy way out of the problem of conflicts and dilemmas is to say thatthe task of deontic logic is to model the moral or legal situation after it hasbeen resolved by appropriate mechanisms of conflict resolution. But we thenadmit that deontic logic is neither able to describe conflicts nor to analyzethe resolution mechanism:

“It is of no avail to say that logic pictures the situation that arisesafter the normative conflict has been resolved. For rather thatwhat is being resolved, the conflict of norms, is the interestingstate of affairs that must be logically characterized.” (Alchourron& Bulygin [6] p. 25)

Another way out is to give up the idea that the expressions of deontic logicsomehow relate to ‘real’ sets of norms and rules, describe what ought to bethe case according to some system of norms (which can be inconsistent andthereby falsify axiom D), but that deontic logic only describes rules for ratio-nal norm giving: a rational law giver cannot enact norms which make bothOA and also O¬A true, because a rational law giver wants that all the normsare obeyed, but a subject could not obey all norms if according to the normsboth OA and O¬A are true. This is the route that von Wright has takenin his work since [252], and it seems also to be the one taken by Weinbergerwhen he writes that “normative systems which contain conflicting norms canexist in social reality, they are logically flawed, but do not loose their func-tion and social meaning” ([234] p. 25, my translation).26 But lawyers, whenthey explain what ought to be done, may be done or is forbidden to do ina certain situation, when e.g. a law has been violated or such a violationis imminent, do not advise the law giver. They advise their clients, whichare usually the subjects of the law. Thus a solution that views deontic logiconly as a tool for law givers, which they may, unlikely as that seems, use inthe activity of passing a law, would miss a large part of ordinary languagediscourses in which the expressions ‘you are obliged to’, ‘you may’, ‘you areforbidden to’ are used that deontic logic claims to formalize.

26Cf. Hollander [100] p. 98 (my translation): “By concluding that a norm-giver may bereasonable or unreasonable (...), Weinberger reaches the same conclusion as von Wright.”

DEONTIC LOGIC 71

The hard way is to accept the possibility of conflicts and make changes toaccommodate them and explain resolution mechanisms. Approaches to ac-commodate conflicts in traditional deontic logic reach from replacing (D) by(P) ¬O⊥ and dropping (C), and in the semantics use minimal models (neigh-borhood semantics) to pick alternative sets of deontic alternatives instead ofone (‘minimal deontic logic’),27 or to use multi-ideality or multipreferencesemantics that has not one deontic alternative or preference relation but sev-eral, and thus can make OA and O¬A true in a model that makes A truein all ideal or best alternatives according to one relation, and ¬A is true inthe ideal or best alternatives according to another,28 to sequent systems thatreplace (C) by a rule of ‘consistent cumulativity’, which tests consistencyof A ∧ B before cumulating OA and OB to O(A ∧ B) and which can onlybe used before a use of (M), and so avoids the derivation of OB from OAand O¬A,29 to axiomatic systems that leave (C) intact but exchange (M)for a rule of ‘permitted monotony’ for the same reason.30 So there are waysin which deontic logic can be modified to accommodate conflicts, and evenkeep possible worlds semantics, though this semantics may not always bevery intuitive.

As for mechanisms to resolve conflicts, a natural way to do so is byweighing the conflicting obligations and giving preference to the ones withthe highest priority.31 Though it seems not to have been done before, thistoo might be accomplished by using preference semantics and in particularGoble’s [67] multiplex preference semantics, through introducing a preferencerelation over the multiple ideality or preference relations (OA then holds ‘allthings considered’ if A is true in the most ideal of the ideal worlds). However,the severe changes that would be required and the multitude of proposals howto go forward to solve the problem of accommodating conflicts and dilemmasshows that there are no easy solutions along the old ways of thinking aboutdeontic logic. In fact, the difficulties of traditional deontic logic to handlesuch problems have led to doubts about the whole enterprise:

27Cf. Chellas [45] pp. 200–203, 272–277 for its monadic and dyadic variants, also cf.Nortmann [159] who likewise employs minimal models but instead of (C) drops (M).

28Cf. Schotch & Jennings [192], and Goble [66] [67] for such solutions.29Cf. van der Torre & Tan [214] for such a labelled sequent system.30Cf. Goble [69]. In [68] Goble also presents a possible worlds semantics that is sound

and complete with respect to such axiomatic systems.31Cf. Isaac Watts [220] part II, ch. V, sec. III, principle 10: “Where two duties seem

to stand in opposition to each other, and we cannot practise both, the less must give wayto the greater, and the omission of the less is not sinful.”

DEONTIC LOGIC 72

“Various notions from normative reasoning, including the notionsof prima facie obligation, contrary-to-duty obligations, and moraldilemmas have raised serious problems for deontic logicians, prob-lems which have defied solution within standard deontic logic andits extensions. As a result, deontic logic has fallen into ill-reputewith those who must actually engage in normative reasoning.”(Nute and Yu [161] p. 1)

I think that these problems, too, call for a fresh approach to deonticlogic. It seems that a more natural treatment of conflicts, dilemmas andtheir resolution can be achieved by directly specifying the norms that con-flict. Resolution mechanisms based on priorities or rankings can then usean ordering of the norms themselves. Also, there are other problems thatare similarly hard to solve with traditional deontic logic and its possibleworlds semantics. One is the distinction between ‘simple morality’ and thesupererogatory. When norms are explicitly represented, we can distinguishbetween basic rules, binding on all, and the “higher flights of morality ofwhich saintliness and heroism are outstanding examples” (Urmson [215] p.211), where the additional satisfaction of the second type of norms will makean act supererogatory. We can also introduce a count on the number ofobligations satisfied, and say that up from a certain count, an act that sat-isfies the obligations is no longer ordinarily obligatory, but a supererogatoryact. By contrast, possible worlds semantics would require us to constructseries of ideal and ‘more ideal’ worlds to model this notion, a constructionthat seems as superficial as talk about ‘preferences over preferences’, or ‘idealideal worlds’, that might be required to resolve conflicts and dilemmas in apossible worlds setting.

2.6.3 Interpretations of Deontic Logic

The expressions that deontic logic formalizes, like ‘it is obligatory that’, ‘itis permitted that’ or ‘it is forbidden that’, are ambiguous in everyday usage:they may be used to describe or report that something is obligatory accordingto some implicitly referred-to moral or legal code, but also to make somethingobligatory, permitted or forbidden. E.g. when a school teacher says “it isforbidden for you to talk in class during the exam,” this may be becausethere is such a general rule at the school, but it may also be so “because I,

DEONTIC LOGIC 73

the teacher, say so”. The ambiguity of such expressions is well-known.32

To distinguish their descriptive from their normative use, the descriptivelyused expressions have been called normative, or deontic, propositions, asopposed to norms. When devising the first axiom system of deontic logic,von Wright [242] makes quite clear that it is the descriptive use of deonticexpressions which he has in mind:

“The system of Deontic Logic, which we are outlining in thispaper, studies propositions (and truth-functions of propositions),about the obligatory, permitted, forbidden (...). We call the pro-positions which are the object of study deontic propositions.”

Deontic logic, in this first and strict sense, is thus not a logic of norms,but a logic of propositions about norms. According to this descriptive inter-pretation of deontic logic, “deontic sentences (represented by the formulae ofdeontic logic) describe what is regarded as permitted, obligatory, forbiddenetc., in some unspecified normative system” (Føllesdal & Hilpinen [53]). Ifa sentence ‘it is obligatory that ...’ is true, then there must exist normswith respect to which it is true, and so in this ‘descriptive interpretation’ thedeontic propositions can also be seen as statements about the existence ofnorms: norm propositions are “propositions to the effect that certain normsexist” (von Wright [251]). In one formulation or another, this descriptiveinterpretation of deontic logic has come to represent the standard view.33

32Cf. Ayer [24] pp. 105/106, Reichenbach [180] pp. 19/20, von Kutschera [128] pp. 11–14, Hilpinen [92] pp. 299/300. Bentham was the first to succinctly point at this ambiguity:“The only mood to which grammarians have thought fit to give the name ‘imperative’ isby no means the only one which is calculated to perform the office of imperation. (...)But in many instances it will be found that the style of imperation is altogether dropped.”He proceeds to give a variety of forms of a law against the exportation of corn, amongthese “the legislator speaking as it were in the person of another man who is consideredas explaining the state which things are in, in consequence of the arrangements taken bythe legislator, ‘It is not permitted to any man to export corn.’ ‘It is unlawful for any manto export corn.’ ‘No man has a right to export corn. Why? because the legislator hasforbidden it.’ ‘The exportation of corn is forbidden.’ ”([29] pp. 104-106, 154)

33Also cf. Kalinowski [114] p. 118 (my translation): “For what does an expression ofthe form ‘it is obligatory that A’ mean? If this is not a norm but a proposition aboutnorms it means that ‘the law giver L has enacted the valid norm N according to which Ais obligatory’ ”; Opa lek & Wolenski [164] p. 384: “Deontic logic is conditioned by norms,because deontic statements are true or false depending on the existence of respective actsof norming.”; Sinowjew [197] p. 266 (my translation): “The truth values of normative

DEONTIC LOGIC 74

The interpretation that the propositions of deontic logic are true andfalse descriptions about what is obligatory, permitted or forbidden, and arenot themselves norms or imperatives, has not been shared by all authors.Castaneda [42] viewed deontic logic as a “modal logic of imperatives andresolutives”. Aqvist [21] argued that it should be possible to interpret de-ontic logic “atheoretically” as a logic of commands, in the sense that OAexpresses a command, and not a proposition about a command; occasionaloddities like the difficulty of interpreting formulas like ¬OA should be ac-cepted as a small price for a logical theory of commands. Such a ‘prescriptiveinterpretation’ of deontic logic was also adopted by Alchourron [2] when heidentified deontic logic with a “logic of norms”, and set out to develop a logicof normative propositions in parallel. Chellas [44] replaced the O-operatorsof deontic logic by the symbol ‘!’, where !A is to be understood as represent-ing a natural language imperative ‘let it be the case that A’, and presentsa logic for such formulas that is equivalent to SDL. For Bailhache [25], de-ontic logic is the logic of ‘deontic norms’ (norms that are created by the useof deontic expressions), where the deontic formulas are then evaluated withrespect to accessible ideal worlds as usual. More recently, Kamp [116] p.233 has demanded that the formulas of deontic logic must be interpretableboth descriptively and prescriptively. Sometimes it is not even quite clear inwhich way standard accounts of deontic logic desire its formulas to be read:Carmo & Jones [40] write that deontic logic is a formal tool needed to ‘de-sign normative systems’ and, just like Bailhache in [26], throughout call thesentences of deontic logic ‘norms’ or ‘deontic norms’ and so seem to adoptthe prescriptive interpretation, but the authors then employ possible worldssemantics to evaluate deontic propositions (norms?) as usual. Føllesdal &Hilpinen [53] speak of deontic propositions constituting or being implied bya normative system (e.g. pp. 13, 29), or of deontic logic formalizing imper-atives (p. 26), but portray the descriptive interpretation of deontic logic asone that they “shall often resort to” (p. 8). In the face of all this confu-sion,34 David Makinson [141] noted that work on deontic logic goes on as if a

propositions are determined according to the usual rules. E.g. a proposition ‘it is forbiddento smoke’ is true if it really is forbidden to smoke (if there exists such a norm), false if itis permitted to smoke, and undetermined if there is neither a prohibition nor a permissionto smoke”.

34Hollander [100] pp. 98–99 uses Hintikka’s concept of a deontically perfect world tomotivate his deontic logic as a logic of norms in the following way: the deontically perfectworld is a place where conflicting norms cannot exist, as opposed to the real world where

DEONTIC LOGIC 75

distinction between norms and normative propositions has never been heardof.

In part, the confusion about what the formulas of deontic logic mean canbe attributed to von Wright himself, who later wrote that while deontic logicis a theory of descriptively interpreted expressions, its laws concern logicalproperties of the norms themselves, and that therefore “the basis of deonticlogic is a logical theory of prescriptively interpreted O- and P -expressions”([246] p. 134). This reflection of logical properties of norms in the logicalproperties of deontic propositions have led von Wright to famously claim that‘logic has a wider reach than truth” ([245] p. vii). Still later, von Wright [251]reaffirmed that “deontic logic is a logic of norm-propositions” and denied thepossibility of a logic of norms, except in the form of advice to a rational lawgiver (cf. [252] p. 132, [256] p. 110).

However, it is quite clear that OA and PA cannot be interpreted as repre-senting imperatives or norms. I have explained in sec. 1.2 why imperatives,and for that reason, also ‘deontic norms’, cannot be meaningfully termed‘true’ and ‘false’. But if norms are neither true nor false, then the Booleanoperators occurring in the formulas of deontic logic such as ‘OA ∧ OB’,‘PA ∨ ¬OA’, ‘OA → PB’, cannot have their usual, truth functional mean-ings as ‘and’, ‘or’, ‘not’, ‘if ..., then’. So a logic of norms, even if one believesit exists, cannot resemble deontic logic,35 and the meaning of (sub-) formulassuch as ‘OA’ must be interpreted descriptively.36 So the descriptive interpre-tation of the formulas of deontic logic is the only tenable one, and is also theone used here.

What the confusion in deontic logic literature as to the meaning of itsformulas illustrates, is a need to better explain the relation between theformulas of deontic logic, as logic of deontic propositions, and the normsthat these propositions are statements about. Von Wright hints at this whenwriting in 1951 [243]:

they can. The logical relations studied are then those that exist between norms in thedeontically perfect world. But deontic alternatives, as described by Hintikka [97], [95], aresets of propositions, not places where norms exist. If deontic propositions are true in aworld that is ideal to ours (it is not just one, as Hollander seems to assume), this is dueto other worlds that are ideal to the ideal world. All the previously mentioned problemsand confusions about deontic logic seem to eclipse here.

35Cf. Makinson [141] p. 30. Keuth [123] and Swirydowicz [208] therefore restrict their‘logic of norms’ to statements of ‘normative entailment’ of the form !A `!B.

36That is, unless one is prepared to leave the common ground and call norms ‘true’ and‘false’, as Kalinowski [111], [114] and Kamp [118], [116] have done.

DEONTIC LOGIC 76

“In this essay the deontic modalities are treated as ‘absolute’. (...)Instead of dealing with propositions of the type ‘A is permitted’,we might consider propositions of the types ‘A is permitted ac-cording to the moral code C’ (...) Extensions of the system Pwill not, however, be studied in this essay.”37

Traditional literature on deontic logic has mostly ignored this relation. Ifmentioned at all, the deontic sentences are called true or false according tosome “unspecified normative system” (cf. Føllesdal & Hilpinen [53] p. 8).But if norms are not somehow explicitly represented, it becomes difficult todiscuss examples. It has e.g. been suggested that if both O(A ∧ B) andO(¬B∧C) true, i.e. there is a conflict, then a cumulation of contents shouldnot be allowed, for O(A∧B∧¬B∧C) would declare something impossible asobligatory, but that instead a disjunction of the contents should be obligatory,i.e. O((A∧B)∨ (¬B ∧C)) should be true. Thus the agent would be obligedto satisfy at least one norm (cf. Jacquette [109]). But the sentences O(A∧B)and O(¬B ∧C) need not be true because two norms demand exactly A ∧Band ¬B∧C. They might be true because two norms demand A∧B∧¬C and¬A∧¬B∧C – but thenO((A∧B)∨(¬B∧C)) would only express an obligationto violate at least one norm. Similarly, in order to cope with conflicts, it hasbeen suggested to exchange (C) for a form of ‘consistent aggregation’: it isallowed to cumulate OA and OB to O(A∧B) if A∧B is consistent (cf. Goble[68] p. 81). But OA and OB might be true because there are two norms thatdemand A ∧ C and B ∧ ¬C. Then O(A ∧ B) is demanding too much: oneof the norms must be violated anyway, so either doing A or doing B mustachieve nothing. It is no wonder that in such discussions it is tempting toslip into a manner of speaking as if OA expresses a ‘deontic norm’, which isviolated by ¬A (e.g. Carmo & Jones [40] p. 317, 321,). But OA is not anorm, it is a sentence that cannot be violated or fulfilled, but just be trueor false; it is, at best, a premiss. Ziemba [258] p. 155 is therefore rightwhen he claims that “the lack of distinction between commands (norms) anddeontic propositions, or propositions on commands, is a source of various evilin deontic logic.”

Taking our cue from von Wright’s suggestion, I think that explainingthe meaning of the formulas of deontic logic by means of not just basically

37Much later, Castaneda [41] indexed deontic operators by adverbial qualifiers to indi-cate according to which set of norms the truth of OiA holds, where i means e.g. a previouspromise or agreement, the rules of a job, or morality.

DEONTIC LOGIC 77

unknown, but an explicitly represented sets of norms will clarify such discus-sions and the understanding of deontic logic.

2.7 A Fundamental Problem of Deontic Logic

In the face of the problems and doubts outlined above, and the multitude ofproposals to modify deontic logic, it has been concluded that

“deontic logic has fallen into ruts. The older rut is the axiomaticapproach, with its succession of propositional and occasionallyquantified calculi. The newer one is possible worlds semantics,with endless minor variations in the details.” (Makinson [141])

In consequence, a fresh start has been called for. A representation of normsin deontic logic could constitute such a fresh start. However, if norms are tobe represented in deontic logic, there is then the question of how to do sowithout falling into the trap of Jørgensen’s Dilemma [110]: how can normsbe a subject of logical reasoning if norms are neither true nor false and socannot be part of any logical inference? This has been called the ‘fundamentalproblem’ of deontic logic:

“It is thus a central problem – we would say, a fundamental prob-lem – of deontic logic to reconstruct it in accord with the philo-sophical position that norms are devoid of truth values. In otherwords: to explain how deontic logic is possible on a positivisticphilosophy of norms.” (Makinson [141])

Jørgensen’s dilemma cannot be dealt with, and the fundamental problemnot solved, by making the object language of deontic logic one of norms.As was explained above: we cannot negate norms, form conjuncts or ad-juncts out of expressions representing norms, for what meaning would theBoolean operators then have? We must let the statements of deontic logicremain true or false statements about norms, deontic or normative proposi-tions that describe what is obligatory, permitted or forbidden according tothe norms. However, the norms according to which the normative proposi-tion may be true or false should be explicitly represented in the semanticsof deontic logic. Deontic logic would then, as it once aspired, not talk aboutideal states, preferred worlds or perhaps preferences over preferences, but

DEONTIC LOGIC 78

more directly about positive law, not to forget the more profane usages ofnormative language like rules set up by mums and dads, employers or schoolteachers, with all their curiosities and inconsistencies.

If the semantics by which we evaluate the truth of deontic formulas ischanged, there are then the questions of what truth definitions should beused for the deontic operators, and in particular what a thus semanticallyreinterpreted deontic logic will be like: will the traditional systems of de-ontic logic still be sound and complete, or in Makinson’s terms: can theybe ‘reconstructed’ with respect to such a new semantics? Or will we arriveat completely new axiomatic systems? The next chapter describes how theformulas of deontic logic can be evaluated with respect to a logical semanticsthat explicitly represents them (I call this ‘imperative semantics), and whatsystems result.

Chapter 3

Imperative Semantics forDeontic Logic

3.1 The Imperatival Tradition

The idea to represent norms or imperatives in the logical semantics for deonticlogic is not entirely new. While the mainstream of deontic logic adhered topossible worlds semantics, quite a number of authors have tried, in one wayor another, to link deontic logic to norms or to imperatives – we mightcall this the ‘normative’ or ‘imperatival tradition’ of deontic logic. What istypical for such approaches is that they have, besides the truth definitions fordeontic operators, a set of objects that are meant to be prescriptions, or thatcontains the propositions that relate to prescriptions in the sense explainedby Weinberger’s Principle (W): the proposition that must be true when theimperative is satisfied, and false if it is violated. The deontic operatorsare then in turn defined with respect to this set, and not a possible worldssemantics. Any proposal that has these features is counted as belonging tothis ‘imperatival tradition’ of deontic logic.

3.1.1 Anderson’s Reduction Revisited

As explained in sec. 2.3, Alan Ross Anderson was the first to propose toreduce deontic logic to alethic modal logic, and define OA by means of aconstant S representing a ‘sanction’:

OA =def 2(¬A→ S)

79

IMPERATIVE SEMANTICS 80

So A is obligatory iff ¬A necessitates the ‘sanction’ S. However, it was soonpointed out that the relation between a wrongdoing and its sanction is notone of necessity, since “in this wicked world, it is not always true that when(say) a robbery is committed someone suffers a sanction” (Nowell-Smith &Lemmon [160]) and Castaneda [42] adds: “Failure to do one’s duty neverentails or strictly implies that punishment will be inflicted. It is commonplacethat many criminals die unprosecuted.” Hence Anderson is accused of thenaturalistic fallacy, concluding from the moral or legal fact that somethingis obligatory the empirical fact that something else (an act of sanctioning)happens with necessity in this world.

Interestingly, in answering these accusations, Anderson did not, as Nowell-Smith & Lemmon suggested, re-interpret his necessity operator ‘2’ in termsof a ‘moral necessity’. Rather, Anderson [17] argues that it is analytic of thenotion of obligation that if an obligation is not fulfilled, then something hasgone wrong.1 S is no longer interpreted as representing a penalty or sanction,as in Anderson [16], but as the statement that “some wrong has been done”(cf. Anderson [13]), that not “all obligations have been fulfilled”, or that“some set of rules is violated”, and consequently the letter ‘S’ is exchangedfor the letter ‘V ’ for ‘violation’. Similarly, von Wright argues in defense ofAnderson that his construction avoids the naturalistic fallacy if the sanctionis interpreted as ‘punishability’, where punishability does not obtain if andonly if all obligations are fulfilled (von Wright [250] pp. 36-37). Anderson’sdefinition can therefore be rephrased as follows:

OA is true iff ¬A necessarily implies V , where V means that somerules are violated or that not all obligations have been fulfilled.

To further refine this definition, we obviously need an explicit set of obli-gating norms, with respect to which the terms ‘violation’ or ‘fulfilment’ canthen be defined. In [75] I have employed a set of imperatives to define con-stants ‘v’ (some imperative is violated) and ‘S’ (all imperatives are satisfied),together with their duals ‘V’ (all imperatives are violated) and ‘s’ (some im-perative is satisfied) in order to reconstruct Anderson’s defense and thus hisdeontic truth definition. The standard system of deontic logic SDL and vari-ants of it can then easily be shown to correspond to such a reconstruction ofAnderson’s reduction of deontic to alethic modal logic.

1Also cf. Goble [64] p. 199: “There is an analytical connection between the statements(i) it is obligatory that p and (ii) if not-p, then V, where the constant ‘V’ is to representsome violation’ or ‘bad state of affairs’ or ‘wrong in the world’.”


3.1.2 Kanger’s Welfare Programs

As explained above (sec. 2.3), Stig Kanger, independently from Anderson,also had the idea of reducing deontic logic to alethic modal logic, but insteadof an Andersonian ‘sanction’ employed a constant Q for his definition of adeontic O-operator, which reads

OA =def 2(Q→ A).

So OA is true iff Q necessitates A. Kanger gives the informal meaning of hisconstant Q as “stating what morality prescribes”. While this may seem anelusive concept, Kanger explains it as follows:

“Consider the universe of discourse, and suppose there are wel-fare programs for this universe. A welfare program consists of aset of propositions expressing what is desirable for this universefrom the viewpoint of human welfare. (...) Now the propositionOughtA is true in the universe of discourse if and only if A isentailed by each (...) welfare program for this universe.”

Refining this idea, let P = {P1, ..., Pn} be a set of sets of propositions, whereeach P ⊆ LPL represents some ‘welfare program for the universe’. Then thefollowing truth definition for a deontic O-operator corresponds to Kanger’sexplanation:

OA is true iff ∀P ∈ P : P |=PL A.

In a proposal endorsed by Barcan Marcus [27], T. J. Smiley [198] elaboratedon Kanger’s ideas and explained that in Kanger’s definition, Q should beseen as expressing the content of an arbitrary moral code, in which casethe definition is “tantamount to saying that something is obligatory if it isentailed by the code, permitted if it is consistent with it, and forbidden ifit conflicts with it”. Smiley identifies Kanger’s constant Q with a code of‘obligatory propositions’, i.e. Q ⊆ LPL and defines

OA is true iff there are propositions p1, ..., pn in the code suchthat p1, ..., pn necessarily imply A.

Smiley then explains that this definition yields von Wright’s classical system,given that there is an additional consistency assumption that excludes thatthere are obligatory propositions p1, ..., pn such that ¬(p1 ∧ ... ∧ pn) is atautology. This seems to be the first reconstruction of a deontic axiomatic


system by means of a semantics that explicitly represents norms. I examineSmiley’s definition in more detail in [75], [76]. Regarding the truth definitioncorresponding to Kanger’s explanation, it seems not to have been exploredanywhere so far, but it can easily be seen as corresponding to a normalmodal logic, and to SDL iff each welfare program P ∈ P is assumed to beconsistent.2

3.1.3 Ziemba’s Deontic Syllogistics

Zdzis law Ziemba [258], [259] defines a ‘deontic syllogistics’ by use of a set Cof conditional commands. A command ‘let every A be B!’ is formalized by∀(A!B), where this is taken to be the ‘name’ of the command. A describes aproperty of a certain person over a certain period of time, e.g. the propertyof being in a store with inflammable materials. If the person has the propertyA, she is addressed by the command ∀(A!B). B is then another propertyof this person, e.g. the property of this person’s not smoking over the sameperiod of time. So ∀(A!B) is the command that forbids this person to smokewhile being in the store. A person is fulfilling a set of commands C if for allcommands ∀(A!B) of the set that address her, she is B.

Deontic statements are then defined true and false with respect to thisgiven set of commands C . Thus the statement x prohCA, that being A isforbidden for x according to C , is defined true iff x is addressed by somecommand in C and it is not possible for x to be fulfilling all commands in Cand be A. Being obliged to be A is then defined as being prohibited to be¬A, and being permitted as not being prohibited to be A.

Ziembas main idea is that to some person, by virtue of some set of com-mands, a certain conduct is forbidden when this person is an addressee ofa command belonging to this set, and that if the person fulfills the con-duct, she inevitably disobeys some command of the set of which she is anaddressee. Ziemba [259] p. 382 is thus “distinguishing a command froma deontic proposition stating that this particular person has an obligation,that he is forbidden or permitted to do so and so with respect to such andsuch a set of commands. These commands are not logical propositions, andthey are neither true nor false.” Though I will not use the syllogistics thatZiemba proposes, he explains the meaning of his deontic statements solely

2Regarding preference semantics, the proposal not to use one, but several sets of normscorresponds to Goble’s [66], [67] multipreference semantics that uses not just one, butmultiple preference relations to explain OA in terms in what is best according to all.


by use of an explicitly given set of commands, and so his proposal belongsto the imperatival tradition of deontic logic.

3.1.4 Stenius’ Logic of Normative Systems

Stenius [205] distinguishes a norm sentence (a sentence used to express anorm) from a factual interpretation that something is obligatory accordingto some (possibly unspecified) system of norms S. The first is expressed by‘Op’, the second by ‘Op ∈ S’ (read as ‘p is obligatory in S’).

Stenius then proposes several principles for a logic of normative systemsS. The two basic principles are:3

If (p1 ∧ ... ∧ pn)→ q is a tautology,then ((Op1 ∈ S) ∧ ... ∧ (Opn ∈ S))→ (Oq ∈ S) is a theorem.

and

If p is a tautology then Op ∈ S is a tautology.

If we use Op to abbreviate Op ∈ S for some fixed system of norms S, andPp forO¬p /∈ S, thenO behaves like a normal modal operator, i.e. the logic ofthis operator is described by the axiom scheme (K) O(p→ q)→ (Op→ Oq)and is closed under the rules (N) ‘if p is a tautology then Op is a theorem’ andmodus ponens. Other principles are rejected by Stenius, and in particular the‘deontic’ axiom scheme (D) Op → Pp is not viewed as valid, since althougha system of norms according to which Op ∈ S and O¬p ∈ S are true and soit is impossible to obey might be called ‘unreasonable’, to “say that such asystem exists is not a logical contradiction”.

While Stenius interprets the ‘factual’ or ‘deontic statements’ Op (or Op ∈S) as statements about a normative system, the logic of these deontic state-ments only mirrors what statements of the form ‘Op’ are already containedin S. The principles he lays out are primarily principles of closure for S, notprinciples of a deontic logic. It is therefore crucial for his proposal that e.g.two norms Op and Oq that already are in S can be seen as ‘deriving’ anothernorm O(p ∧ q), i.e. a norm that must – by the first principle – then also bein S. Thus his proposal hinges on the question whether there really are such

3Note that p and q are here proposition variables, i.e. sentences from LPL, rather thanproposition letters. In this section I try to keep to the original texts as closely as possible.


logical relations between norms.4 In this respect his proposal and relatedproposals5 differ from other proposals discussed in this section, and also theproposal explained in the following sections: I do not assume any theory oflogical relations between norms or imperatives, for the simple fact that it isdoubtful and heavily disputed whether such relations exist, and that I agreewith a position of ‘imperativological scepticism’ that, at least for imperatives,altogether denies them. If, contrary to this view, such relations do exist, thennothing is lost: we can close the set of norms or imperatives by any acceptedprinciple or rule and still use the formulas of deontic logic to describe whatis obligatory according to this set, just as Stenius proposes. However, if suchlogical relations do not exist, then the interpretation of deontic logic as one ofexpressions about sets of norms or imperatives will still remain meaningful.

3.1.5 Alchourron & Bulygin’s Expressive Conceptionof Norms

In Carlos E. Alchourron & Eugenio Bulygin’s [5], [8] ‘expressive conceptionof norms’, norms are taken to be sentences used to prescribe that some-thing ought, ought not or may be (done). The existence of such sentencesis dependent on certain empirical facts, and as there are no logical relationsbetween facts, there is no room for a logic of norms. However, as Alchourron& Bulygin point out, this does not preclude the possibility of a deontic logicas a logic of normative propositions. For the propositions that have beencommanded by some authority (called ‘Rex’ by Alchourron & Bulygin) canbe used to form a set, the ‘commanded set’ A. The following first definitionfor what is obligatory according to this ‘commanded set’ is then proposed:

p is obligatory in A is true if p has been commanded by Rex andso is a member of the commanded set A.

Alchourron & Bulygin then proceed to expand the set A and define theconcept of a ‘normative system’ as the set of all propositions that are conse-quences of the commanded propositions, i.e. the system corresponding to A

4For this reason, Stenius [206] later retracted his proposal and only called a system ofnorms ‘well-formed’ if it is not closed under consequences.

5In a very similar way to Stenius, Keuth [123] and Swirydowicz [208], [209] have defineddeontic operators via a normative system closed under rules, where Swirydowicz usesZiemba’s [258], [259] dyadic formalization of imperatives which he enriches with logicalrelations between the imperatives.


is Cn(A) (where Cn means classical consequence). Their second, correcteddefinition for the truth of normative propositions is then the following:

‘It is obligatory that p in A’ is true if and only if p is a mem-ber of the system Cn(A) – i.e. if and only if p belongs to theconsequences of A.

This second definition is similar to one employed by Niiniluoto [157] p. 113:

Op is true if and only if some norm-authority x has commanded!p or p is a logical consequence of some sentences q1, ..., qn suchthat x has commanded !q1, ..., !qn.

In [76] I consider an O-operator O1 that is similar to Alchourron & Bulygin’sfirst proposal, in that O1A is true if and only if it is explicitly commandedto bring about A. For Alchourron & Bulygin’s second definition and thedefinition used by Niiniluoto it is easy to see that both correspond to anormal modal system, and that they correspond to standard deontic logicSDL if it is additionally assumed that the norms are consistent, i.e. O⊥ isnot true. SDL is thus easily reconstructed with respect to a semantics thatmodels norms.

3.1.6 Van Fraassen’s Proposal

Conflicts of obligation are difficult to accommodate in standard deontic logic,the equivalent formula ¬(OA∧O¬A) of the characteristic axiom (D) excludesthem. Therefore Bas van Fraassen [57] introduced a semantics that definesthe truth of deontic operators with respect to explicitly given, possibly con-flicting imperatives. The basis of his semantics is a set of imperatives in forceI = {i1, ..., in}. To every imperative there is a class i+ ⊆ B of valuationsv (van Fraassen calls these ‘possible outcomes’) such that i is fulfilled in v.His first proposal for the definition of deontic O-operator is then:

OA is true iff, for some imperative i that is in force, i+ is part ofthe set of possible outcomes in which A is true.

So something is obligatory if it is necessary for one imperative to be fulfilled.Van Fraassen’s definition fails to include that as obligatory what is de-

manded not by one, but several (consistent) imperatives collectively. To doso, he proposes the following, second definition of a deontic O-operator:

OA is true iff there is a possible state of affairs v in ‖A‖ whosescore is not included in the score of any v′ in ‖¬A‖.


Here, the ‘score’ of an outcome v is defined as the subset of the imperativesthat are fulfilled at v. So it is obligatory to do A if doing A makes it possibleto satisfy a set of imperatives that cannot all be satisfied by doing ¬A. Thiscan also be expressed in another way: A is obligatory if A is necessary tosatisfy some maximal non-conflicting subset of the imperatives.

With respect to van Fraassen’s idea, John F. Horty [101] proved that thesame definition can be provided via an application of Reiter’s [181] defaultlogic. Here, a default is a triple A : B/C such that A is the antecedentor ‘prerequisite’ that must be established to draw the conclusion C, andB, the ‘justification’, is a formula that a conclusion set must be consistentwith in order for the conclusion to be drawn (we might also think of ¬B asexceptional circumstances in which drawing the conclusion C is not permittedeven if A is present). A default theory ∆ = 〈W,D〉 consists of a set of offormulas W (the facts) and a set of defaults D. The set of ∆-conclusions ofan arbitrary set of formulas Γ (we write Concl∆(Γ)) is the smallest set suchthat (i) W ⊆ Concl∆(Γ) (the facts are conclusions), (ii) Cn(Concl∆(Γ)) =Concl∆(Γ) (the set is logically closed), and (iii) for each default A : B/C ∈ ∆,if the prerequisite A ∈ Concl∆(Γ) and the negation of the justification ¬B /∈Γ, then the conclusion C ∈ Concl∆(Γ). The ‘appropriate conclusion sets’ (or‘extensions’) of the default theory ∆ are then all sets of formulas E such thatConcl∆(E) = E (the fixed points of the conclusion operator). Let I be abasic imperative structure as defined above, and define the associated defaulttheory ∆ = 〈W,D〉 to be such that W = ∅ and D = {> : A/A | !A ∈ I}.Horty shows that van Fraassen’s second definition is equivalent to defining

OA is true iff A ∈ E for some extension E of the default theory∆ associated with I

In [102], Horty then presents a variation of van Fraassen’s definition thatmakes OA relative not to some, but all extensions of the associated defaulttheory:

OA is true iff A ∈ E for all extensions E of the default theory∆ associated with I

This definition can then be rephrased as follows: A is obligatory if it is neces-sary to satisfy any maximal non-conflicting subset of the imperatives. Bothdefinitions provide an interesting way to continue reasoning about obliga-tions in situations when there are conflicts or dilemmas, a way that standarddeontic logic does not support. In [76] and [77] I have explored the definitionsin more detail.


3.1.7 Input/Output Logic

In answering his own call [141] for a reconstruction of deontic logic in accordwith the position that norms are neither true nor false, David Makinson,together with Leon van der Torre [142], [143], develops input/output logicas a tool to “extract the essential mathematical structure behind this recon-struction” (cf. [144]). In their view, input/output logic should be seen notas a new kind of logic, but as a way of handling norms: given a set of norms,it must e.g. be determined which of these are still operative in a situationthat already violates (or satisfies) some of them. The set of norms G is takenas a ‘black box’ or transformation device, and the task of logic is then thatof a ‘secretarial assistant’, preparing the factual information that goes intothis device (the ‘input’ A) and coordinating the production of the ‘output’,written as out(G,A).

The simplest input/output operation that Makinson & van der Torresuggest is that of ‘simple-minded output’. Let G be a set of conditionalnorms (a, x), where a and x are sentences in the language of propositionallogic LPL, a describing some condition or situation, while x represents whatthe norm tells us to be obligatory in that situation. A set of sentencesA ⊆ LPL serves as explicit input. The operation first expands A to itsclassical closure Cn(A). This is passed into the transformer G which deliversas immediate output the set G(Cn(A)) = {x | (a, x) ∈ G and a ∈ Cn(A)}.Simple-minded output out1(G,A) is then again the classical closure of thisset, i.e. out1(G,A) = Cn(out1(G,A)). This operation is depicted by thefollowing figure from [144]:

Figure 3.1: Simple-Minded Output

So simple-minded output provides the propositions that must be true if the


conditional norms that are ‘triggered’ by the input, i.e. whose conditionsare true given the input, are to be satisfied. Makinson & van der Torrethen refine this simple-minded output and provide further output operationsthat e.g. provide reasoning by cases (if there are two norms (a, x) and (b, x)in G, then x is in out(G, {a ∨ b})), normative detachment (if there are twonorms (a, x) and (x, y) in G then y is in out(G, {a})), or permit ‘throughput’(if there is a norm (a, x ∨ y), then y is in out(G, {a,¬x})). Finally, theoperation may employ restraints to e.g. ensure that only such a (maximalsub-) set of norms is used that provides only consistent output (output suchthat ⊥ /∈ out(G,A)).

In the above outline, the ‘output’ is a set of descriptive sentences thatmight be seen as describing what is obligatory given the norms in G and thesituation described by the sentences in A (deontic interpretation). Hence wemay use an input/output operation out(G,A) to define a deontic logic via

G,A |= Ox iff x ∈ out(G,A)

i.e. Ox is true given the set of norms G and the factual descriptions A iff xis in the output out(G,A) of the used output operation. But the operationmight also be viewed as providing new norms conditional on the situationdescribed by a singular sentence a as input, where the new set of normsmay be defined as out(G) = {(a, x) | x ∈ out(G, {a})}, i.e. the operationprovides the norm (a, x) just if x is in the output ∈ out(G, {a}) (normativeinterpretation).6 In the first interpretation the head depicted in the abovefigure acts as a kind of lawyer or legal advisor, telling the agent what oughtto be done according to the norms in the situation described by A. In thesecond interpretation the head acts as a kind of judge, or subsumtion machine(Subsumtionsautomat), handing out norms for specific situations as a resultof the (general) laws known to the judge, a picture that is remindful of theLatin proverbs iura novit curia and da mihi facta, dabo tibi ius (the courtknows the law, give me the facts and I will give you the law). Makinson & vander Torre leave it open how the output should be interpreted, in particular nodeontic operators are defined, but if the aim is to reconstruct deontic logic,then obviously the first interpretation should be preferred. It is easy to seethat then e.g.

6Cf. Makinson & van der Torre [142] p. 385. Swirydowicz [208] defines a ‘normativeconsequence relation’ very similar to Makinson & van der Torre’s ‘basic output’ to providea method to derive norms from norms.


(Conj) OA ∧OB ↔ O(A ∧B)(N) O(A ∨ ¬A)

or statements such as ‘if G,A |= Ox then G,A ∪ {a} |= Ox’ (strengtheningof the input) all hold for the simple-minded output operation out1. Themain question will then be which kind of output operation provides the mostadequate results for a correspondingly defined deontic O-operator. In [80] Ihave considered truth definitions of dyadic deontic operators that resemblethe operations of input/output logic, and discussed some of their properties.I will to return to the results of this work shortly, but will now explain myown approach to model norms in a semantics for deontic logic.

3.2 Imperative Semantics for Deontic Logic

If deontic propositions are to be evaluated not with respect to possible worlds,but to given sets of norms, then these must be somehow represented in thelogical semantics of deontic logic. There are many different types of normslike individual and general, norms that empower, grant licenses, or defineterms within other norms, act as basic principles, direct procedures, rule thecreation of norms, etc., and though it should be possible to finally arriveat a logical semantics that represents all of them, it is better to start frommost basic terms. These seem to be simple, obligating norms that requirean individual to perform some action or achieve some state of affairs, thesort of norms that are, in ordinary language, usually created by the use ofan (unconditional) imperative expression. In fact, I will simply employ a setof imperatives I. Of course not all imperatives create norms. Imperativesmay fail to create a normative relationship, to bind the addressee to do thebidding of the speaker, e.g. if uttered by a child or passerby, and sometimesthe subject can choose to accept them, e.g. when we enter a game or when acheerleader says “everybody, clap your hands”. But we can simply stipulatethat I is a set of imperatives that are accepted by the subject and, at leastin the subject’s view, create a binding obligation.7

7By stipulating that the imperatives in I are ‘binding’ or ‘valid’, a notion that may inturn depend upon some acceptance by the subjects and/or a ‘legal community’, I hope toside-step all discussions on whether normative propositions are really ‘true’ or ‘false’, orrather just ‘acceptable’ or ‘non-acceptable’ (cf. Mazzarese [148], Niiniluoto [158]), for fromthe same viewpoint that accepts these imperatives as binding, some normative propositionsthat something is obligatory, permitted or forbidden according to the imperatives must


To each imperative corresponds a descriptive sentence that describes thesituation that obtains if and only if the imperative is satisfied. If there wereno such sentence, then the subject could not understand what the personusing the imperative wants her to do, and therefore not create an obligation.This is Weinberger’s principle (W), and I have already explained in sec. 1.3that the fact that there is such a correspondence between imperatives andindicatives seems to be unilaterally accepted. So in addition to the set I thereis a function f : I → LBL that associates with each imperative a formulafrom the language of some (descriptive) basic logic BL. For an imperativei for which f(i) is A, I write ‘!A’, in accordance with tradition.8 As forthe basic logic, while any logic suffices that provides a consequence relationΓ `BL A to express that A can be derived from Γ, I will mostly let thebasic logic be propositional logic PL. This, the couple I = 〈I, f〉, is thebasis of the semantics I will use. I call I a basic imperative structure and asemantics that uses such structures basic imperative semantics. Extensionsof basic imperative semantics are possible and straightforward. E.g. in [78],[80] I have added a relation < on I to express priorities between imperatives.Other functions may be employed to indicate authorities or subjects of theimperatives.

One might be tempted to reduce the semantics and replace I by f(I),i.e. by a set of formulas of the basic logic LBL.9 But our semantics is aboutimperatives, not indicatives, and there might be different imperatives that arefulfilled under the same condition, but are e.g. of different rank or priority,and so it might be important to distinguish them. Also note that, unlike inother approaches like that of Alchourron [3], it is not assumed that the setof imperatives I is ‘closed’ under any ‘logical rule’. There is no commitmentto any ‘logic of imperatives’, theory of ‘implicit commanding’, etc.

I do not provide any further analysis of the ‘content’ of an unconditionalimperative than what is explained above. Weinberger later argued that (W)should be accepted only on the condition that the corresponding indicativeis not contradictory and not tautological,10 but I will not assume generallythat such restrictions hold. Also, it is clear that imperative sentences arecompositional and regularly contain connectives like ‘not’ ‘and’ or ‘or’, as

then be true or false.8Cf. Hofstadter & McKinsey [99], Reichenbach [180].9I have done so for better legibility in the largely formal papers [76] and [77].

10Cf. already Weinberger [222] p. 121, and then similarly [225] pp. 17–19, [229] p. 69,[231] p. 229–231.


in ‘do not eat with your fingers’, ‘buy apples and walnuts’, ‘write her aletter, or at least a postcard’, and these imperatives are then satisfied bynot eating with one’s fingers, buying apples and buying walnuts, and writingher a letter or writing her a postcard. But Weinberger’s principle as wellas the analysis of Hofstadter & McKinsey [99] show that in terms of thesentences that describe the satisfaction of an imperative, a formalization ofthese connectives within imperatives would only amount to a doubling of theusual Boolean operators: if we employ the name !(p1×p2) for the imperative‘buy apples and walnuts’, its associated descriptive sentence f(!(p1 × p2)) isstill p1 ∧ p2, i.e. the imperative is still satisfied just in case its addresseebuys apples and buys walnuts. Also, we may want to use expressions thatare imperatives not in the grammatical, but in a functional sense in ourexamples, like ‘I must ask you to pack your gear and leave the property,’‘can you close the window or turn off the ventilation, please?’ or ‘you oughtto go now,’ and then, with the aid of principle (W), we need not worry abouteach expression’s proper formalization too much.

I will, however, make two assumptions about the contents of the imper-atives: i) that imperative sentences are not separable, and ii) that they areindependent. Sometimes even doing only part of what has been requestedor commanded is seen as something good and (partly) following the order,while at other times failing a part means that satisfying the remainder nolonger makes sense. E.g. if I am to satisfy the imperative “buy apples andwalnuts”, and the walnuts are needed for biscuits and the apples for dessert,then it makes sense and may be required to get the walnuts even if applesare out. (We just have the biscuits for dessert.) If, however, both land in aWaldorf salad, then it might be unwanted and a waste of money to buy thewalnuts if I cannot get the apples. As the example illustrates, there is no‘logical’ method to distinguish one case from the other. The following maybe used as a ‘test’ for separability:11

(Sep) If by use of an imperative sentence i someone is required, amongother things, to do A and B, and is still required to do A even ifB cannot be done, then i is separable.

Instead of a separable imperative, we may just use two or more inseparableimperatives to express what is required. However, ordinary usage of twoimperative sentences does not guarantee their independence. Suppose a flight

11This resembles Hamblin’s [74] p. 184 definition of separability.


instructor tells a student to do the following at takeoff:12

“Add power slowly at first and then gradually, smoothly bringin full power. Take a quick glance in your mirror just as youadd power on the roll. Pay attention to the glider’s nose. Whenwe add power too quickly, the glider’s nose pitches up and thetail slams down. If you think you have added power too quickly,check your mirror. If the glider’s nose is high in the air, then goahead and smoothly add full power.”

Imagine the mirror is broken and so the flight student cannot check theglider’s nose when going up: is she still required to add power? Or shouldshe, as I think, rather not start with a broken mirror? In the second case theseparate imperatives ‘combine inseparably’, i.e. it is, from the standpoint ofthe authority, not required to do one thing when the other one cannot bedone.13 The following test may be used to determine dependent imperatives:

(Dep) If two imperative sentences i1 and i2 have been used, but theyare such that if what i1 demands cannot be carried out, then thesatisfaction of i2 is no longer required, then i1 and i2 are dependenton each other (combine inseparably).

Two or more separate imperatives that combine inseparably may then berepresented by one imperative that demands all of what the separate imper-atives demand.

That sentences may be independent grammatically, but should not beformalized so, is a phenomenon that also holds for descriptive sentences. E.g.the statement ‘Imagine me on the panel! I will most likely make a fool out ofmyself!’ should not be treated as an imperative sentence and an (unqualified)indicative prediction, but rather be reformulated as a conditional ‘If I get puton the panel, I will most likely make a fool out of myself.’ This then is anencouragement to treat such problems as problems of formalization (propersemantical representation), rather than as problems of logic.

The assumption that the imperatives in I are inseparable and indepen-dent becomes important when treating conflicts and dilemmas (where not allimperatives can be satisfied collectively) or contrary-to-duty circumstances(where some imperative has been violated). E.g. having spent most of the

12The quote is from the manual of the Schweizer 2-33 sailplane.13Again, the terminology is owed to Hamblin [74] p. 184.


money on ice cream, I might not be able to buy both the apples and the wal-nuts, and so find myself in a dilemma. I might even have spent so much onice cream that I can’t buy the walnuts with the change: I am in a contrary-to-duty situation. But it might still be enough to purchase the apples. De-pending on how the imperative is to be understood, I might still be underan obligation to buy the apples or the walnuts (in the first case), or just theapples (in the second), or nothing at all (if I was shopping to help prepare aWaldorf salad).

The above approach to modeling normative situations is very basic. Asnoted above, there are all kinds of norms for which the core concept wouldhave to be amended and adjusted – think of explicit permissions (licenses),or of general norms: in order to model, within the basic concept, a generalnorm not to lie, one would have to add to the set, for every subject and anyoccasion to lie, an individual, independent imperative that obliges the subjectnot to lie on this occasion. While some adjustments of the basic conceptsare treated below, a more fundamental criticism of my approach would bethat it does not seem to account for norms that are not created by use ofspeech. One may think e.g. of natural law, Kant’s categorical imperative, ormore down-to-earth rules of courtesy, custom or tradition. Now, as long asthese can be rephrased to be inseparable and independent, and correspondto descriptions of situations in which they are satisfied or violated in thesense explained by Weinberger’s principle (W), nothing hinders us to makeI a set of such norms. The set I is not an ontological category, though itis explained in terms of imperatives, uttered by some authority, that are,by some subject, believed to be binding. This seems to me the best wayto reconstruct everyday normative reasoning, when we consider individualdemands from possibly different sources and try to satisfy them as best aswe can. However, I admit that when obligations are considered that are(only) justified by an appeal to prohairetic notions – something is obligatorybecause it is the best thing to do, it brings about the greatest good for thegreatest number, in all possible situations everyone is better off with it donethan not done, etc. –, then the more natural model may be one in terms ofideality or preference relations between possible worlds after all.


3.3 The Reconstruction of Deontic Logic

Heeding Makinson’s call for a solution of the ‘fundamental problem’ anda ‘reconstruction of deontic logic’ as a logic concerned with norms, but inaccord with the philosophical position that norms are devoid of truth values,we can apply the semantics described in the previous section to give truthdefinitions of monadic and dyadic deontic operators, and define systems ofdeontic logic that are sound and complete with respect to such a semantics.Among these are also the standard systems of monadic and dyadic deonticlogic as described in chapter 2, and so it is demonstrated that these can be‘reconstructed’ with respect to a semantics that explicitly represents norms,but not as entities that can be true or false. My project to apply imperativesemantics to solve the ‘fundamental problem’ was carried out in a series ofresearch papers since Makinson first stated it at the ∆EON’98 workshopin Bologna, Italy, and here the main results of the individual papers aresummarized.

3.3.1 Sets, Sentences, and Some Logics about Imper-atives [2001]

The first paper in the series, written for the ∆EON’00 workshop held inToulouse, France, 20–22 January 2000 and subsequently published in revisedform in 2001 [75], describes the basic imperative semantics and outlines howit may be applied to solve Makinson’s ‘fundamental problem’.

The first result is almost trivial: Let I = 〈I, f〉 be a basic imperativestructure and define

I |= OA iff ∃{i1, ..., in} ⊆ I : f(i1), ..., f(in) |=PL A

So according to this truth definition, OA is true if and only if there are imper-atives in the set I such that their associated descriptive sentences classicallyimply A, i.e. A is necessary to satisfy some imperatives. Letting the truthdefinitions for Boolean operators be as usual, if I is assumed to be a setof non-conflicting imperatives, i.e. f(I) 2PL ⊥, then von Wright’s classicalaxiomatic deontic system is sound and complete with respect to this seman-tics. If I is additionally assumed to be non-empty, i.e. there is at least oneimperative i ∈ I, then standard deontic logic SDL, restricted to a languagethat does not permit ‘mixed formulas’ and nested deontic operators, is soundand complete with respect to this semantics.


The paper then proceeds to a reconstruction of Anderson’s reduction.Taking our cue from Anderson’s idea that his constant means that someviolation has occurred, and Kanger’s idea that his constant represents the factthat all propositions of a ‘welfare program’ are satisfied, the paper presentsfour different constants defined with respect to a basic imperative structureI = 〈I, f〉 and valuation v ∈ B:

I, v |= S iff ∀i ∈ I : v |= f(i)I, v |= s iff ∃i ∈ I : v |= f(i)I, v |= V iff ∀i ∈ I : v |= ¬f(i)I, v |= v iff ∃i ∈ I : v |= ¬f(i)

So according to these definitions, S is true at a valuation v if all imperativesare satisfied at it (this models Kanger’s constant), s is true if some imperativeis satisfied at v, V is true if all imperatives are violated at v, and v is true ifsome imperative is satisfied at v (this is Anderson’s constant). The following(Kangerian) definition of a deontic O-operator is then employed:

OA =def (�s ∨ �v) ∧2(S→ A)

Here, the operator ‘2’ represents (logical) necessity, and 2A means that A istrue at all valuations v ∈ B. Possibility ‘�’ is defined as ¬2¬ as usual. Sinceimperatives that demand contradictions or tautologies to be realized are notper se excluded, the formula �s ∨ �v must be used to ensure that the setof imperatives is non-empty, i.e. there are imperatives to satisfy or violate.So OA is true iff the set of imperatives is non-empty and A is necessary tosatisfy all imperatives. Since S is equivalent to ¬v, an Andersonian definitionof OA

OA =def (�s ∨ �v) ∧2(¬A→ v)

may be equivalently employed to define OA as true if A is necessary to avoida violation. It is then shown that the deontic fragment of a system thataxiomatizes these constants and operators (the axioms are immediate) equalsthe classical system of deontic logic if an axiom �S is employed to ensure thatthe set of imperatives is again non-conflicting, and equals standard deonticlogic SDL if additionally an axiom �s∨�v is employed to ensure that the setof imperatives is non-empty.

Finally the paper presents a first approach to the problem of reconstruct-ing the dyadic deontic systems of deontic logic with respect to imperativesemantics. For this, a ‘naıve’ contraction operator n is defined that re-moves those sentences from a set Γ ⊆ LPL that derive a sentence A, i.e.ΓnA = Γ − {B ∈ Γ | |=PL B → A}. This contraction operator is then used


to remove all those imperative-associated sentences from f(I) that are al-ready violated in a situation described by C, where C represents a possiblycontrary-to-duty situation. This contraction is ‘naıve’ as, of course, theremay be imperative-associated sentences left that each can be individually butnot collectively true together with C. But I wanted to distinguish contrary-to-duty situations from conflicts and dilemmas, and preserve the possibilityof the latter and the possible truth of O⊥ in such situations. So consequentlythe truth of a dyadic deontic O-operator is defined with respect to a basicimperative structure I = 〈I, f〉 as follows:

I |= O(A/C) iff f(I)n¬C ∪ {C} |=PL A

According to this definition, O(A/C) holds if A is necessary to satisfy allthe imperatives that are not violated in the circumstances described by C.Curiously, the dyadic deontic logic that is proven to be sound and completewith respect to such a truth definition is just like the system DDLH describedin sec. 2.5 (i.e. like Hansson’s DSDL3) except that it lacks its typical deonticaxiom (DD-R). We can also make the full set DDLH sound and complete byassuming that dilemmas are excluded for all possible circumstances, i.e. thatif 2PL ¬C then also f(I)n¬C 2PL ¬C. However, this would commit us toassuming that for any two imperatives !i1, i2 ∈ I, either |=PL f(i1) → f(i2)or |=PL f(i2)→ f(i1) – otherwise the situation ¬(f(i1)∧ f(i2)) constitutes adilemma. To assume that the demands of all imperatives are thus ‘chained’rather corresponds to the nested system of spheres provided by preferencebased semantics, but it seems unlikely that any real-world set of commandswill ever meet this assumption.

3.3.2 Problems and Results for Logics about Impera-tives [2004]

My second paper on imperative-based semantics, written for the ∆EON’02workshop held in London, UK, 22–24 May 2002 and subsequently publishedin 2004 [76], explores possible definitions for monadic deontic operators inmore detail.

In the first part the following truth definitions for deontic O-operatorswith respect to a basic imperative structure I = 〈I, f〉 are considered:


I |= O1A iff ∃i ∈ I : f(i) = AI |= O2A iff ∃i ∈ I : |=PL f(i)↔ AI |= O3A iff ∃i ∈ I : |=PL f(i)→ AI |= O4A iff ∃i1, ..., in ∈ I : |=PL (f(i1) ∧ ... ∧ f(in))→ AI |= O5A iff f(I) |=PL A

So O1 takes obligatoriness very literal, describing A as obligatory only if therereally is an imperative that demands a state of affairs properly described byA. O1 is the operator first proposed by Alchourron & Bulygin [5]. O2 is sim-ilar, but allows A to be replaced by any logically equivalent proposition. ForO3A to be true it is sufficient that A is necessary to satisfy one imperative.This is the first of the definitions for OA that van Fraassen [57] proposed.The definition of O4 is similar to Smiley’s [198] definition of a deontic op-erator: O4A is true iff there is non-empty set of imperatives such that Ais necessary to satisfy them. We have already seen in the first paper thatthis definition, with an assumption of the imperatives being non-conflicting,yields von Wright’s classical deontic system (with the language restricted sothat O-operators do not occur iterated or nested). Finally O5 is similar tothe ‘corrected’ definition used by Alchourron & Bulygin [5] in that O5A istrue iff A is necessary to satisfy all imperatives (regardless if there are any).This definition, together with the assumption that the imperatives do notconflict, then yields SDL (with the language restricted as above). Sound andcomplete axiomatic systems are given for all operators.

I then consider what happens if we mix the operators. Obviously we canthen express truths such as ‘OiA → OjA’ for 1 ≤ i < j ≤ 5. A maybeunexpected finding is that though all possible mixtures of operators havesound and (weakly) complete systems (the additionally required axioms areforthcoming), no system but the one that only uses operators O4 and O5 isalso ‘strongly complete’: the semantics is not compact, i.e. there are infinitesets of formulas such that each finite subset of these can be modeled usingbasic imperative semantics, but not the whole sets. A similar experienceawaits us for the use of semantic restrictions. Four such restrictions areconsidered:[R-0] I 6= ∅ (Non-Triviality)[R-1] ∀i ∈ I : 2PL ¬f(i) (Excluded Impossibility)[R-2] f(I) 2PL ⊥ (Collective Satisfiability)[R-3] ∀i ∈ I : 2PL f(i) (Excluded Necessity)

It turns out that for O1 and O2, no axiomatic system is strongly complete


if the set of imperatives is restricted by [R-0], and for O3 and O4 no systemis strongly complete with respect to a set of imperatives restricted by [R-3]. Logics based on O5 are strongly complete with respect to semanticsthat use either [R-0] or [R-3], but not when both restrictions on the setof imperatives are applied collectively. For all other variations there aresound and (strongly) complete systems. The lesson to be learned here is thatthough all of the above restrictions seem somehow intuitive (exclude empty‘sets of imperatives’, exclude imperatives that demand the impossible orthe logically necessary, exclude conflicts), one cannot arbitrarily apply themwithout risking to loose mathematical properties that might be desirable fora corresponding deontic theory.

The next part of the paper focusses on a truth definition that is similar tovan-Fraassen’s [57] second truth definition. Phrased with respect to a basicimperative structure I = 〈I, f〉, it reads:

I |= OFA iff ∃Γ ∈ If⊥: f(Γ) `PL A

Van Fraassen intended to permit conflicts, i.e. cases in which OFA andOF¬A are both true, but not OF⊥. To do so, the truth of OFA is definedwith respect to maximal non-conflicting subsets of imperatives: IfA is theset of all maximal sets of imperatives such that the sets of correspondingdescriptive sentences do not derive A i.e. I fA contains all Γ ⊆ I suchthat (i) f(Γ) 0PL A and (ii) there is no ∆ ⊆ I: Γ ⊂ ∆ and f(∆) 0PL A.OFA is then defined true iff A is necessary to satisfy the imperatives in somemaximal non-conflicting subset of I.

What surprised van Fraassen himself is that such a definition correspondsto a sound and (as I show: only weakly) complete deontic axiom system thatemploys the axiom schemes (Ext), (M), (N) and (P) P> for the OF -operator.But his truth definitions also yields the following truth: if !A, !B ∈ I, andA ∧ B is consistent, then OF (A ∧ B) is true, i.e. if two imperatives do notconflict then what they conjunctively imply is also obligatory. So a limitedform of cumulativity is admitted, but axiomatically this is not expressed.I show in the paper how this difficulty can be overcome by using the O2-operator as defined before.14 However I want to note here that instead alsothe following OC-operator could be used:15

14E.g. we have that if 0 ¬(A ∧B), then O2A ∧O2B → OF (A ∧B).15I owe this observation to Lou Goble (private correspondence). Note thatOC is stronger

than Kenny’s [122] operator of ‘satisfactoriness’ (cf. above p. 34): a definition of Kenny’soperator is I |= OKA iff ∃i ∈ I : |= A→ f(i); and then OCA→ OKA is valid. Also note


I |= OCA iff ∃i1, ..., in ∈ I : |=PL (f(i1) ∧ ... ∧ f(in))↔ A

So according to this operator, it is obligatory that A iff A is necessary andsufficient to satisfy one or more imperatives. A mixed axiomatic systembased on the operators OC and OF and defined via the axiom schemes (Ext)and (C) (for OC), and (Ext), (M), (N) and (P) (for OF ) as well as

(F1′) If 0PL ¬A then OCA→ OFA is a theorem.(F2′) OCA ∧OFB ∧ ¬OF (B ∧ A)→ OF (B ∧ ¬A)

is then sound and (weakly) complete with respect to this semantics. (F1′)expresses the already noted truth that if A is what one or several imperativesdemand, and is also consistent, then it is obligatory according to the truthdefinition for OF . (F2′) is the corresponding truth, so far unnoticed in deonticlogic literature, that if A, though demanded by some imperative, cannot beagglomerated with a sentence B for which OFB is true, i.e. ¬OF (B ∧ A) istrue, then the fact that OFB holds must be due to a conflicting imperativethat demands B and the contrary of A, and so OF (B ∧ ¬A) must be true.In the paper I then introduce a ‘sceptic’ operator OS first formally discussedby John F. Horty [102],

I |= OSA iff ∀Γ ∈ If⊥: f(Γ) `PL A

that expresses that A is obligatory iff its truth is necessary for the satisfactionof any maximal consistent subsets of the imperatives, and show that SDL issound and complete with respect to this truth definition. I also introduceaxioms for a combined system for the O2, OF and OS-operators that is thenproved to be sound and (weakly) complete.

In the final part of the paper I address the question of nested or iterateddeontic operators. The language of deontic logic is now, like LDL definedin sec. 2.2, completely unrestricted, i.e. we may have mixed formulas like‘B ∧OA’, or formulas in which deontic operators occur iterated as in ‘OOA’or nested as in ‘O(A ∧ O¬B)’. In the semantics, the imperative-associatedsentences use the same language LDL, and e.g. !O¬A means an imperative‘let it be forbidden that A’ which is satisfied iff it is forbidden that A. I provethat for all of the previously defined O-operators O1−O5 (defined now usingdeontic logic as basic logic), the earlier systems are still sound and complete,and in particular the axiom schemes proposed for such a language, like

(DT) O(OA→ A),(D4c) OOA→ OA,

that the O-operator of S. O. Hansson’s [85] preference-based deontic logic PDL behavessimilarly to OC (cf. van der Torre [213] pp. 16-17).


all fail: an order to see to it that parking on highways is forbidden doesnot by itself make it forbidden. In this respect, deontic semantics based onimperatives and deontic semantics based on ideality or preference part ways.

3.3.3 Conflicting Imperatives and Dyadic DeonticLogic [2005]

My third paper on imperative-based semantics, written for the ∆EON’04workshop held in Madeira, Portugal, 26–28 May 2004 and published in 2005[77], explores the two truth definitions proposed by van Fraassen and Hortyto cope with possible conflicts of obligations in a dyadic deontic setting.

The following truth definitions of dyadic deontic operators, relative to abasic imperative structure I = 〈I, f〉, stand in the center of the paper:

I |= OF (A/C) iff ∃Γ ∈ If¬C: f(Γ) ∪ {C} `PL AI |= OS(A/C) iff ∀Γ ∈ If¬C: f(Γ) ∪ {C} `PL A

So OF (A/C) is true if A is required to satisfy some imperatives whose de-mands are collectively consistent with the current, possibly contrary-to-dutysituation described by C. Correspondingly, OS(A/C) is defined true if Ais required to satisfy any maximal subset of imperatives such that theirdemands are collectively consistent with the situation C. Again, the OF -operator allows conflicts, it may be that both OF (A/C) and OF (¬A/C)are true, whereas OS cautiously only pronounces that as obligatory whatis necessary to satisfy any maximal set of imperatives whose demands areconsistent with the situation C, and so ¬(OS(A/C) ∧ OS(¬A/C)) holds forall non-contradictory C.

Similar definitions to the ones above have already been used in other con-texts, and the paper proceeds to point out similarities to Kratzer and Lewis’spremise semantics [125], [138],16 Poole systems without constraints [173], the‘X-logics’ of Siegel and Forget [194], [55], and the ‘epistemic states’-semanticsof Bochman [30], [31], [32] that is similar to the cumulative models in Kraus,Lehmann and Magidor [126]. It is then proved that to the definitions abovecorrespond sound and (I prove: only weakly) complete axiomatic system,where the axioms for OF are just like that for DDLL except that (DC) does

16Lewis [138] (p. 233) states in a footnote that for deontic conditionals, the premises ofthe premise semantics might be understood to be “something that ought to hold”, and soseems the inventor of this kind of imperative semantics.


not hold, (DD) is replaced by (DP) P (>/C), and (CCMon) is used as addi-tional axiom scheme, and the axioms for OS are exactly like those for DDLH ,except that (RMon) is exchanged for (CCMon), and finally the following‘mixed’ axiom schemes are added:(RMonFSS) P F (D/C)→ (OS(A/C)→ OS(A/C ∧D))

(RMonSSF ) P S(D/C)→ (OS(A/C)→ OF (A/C ∧D))

The OS-axioms are equivalent to those used by Kraus, Lehmann and Magidor[126] for their system P except that the ‘deontic’ axiom (DD-R) is added.The OF -axioms are those of Bochman [30], except that his axiom (Pres) isstrengthened to (DP). The mixed schemes are again those of Bochman.

Besides the new and relatively complex completeness proof, the paperpoints out a link to Goble’s deontic multiplex preference semantics [66], [67]which – in terms of possible worlds ordered by several preference relations– similarly attempts to accommodate conflicting obligations. The dyadicdeontic formula O(A/C) is defined true by multiplex preference semanticsiff there is at least one preference relation such that the best C-worlds areA-worlds (this corresponds to OF ) or if according to all preference relationsthe best C-worlds are A worlds (this corresponds to OS). I show that theconstruction of a set of imperative-associated sentences described by theearlier completeness proof, that makes true a formula of deontic logic A, canalso be used to define preference relations that make true A by use of thesedefinitions, and that – except for a small change in the restrictions applyingto (DN) and (DD)/(DP) – the above axiomatic system is then also weaklycomplete with respect to such a multiple preference semantics.

3.3.4 Deontic Logics for Prioritized Imperatives [2006]

My fourth paper on imperative semantics was originally written for the work-ing group ‘Law and Logic’ at the XXII. World Congress of Philosophy of Lawand Social Philosophy (IVR 2005), held 24–29 May 2005 in Granada, Spain;a revised version was published in 2006 [78].

While the previous paper considered conflicts between imperatives andhow deontic operators, in particular OF and OS, can be defined that takethem into account and ‘localize’ these conflicts, this paper studies methods toresolve conflicts. It is commonplace that a priority relation, or ranking, be-tween the imperatives may help to resolve conflicts. W. D. Ross [189], to thateffect, distinguished prima facie duties, possibly conflicting obligations that


appear to hold before we look for some ordering, provided e.g. by consider-ing which are more pressing, ranks of authorities, gross differences in utilitiesetc., from what we view as our ‘duties sans phrase’ after all these things havebeen considered. While e.g. the O3-operators employed previously might beseen as describing prima facie duties only, letting it be sufficient for OA thatA is necessary to satisfy one imperative, I now consider prioritized impera-tive structures I = 〈I, f, <〉 that additionally employ a priority relation <between imperatives, where i1 < i2 means that i1 takes priority over i2, andhow these might be used to resolve conflicts to arrive at a definition for anO-operator that describes ‘all-things-considered’ obligations.

There have been quite different proposals as to how priorities might beused to resolve conflicts. I adopt an algorithm first introduced by Rescher[183] and further developed by Brewka [35], [36] for the purpose of deter-mining ‘preferred subtheories’ in theory revision, a proposal that makes veryweak demands on ‘<’ (it is irreflexive, transitive and well-founded, i.e. in-finite descending chains are excluded). This then is compared with othermethods suggested in the literature, such as Horty’s [104] definition of ‘bind-ing imperatives’, Alchourron & Makinson’s [9] criterion of ‘least exposure’,Prakken’s [175] ‘hierarchical rebuttal’, Sartor’s [190] ‘prevailing relation’, andAlchourron’s [4] use of ‘safe contraction’. These methods then turn out to beeither equivalent, or to produce counterintuitive results in examples whereBrewka’s method does not.

The dyadic deontic O-operator is then defined with respect to such pri-oritized imperative structures I = 〈I, f, <〉 as follows:

I |= O(A/C) iff ∀Γ ∈ I ⇓ ¬C: f(Γ) ∪ {C} `PL A

This is similar to the definition of the dyadic OS-operator used in the pre-vious paper (a definition similar to that for OF could have been used justas well), but instead of defining O in terms of what is required to satisfythe imperatives in all maximal subsets Γ of I such that f(Γ) does not imply¬C, i.e. all sets in I f¬C, we use all sets Γ in a set I ⇓ ¬C of maximalpreferred subsets such that f(Γ) does not imply C. Very roughly, for anysituation C, any set in I ⇓ ¬C is defined by first adding a maximal set ofthe most important imperatives such that their demands do not derive ¬C,then adding to it a maximal set of the second most important imperativesthat can be added without the set of corresponding demands now deriving¬C, and so on. The axiomatic system that corresponds to such a definitionis then – not surprisingly, because it was not assumed that all conflicts can


be resolved by the use of priorities – still the system P of Kraus, Lehmann& Magidor [126] with the dyadic deontic axiom (DD-R) added, as provedbefore.

Finally the paper examines what properties of the priority relation mustbe assumed so that it resolves all conflicts that might arise for possible situa-tions C in a set of imperatives I. This is in line with von Wright’s proposal in[249] pp. 68, 80 that an axiological order can “provide a safeguard against anygenuine predicament”. It obviously suffices to let the (irreflexive, transitive,well-founded) priority relation be total, i.e. for any two distinct imperativesi1, i2, either i1 < i2 or i2 < i1, but the paper shows that this requirement ofa total ordering may be relaxed somewhat. It is then proved that if the setof imperatives is thus ‘uniquely prioritized’, then the axiomatic system thatis sound and complete with respect to the above truth definition is DDLH ,i.e. Hansson’s logic DSDL3. It is easy to prove that if for such uniquelyprioritized imperatives a truth definition similar to OF is used, i.e.

I |= O(A/C) iff ∃Γ ∈ I ⇓ ¬C: f(Γ) ∪ {C} `PL A,

then the corresponding axiom system is the Lewis-type dyadic deontic logicDDLL. This completes the reconstruction of the deontic standard systemspresented in chapter 2: the standard dyadic deontic logics DDLH and DDLL

can be reconstructed by use of an imperative semantics that assumes thatall conflicts and dilemmas are solvable by use of a priority relation.

Dyadic deontic logic has often been linked to conditional prescriptions,but the reconstruction shows that it is not necessary to take special consider-ations about conditional imperatives into account: when the primary imper-ative is !A, a contrary-to-duty imperative may be formalized as !(¬A → B)or !¬(¬A ∧ B) to produce the result that O(A>) and O(B/¬A) are true.That dyadic deontic logic can be viewed as being primarily concerned withcontrary-to-duty situations, and not so much with conditional prescriptions,may be seen as an interesting point of the 2004 and 2006 research papers.

3.3.5 Prioritized Conditional Imperatives:Problems and a New Proposal [2007]

Imperatives may be conditional. E.g. consider the instruction ‘when thewater boils, add the fat and the flour’. Using a material conditional in theimperative’s associated sentence, and e.g. representing it by !(C → (A∧B)),will not do: otherwise one could fulfill the instruction simply by not letting


the water boil. But if the water does not boil, e.g. because there is a powerfailure or one decides not to continue with the recipe, one has not doneanything right or wrong: the opportunity to satisfy or violate the instructiondid not arise.17 Much of the literature on imperatives and deontic logic isdevoted to understanding the reasoning about conditional norms, which vonWright [253] called the ‘touchstone’ (Prufstein) for normative logic.

In a paper written for the interdisciplinary seminar on Normative Multi-Agent Systems NorMAS07 held 18–23 March 2007 at Dagstuhl, Germany,and subsequently published in revised form in [80], I have formalized condi-tional imperatives by introducing an additional function g : I → LPL to basicimperative semantics that associates with each imperative the description of acondition that provides the opportunity for satisfying or violating the imper-ative (I write A⇒!B for an imperative i such that g(i) = A and f(i) = B).Given a set of factual descriptions W ⊆ LPL, an imperative is called ‘trig-gered’ if its antecedent is true given W – then the opportunity for its satis-faction or violation arises.18 The set Triggered(W, I) = {i ∈ I | W `PL g(i)}is then the set of imperatives that are triggered given the facts W . A trig-gered imperative g(i) ⇒!f(i) is satisfied iff W |=PL g(i) ∧ f(i), i.e. if alsoits consequent is true, and violated iff its consequent is false given the facts,i.e. iff W |=PL g(i) ∧ ¬f(i). This makes it possible that a conditional im-perative g(i)⇒!f(i) is neither satisfied nor violated. When an imperative isneither satisfied nor violated because the antecedent is false, the imperativeis called ‘obeyed’.19 A subset Γ ⊆ I of imperatives is ‘obeyable’ given thefacts W if m(Γ) ∪W 0PL ⊥, where m(i) = g(i) → f(i) is the ‘materializa-tion’ of i, i.e. the material conditional corresponding to the imperative. SoΓ is obeyable given W iff it is not the case that for some {i1, ..., in} ⊆ Γ wehave W `PL (g(i1)∧¬f(i1))∨ ...∨ (g(in)∧¬f(in)): otherwise we know thatwhatever we do, i.e. given any maxiconsistent subset V of LPL that extendsW ⊆ V , at least one imperative in Γ must be violated.

The truth definitions of deontic operators are then defined with respectthese conditional imperatives, and in particular with respect to those impera-

17This is Castaneda’s [41] distinction between those contents of an obligation statementthat work as circumstances and those that work as ‘deontic foci’.

18Cf. Rescher [184], Sosa [200], van Fraassen [57] Also cf. Greenspan [72]: “Oughts donot arise, it seems, until it is too late to keep their conditions from being fulfilled.” Horty[104] seems to have introduced the term ‘triggered’.

19The terminology differs here: Downing [48] uses the term ‘complied by’ instead of‘obeyed’, while Vranas [?] calls the imperative ‘avoided’.


tives in the set of imperatives I that are triggered. Several options arise, andin parallel to input/output logic as developed by Makinson & van der Torre[142], [143], I discuss the following definitions for O-operators with regard tothus extended ‘conditional imperative structures’ I = 〈I, f, g〉:(td -cd1) I |= O(A/C) iff f(Triggered({C}, I)) `PL A(td -cd2) I |= O(A/C) iff ∀V ∈ LPL⊥¬C : f(Triggered(V, I)) `PL A(td -cd3) I |= O(A/C) iff f(Triggered∗({C}, I)) `PL A

(td -cd4) I |= O(A/C) iff ∀V ∈ LPL⊥¬C : f(Triggered∗(V, I)) `PL A

Truth definition (td -cd1) corresponds to Makinson & van der Torre’s ‘simple-minded output’ out1. It pronounces A as obligatory if A is necessary to satisfyall imperatives that are triggered by C. Definition (td -cd2), like ‘basic out-put’ out2, allows for ‘reasoning by cases’, it e.g. makes O(A/C ∨ D) truein case there are two imperatives C ⇒!A and D ⇒!A and the situation isdescribed by C ∨ D (in the definition, each V ∈ LPL⊥¬C is a maximalsubset of the language that is consistent with C). Definition (td -cd3), corre-sponding to ‘reusable output’ out3, allows ‘normative detachment’ and makese.g. O(A/C) true if there are two imperatives C ⇒!B and B ⇒!A and thesituation is described by C (in the definition, the set Triggered∗(W,Γ) of ‘it-eratively triggered’ imperatives means the smallest subset of Γ ⊆ I such thatfor all i ∈ Γ, if f(Triggered∗(W,Γ))∪W `PL g(i) then i ∈ Triggered∗(W,Γ)).Definition (td -cd4), corresponding to Makinson & van der Torre’s reusablebasic output out4, then permits both, reasoning by cases and normative de-tachment.

It seems important to be able to use ‘circumstantial reasoning’, i.e. em-ploy the information about the situation not only to determine if an impera-tive is triggered, but also for reasoning with its consequent. E.g. if the set ofimperatives is {A⇒!(B ∨ C)}, with its imperative interpreted as expressing“if you have a cold, either stay in bed or wear a scarf”, one would like toobtain O(C/A∧¬B), expressing that given that I have a cold and don’t stayin bed, I ought to wear a scarf. To allow for such circumstantial reasoningfor all the above definitions, and corresponding to the ‘throughput versions’out+ of input/output logic, the above definitions may be changed into

(td -cd1+) I |= O(A/C) iff f(Triggered({C}, I)) ∪ {C} `PL A(td -cd2+) I |= O(A/C) iff ∀V ∈ LPL⊥¬C: f(Triggered(V, I)) ∪ V `PL A(td -cd3+) I |= O(A/C) iff f(Triggered∗({C}, I)) ∪ {C} `PL A(td -cd4+) I |= O(A/C) iff ∀V ∈ LPL⊥¬C : f(Triggered∗(V, I))∪V `PL A

Though all these modifications of the original ‘simple-minded’ definition (td -


cd1) may be justified by corresponding intuitions, we are, unfortunately, notcompletely free in combining them: both (td -cd2+) and (td -cd4+) collapseinto treating an imperative A ⇒!B equally to one that demands a materialconditional !(A → B), and (td -cd4) permits ‘ghostly contrapositions’ that Iargue do not seem adequate for a logic about conditional imperatives. Soreconstructing deontic logic with respect to conditional imperatives presentssome difficult choices.

The paper then focusses on how these definitions may be changed toaccommodate possible conflicts of imperatives, but where (similar to theprevious paper) a priority relation holds between the (now conditional) im-peratives that can be used to resolve some of them. Instead of conditionalimperative structures I = 〈I, f, g〉 we now use prioritized conditional imper-ative structures I = 〈I, f, g, <〉 to evaluate the truth of a formula O(A/C).The question is then how the definitions must be changed to reconstruct‘real-life’ reasoning used to resolve conflicts between conditional imperativesby means of priorities. In this respect, a challenge is posed by the so-called‘order puzzle’. Consider the imperatives in the following ‘drinking and driv-ing’ example, ranked in descending order:

(1) Your mother says: if you drink anything, then don’t drive.(2) Your best friend says: if you go to the party, then you do the driving.(3) Some acquaintance says: if you go to the party, then have a drink

with me.

and assume that the situation is such that you go to the party. Intuitively,you should satisfy the second-highest ranking imperative used by your bestfriend, i.e. you should do the driving, but not satisfy your acquaintance’sdesire to have a drink with her, as this would then violate your mother’s orderwhich ranks highest. So you should drive, but it is not true that you shoulddrink. Formalizing the ordered imperatives by A⇒!¬B < C⇒!B < C⇒!Aand the situation by C, it turns out that, curiously, existing approaches toformalize reasoning with prioritized conditionals, proposed in Horty [105],Marek & Truszczynski [146], Brewka [37] and Brewka & Eiter [38], yieldeither O(A∧B/C) (so you ought to drink and drive) or at least O(B/C) (soyou should drink).

Several methods are discussed to resolve the order puzzle in accord withintuitions. I propose that regarding the truth definitions above, the relevantsubsets of the possibly conflicting imperatives in I are those that are ‘max-imally obeyable’ in the sense explained above: to help determine what the


agent ought to do, the method should at least not hand out sets of imper-atives of which it is already known beforehand that at least one imperativemust be violated. In order to respect priorities and use them to solve someor possibly all of the conflicts, Brewka’s original algorithm from [35], [36](cf. above sec. 3.3.4) is then employed to construct the ‘preferred’ maxi-mally obeyable subsets by testing at each step whether adding a next-lowerimperative leaves the constructed set obeyable. Instead of I, these preferredmaximally obeyable subsets are then used in the above definitions to describeas obligatory what is, for all these sets, necessary to satisfy all the impera-tives in the set that are triggered, triggered∗ etc. I show that all of the abovedefinitions produce the intuitively ‘right’ result for the order puzzle, and sothe proposed method is also neutral with respect to the question which ofthe above definitions should be chosen.

3.4 The Paradoxes of Deontic Logic Recon-

sidered

For a long time deontic logic has been plagued by so-called ‘paradoxes’,seemingly counterintuitive formulas that are nevertheless valid in the mainsystems of deontic logic. It might be questioned whether these are at allsolvable, or do not rather stem from different uses, in quite different contextsof normative reasoning, of the natural language phrases ‘it is obligatory that’,‘it ought to be that’, ‘it is required that’, ‘it is morally necessary that’, ‘ithas been commanded that’ that seemingly all are to be encompassed by theone O-operator of deontic logic. A reaction to this situation may be, aswas proposed above, to define not one, but several O-operators that ‘talkdifferently’ about the norms. The paradoxes of deontic logic then serve asbenchmarks how well one particular definition of an O-operator reflects theintended usage of the natural phrases in these benchmark examples.

3.4.1 Ross’s Paradoxes

The first paradox, Ross’s paradox, was originally levelled not at deontic logic,but at the logic of imperatives as designed by Jørgensen [110] in accord withDubislav’s proposal in [50] (cf. above sec. 1.5). Ross [188] showed that ifsuch imperative inferences are accepted, then it is possible to ‘derive’ from


the imperative ‘post the letter’ the imperative to ‘post the letter or burn it’,which seems counterintuitive.

In deontic logic, both in von Wright’s classical system and in the standardsystem SDL, all instances of the scheme OA→ O(A∨B) are theorems. If p1

is meant to formalize the sentence ‘John posts the letter’, and p2 the sentence‘John burns the letter’, then Op1 → O(p1 ∨ p2) is an instance of this schemeand formalizes the natural language expression ‘if John ought to post theletter, then John ought to post the letter or burn it’, which might be seen assimilarly problematic as the original ‘imperative inference’.20 In sec. 3.3.2 wehave seen that definitions of O-operators of the type O4 and O5 may be usedto ‘reconstruct’ the classical and the standard system of deontic logic, andso the formula Op1 → O(p1∨p2) is valid for such readings of the O-operator.

However, it should be noted that using the definitions of these operators,Ross’s paradox appears much less problematic than it did for imperativeinferences. Though both O4 and O5 describe the normative situation, thetruth of Op1 → O(p1 ∨ p2) for both operators cannot be used to claim thatby burning the letter some, if only derived, imperative is satisfied. A goodmodel of the normative situation in case of Ross’s paradox is to let theset of imperatives be {!p1}, i.e. there is (only) the imperative to John topost the letter in the relevant set of imperatives. The definitions of O4

and O5 then make true Op1, so John ought to post the letter, and, sinceboth operators define OA as true if and only if A is necessary to satisfysome (or all) imperatives, also O(p1 ∨ p2) is true, for any situation that issatisfactory with regard to the only imperative !p1 will also make p1 ∨ p2

true. However, we do not have that bringing about p1 ∨ p2 is also sufficientto satisfy some imperatives: in particular, there is no imperative in the set{!p1} that will get satisfied by bringing about p2, i.e. burning the letter. Thiswas different for Ross’s original setting, where burning the letter satisfies thederived imperative !(p1∨p2), and so seemingly forces us to acknowledge thatin this case, John did at least something right, did something that satisfiedan imperative etc. Since the set of imperatives is not ‘closed’ by any ‘logicof imperatives’ (there is none), there is also no imperative in the set thatgets satisfied by burning the letter. So I think that in this way, imperativesemantics really takes the edge off Ross’s paradox.

20Cf. Danielsson [47] for a recent claim that such a theorem is problematic and shouldbe avoided. However, his axiomatic proposal to eliminate the paradox does not work, asshown in [79].


It seems to me that defendants of the view that Op1 → O(p1∨ p2) shouldnot be true, use a different interpretation of ‘ought’ than that which was usedto define the operators O4 and O5: a reading of OA for which it does notsuffice that bringing about A is necessary for satisfying the imperatives, butin that A must be also sufficient for satisfying at least one imperative. Inthis reading, the truth of O(p1 ∨ p2) is unacceptable: just posting the letteror burning it will not suffice for John to do as he was commanded to do. Forsuch ‘strong’ readings of ‘ought’, the operators O1, O2 and OC may be usedthat were also defined in sec. 3.3.2: for these operators OA is defined trueonly if A is necessary and sufficient to satisfy at least one of the imperatives.Of course this is not only the way a deontic operator can be semanticallydefined to avoid the truth of the formula Op1 → O(p1 ∨ p2): we can e.g.define an operator OR in the following way:

I |= ORA iff f(I) `R A

Here ‘`R’ means Weingartner and Schurz’s relevant ‘R-consequence’ that wasexpressly tailored to avoid Ross’s paradox (see above p. 31). If one thinksthat this ‘really is’ how the reasoning about obligations works in naturallanguage, then such a definition would be appropriate. The set of proposalsfor how to define O-operators in imperative semantics is not closed. But Ithink that a reading of ‘ought’ in the sense that ‘bringing about A is requiredto satisfy the imperatives’ (but not necessarily sufficient to satisfy them) isused at least sometimes in natural discourses and that the paradox onlyoccurs when we confuse this weak sense of ought with its strong sense. Sowith its O-operator, standard deontic logic may catch at least sometimes thenatural meaning of ‘ought’.

Ross’s second paradox, also called ‘Weingartner’s Paradox’ or the ‘windowparadox’ in sec. 1.5, receives a similar solution. Here it is seen as paradox-ical that from an imperative ‘close the window and play the piano’ a singleimperative to ‘close the window’ or to ‘play the piano’ can be ‘inferred’. Thesituation may be modeled by a set {!(p3∧p4)}, where its singular imperativedemands that John closes the window (formalized by p3) and plays the pi-ano (formalized by p4). As explained before, the imperative !(p3 ∧ p4) is, asit should be, modeled as inseparable: nothing ‘right’ is done by closing thewindow or playing the piano alone. Although e.g. the operators O4 and O5

used to ‘reconstruct’ the classical and the standard system of deontic logicmake true all of O(p3 ∧ p4), Op3 and Op4 (all of the sentences in the scope ofthe O-operator describe what is necessary to satisfy the imperatives), neither


p3 nor p4 alone are also sufficient to satisfy an imperative. So again I thinkthis takes the edge off the paradox. Someone who views it as paradoxicalthat we may conclude ‘John ought to play the piano’ from ‘John ought toclose the window and play the piano’ seems to be using a stronger ‘ought’than expressed by these operators, and for such a ‘strong ought’ one of theoperators O1, O2, or OC may be used, as they only make OA true in themodel for formulas A that are (at least) equivalent to p3 ∧ p4. But consideragain the setting of the ‘window paradox’ (cf. sec. 1.5). Suppose that β,after α issued the command ‘close the window and play the piano’ to x, andnot hearing anything for a while, enters x’s room, saying ‘Did you not listento α? You ought to play the piano.’ x replies: ‘It is true that I could playthe piano, and I could even first close the window in order to not disturb theneighbors. I heard what α said and I understand it. I also accept that I haveto do what I am ordered to do by α. But it is not true that I ought to playthe piano.’ I think that this would be a very strange reaction to what α andβ have said. So I think that at least in some interpretation, given the orderto x ‘close the window and play the piano’, the statements are true that ‘xought to play the piano’, ‘x is required to play the piano’, ‘x was orderedto play the piano’ etc. So logics that formalize this reading should not bediscarded. It suffices that the semantics provides methods for all parties todefine ‘their’ reading of ‘ought’ in this paradox.

3.4.2 Contrary-to-duty Paradoxes

Prior’s paradox of commitment [177] as well as Chisholm’s paradox [46] wereexpressly directed at the first systems of deontic logic. However, in discussingthe paradoxes it might be helpful to keep apart the prescriptions that underlythe respective settings from the deontic propositions that are true then.

Prior’s paradox pointed at the difficulty of defining conditional obligationin (monadic) deontic logic: from a statement O¬p1 we derive O(p1 → p2),which may be interpreted as, given that p1 describes something forbidden,like stealing, the realization of p1 commits the agent to p2, where p2 may de-scribe anything (e.g. committing adultery, etc.). Let {!¬p1} be the relevantset of imperatives, it contains just one imperative to Jane not to steal somecertain goods. The operators O4 and O5 used to reconstruct the classicaland the standard system of deontic logic both make O(p1 → p2) true, whichexpresses that it ought to be that Jane either does not steal or commits adul-tery. However, note that there is no secondary, contrary-to-duty imperative


in the set of imperatives. In this sense, it is not true that Jane is under aconditional obligation to commit adultery. Now consider what happens ifthe set does in fact contain such a conditional, contrary-to-duty imperative,e.g. one that in this case commands Jane to return the stolen goods. Letp1 ⇒!p3 be this contrary-to-duty imperative, i.e. the set of imperatives isnow {!¬p1, p1 ⇒!p3}. The first thing to note here is that the first imperativedoes not ‘derive’ the second imperative – there is no such inference relationbetween imperatives. For modeling obligations that arise from conditionalimperatives in possibly contrary-to-duty circumstances, a number of defini-tions were proposed in [80]. They also take into account possible priorityrelations, but we need not worry about these here. When we are not inter-ested in how obligations in one situation may influence obligations in someother situation, we can also draw the situation description outside the scopeof the dyadic operator, i.e. make the situation part of the model and notthe operator. The simplest definition of OA with respect to a conditionalimperative structure I = 〈I, f, g〉 (cf. sec. 3.3.5) and a set of facts W is thenthe following:

(α) I,W |= OA iff ∀Γ ∈ Ifm W : f(Triggered(W,Γ)) `PL A

The set I fm W is the set of ‘maximally obeyable’ subsets of the set ofimperatives I regarding the situation described by W , which is defined asthe set of all maximal sets Γ ⊆ I such that {g(i)→ f(i) | i ∈ Γ}∪W 0PL ⊥.The set Triggered(W,Γ) = {i ∈ Γ | W `PL g(i)} contains all conditionalimperatives that are ‘triggered’ by the facts, i.e. their antecedent made true.OA is then defined true if the truth of A is required for the satisfaction ofthe triggered imperatives in all maximal subsets of conditional imperativesthat in this sense do not conflict with the facts. Applying this to the setof conditional imperatives {!¬p1, p1 ⇒!p3}, if there is no special informationabout the situation, i.e. the sets of facts W is empty, then OA is true justfor propositions A that are necessary for the truth of ¬p1, i.e. Jane mustnot steal, or do anything that includes stealing. If we obtain the additionalfact that Jane has violated the imperative to steal the goods, then the setof facts is changed to W = {p1}, and then the obligations also change:the imperative not to steal is violated and therefore removed from the setof relevant imperatives used to deliberate what is obligatory from now on,and the imperative to return the goods has been triggered. The result isthat OA is now true only for such A that are necessary for the truth of p3,i.e. for returning the stolen goods. We do not have Op2 in the modeled


contrary-to-duty situation, i.e. it is not true that Jane ought to commitadultery given that she has stolen the goods. O(p1 → p2) is still true in theinitial situation, but only because not-stealing-or-committing-adultery is anecessary condition of not stealing. It does not express a commitment tobring about p2. If one is unhappy with such a ‘weak’ notion of ought, andprefers a stronger version that makes only obligatory what is necessary andsufficient to satisfy one or more imperatives, then akin to the operator OC

defined before the following definition may be used,

(β) I,W |= OA iff ∀Γ ∈ I fm W : ∃i1, ..., in ∈ Triggered(W,Γ) :`PL (f(i1), ..., f(in))↔ A

and then in the initial situation OA holds for just those A that are equivalentto ¬p1 (i.e. not stealing), and in the contrary-to-duty-situation that W ={p1} then OA holds for just the A that are equivalent to p3, i.e. to returningthe stolen goods.

Chisholm’s paradox receives a similar resolution. Chisholm’s [46] paradoxconsists of the following sentences:

(1) It ought to be that a certain man go to the assistance of his neighbors.(2) It ought to be that if he does go he tell them he is coming.(3) If he does not go, then he ought not to tell them he is coming.(4) He does not go.

The sentences were originally seen as paradoxical because whatever formal-ization is chosen in standard deontic logic, they will either be inconsistentin SDL, or one of its four sentences will derive one of the others. For itstreatment in imperative semantics, let the set of relevant imperatives be{!p1, p1 ⇒!p2,¬p1 ⇒!¬p2}. It is easy to see that in a the situation withoutthe fact described by (4), i.e. with W = ∅, definition (α) makes true OAjust for those A that are necessary for the truth of p1, i.e. for the man’sgoing. Among these is Op1, but also O(¬p1 → ¬p2). If one does not like thelast statement to be true and prefers a stronger ‘ought’-operator, one canswitch to definition (β), and then OA is true just for those A that are eitherequivalent to p1, i.e. the man is just under an obligation to go and assisthis neighbors. It is disputed whether already the obligation to tell that heis coming should be considered as ‘triggered’ in the initial situation: thoughthe man has an obligation to go and thus to make true the antecedent ofthe conditional imperative to call the neighbors, we do not know yet for sureif the man will in fact go, and then telling and not going might be consid-ered worse then not telling and not going. But for those who accept such a


‘deontic detachment’, (α) may be changed into

(γ) I,W |= OA iff ∀Γ ∈ Ifm W : f(Triggered∗(W,Γ)) `PL A,

where the definition of Triggered∗ considers also those imperatives as ‘trig-gered’ whose antecedents must be true if the other triggered imperatives areto be satisfied (cf. sec. 3.3.5 for the formal definition). Definition (γ) thenmakes OA true for all propositions A that are necessary for the truth ofp1 ∧ p2, i.e. for going and calling, among these O(p1 ∧ p2), Op1, Op2, butalso O(p1 → p2) and O(¬p1 → ¬p2). Again, if the last two truths are notaccepted, and a stronger version of ‘ought’ is preferred, then the definitionmay be changed again into

(δ) I,W |= OA iff ∀Γ ∈ I fm W : ∃i1, ..., in ∈ Triggered∗(W,Γ) :`PL (f(i1), ..., f(in))↔ A,

and then OA holds only for those A that are equivalent to either p1 ∧ p2, p1

or p2, i.e. only going, calling and going-and-calling is obligatory. Now changethe set of facts into W = {¬p1}: Chisholm’s sentences tell us that the man isnot going to go. The set of relevant imperatives shrinks to {¬p1 ⇒!¬p2}: itis the only imperative triggered and still possible to satisfy given our updatedknowledge. Both definitions (α) and (γ) make OA true just for those A thatare necessary for the truth of ¬p2, i.e. for not telling the neighbors that heis coming, and the stronger definitions (β) and (δ) define OA true only forthose A that are equivalent to ¬p2. So not calling his neighbors is the onlyobligation left. If we add the definition

I,W |= A iff W `PL A,for all non-deontic, propositional formulas A ∈ LPL, then in this situationalso ¬p1 is described as true, which is what also might be demanded regardingChisholm’s paradox. Thus I think Chisholm’s paradox is adequately treatedin imperative semantics.

3.4.3 Paradoxes of Defeasible Reasoning

Finally, there have appeared paradoxes for deontic logic that also addresscontrary-to-duty imperatives, but in these scenarios the contrary-to-duty sit-uation gives rise to a special obligation to violate a primary rule. One of theseparadoxes, termed ‘the paradox of the considerate assassin’ describes the fol-lowing situation: the set of imperatives is {!¬k, !¬c, k ⇒!c}. The imperative!¬k is intended to mean that you should not kill the witness, !¬c means


that you should not offer anyone a cigarette (including the witness), andk ⇒!c means that if you kill the witness, you should offer her a cigarettefirst. Prakken & Sergot [176], who first introduced this example (though in adeontic setting, not a setting with explicitly represented imperatives), arguethat the solution should make Oc true for the situation k, as this appliesthe imperative that is more specific for the given circumstances. However,it is easy to see that e.g. truth definition (α), with the set of imperatives asdescribed and the set of facts W = {k}, does not make Oc true: the set of‘maximally obeyable subsets’ of I in this situation is {{!¬c}, {k ⇒!c}}, i.e.the assassin has the choice to either keep to the rule of not offering cigarettes,or offering, in this special case, a cigarette. Both commands apply to thesituation, and it has not been explicitly specified that the second should takepriority over the first. Hence only O(c∨¬c) is true, and thus only tautologiesare described as being obligatory.

A similar idea underlies the second example, also first proposed by Prakken& Sergot [176] and slightly modified by Makinson [141]. Here the set of im-peratives is {!¬f, f ⇒!(f ∧ w), d⇒!(f ∧ w)}: There is a general prohibitionof fences !¬f except if there already is one – in that case it should be white,i.e. f ⇒!(f ∧ w) – or if the owner has a dog, in which case the ownershould have a white fence, i.e. d ⇒!(f ∧ w). Again, the more specific im-perative is intended to be applied in the situation where there already is afence or there is a dog, and so for the situation f the sentence O(f ∧ w)should be true, and likewise for the situation d. For the situation f , defini-tion (α) produces the right result: the set of ‘maximally obeyable’ subsetsin this situation is {{f ⇒!(f ∧ w), d ⇒!(f ∧ w)}}, the only triggered im-perative in this set is f ⇒!(f ∧ w), and so Of ∧ w) is defined as true.However, things are different for the situation where there is (only) a dog,i.e. W = {d}: the imperative that there is to be no fence is not yet violatedand so still considered relevant, and so the set of maximally obeyable sub-sets is {{!¬f, f ⇒!(f ∧ w)}, {f ⇒!(f ∧ w), d ⇒!(f ∧ w)}}. In each of thetwo subsets there is just one imperative that is triggered in the situation d,namely !¬f and d⇒!(f ∧w) respectively, and so only O(¬f ∨ (f ∧w)), andwhat else is necessary to make ¬f ∨ (f ∧w)) true, is described as obligatory.In particular, O(f ∧ w) is false.

While it is true that neither definition (α), nor any other definition pro-posed for imperative semantics, includes a ‘specificity test’, I am not con-vinced that this is a defect. For default rules that describe a knowledge base,it might be argued that facts learned for more specific circumstances should


override possibly incomplete more general knowledge, and so the rule thatbirds fly should not be applied to penguins once we have learned that thesedo not fly, but it still should be applied to other birds. For imperatives theredoes not appear to be such a general rule of overriding specificity. A sol-dier who is ordered before a mission to, under no circumstances, use lethalforce, and is then ordered by another officer to, if meeting a certain enemy,kill that enemy, might, upon meeting this enemy, wonder what to do evenif both commands stem from equally ranking officers. In law there is thelegal principle lex specialis derogat legi generali, but this is not universallyapplicable to all sets of norms, particularly not if they stem from from dif-ferent parts of the legal system, and when it competes with other principleslike lex posterior, lex superiori, or standard argument forms like teleologicalinterpretation. But it seems that if there is a case where a more specificimperative should take priority, we can use a priority ordering to expresssuch relations between the imperatives. So, for the first example, let the pri-ority ordering be !¬k < k ⇒!c < !¬c, and let the priority ordering included ⇒!(f ∧ w) < !¬f in the case of the second example (where i < j meansthat the imperative i takes priority over the imperative j). For prioritizedconditional imperative structures I = 〈I, f, g, <〉 the definition (α) may beredefined as follows:21

(ε) I,W |= OA iff ∀Γ ∈PoI(W, I) : f(Triggered(W,Γ)) `PL A

Here, the set Po(W, I) is the set of preferred maximally obeyable subsets asdefined in [80] (cf. above sec. 3.3.5). For total priority orderings like the onesin the above examples (we ignore the additional imperative f ⇒!(f ∧ w) inthe second example, the ‘paradox’ pertaining to this imperative was alreadysolved) it suffices that for this case the (only) set Γ in Po

I(W, I) is constructedby adding the highest-ranking imperative i1 to Γi1 if W ∪ {g(i1) → f(i1)}is not inconsistent, adding the second highest-ranking to Γi2 if Γi1 ∪ W ∪{g(i2) → f(i2)} is not inconsistent, and so on, and letting Γ be the unionof all sets Γi such that i ∈ I. So in the first example, where W = {k},I = {!¬k, !¬c, k ⇒!c}, and the prioritization is !¬k < k ⇒!c < !¬c, theset Γ of relevant (preferred) imperatives is {k ⇒!c}: in the first step, !¬k isnot added since it is already violated in the situation k, in the second stepk ⇒!c is added since it is consistent with k, and in the third step nothingis added since {k → c,¬c} is not consistent with k. Because k ⇒!c is also

21This is similar to definition (td -pcd1-8) that was defined with respect to preferredmaximally obeyable subsets in [80].


triggered given the facts W = {k}, definition (ε) then makes true Oc. So ifthe assassin is going to kill the witness, and the order to offer the witnessa cigarette first takes priority over the general rule not to offer cigarettes,then the assassin ought to offer the witness a cigarette first. The solutionfor the second example, and the situation that there is a dog, is constructedsimilarly. Thus both paradoxes receive a fair solution.

I think that these paradoxes also show the usefulness of imperative seman-tics for deontic logic. Here, priority orderings pertain to given imperatives,not to statements of obligation that are true in relation to them. If we onlyhad these statements, as in traditional deontic logic, a reasoning with prior-ities would be much harder: we would have to ask e.g. whether a formulaOA that derives from a formula O(A ∧ B) of known priority also ‘inherits’its rank, and if it is exactly the same rank or some lower one. Likewise itwould be hard to determine the preferred subsets that are consistent withthe situation: we do not know whether the truth of OA holds because thereis an obligation to realize A, or perhaps an obligation to realize A∧B whichmight no longer be relevant since ¬B is true, or which might already beoverridden by a higher-ranking obligation to realize ¬B – but canceling theobligation from which OA derives would force us to cancel also OA, unless italso derives from some other obligation, but in that case its rank must nowperhaps be changed, etc. But statements of obligation are not true by them-selves, they are true because there exist certain norms or imperatives, and itseems much more natural to say that it is these that get canceled if they canno longer be satisfied (or are already satisfied), or are overridden by othernorms. Imperative semantics makes these relations easier to understand andto model the deontic propositions that are true with respect to the norms.

Chapter 4

Conclusion

In 1998, David Makinson pointed out a ‘fundamental problem’ of deonticlogic: if deontic logic is supposed to be concerned with norms, how can itbe ‘reconstructed’ in accord with the philosophical position that norms aredevoid of truth values? In reaction to Makinson’s call I argue that norms, orsimpler: imperatives, should be explicitly represented in the logical semanticsof deontic logic. The approach proposed here, and the imperatival traditionof deontic logic it follows, thus depicts deontic logic as a logic about norms,not of norms.

If there is a logic of imperatives or norms, then perhaps this approachwould not be much worth. If, in the face of Jørgensen’s Dilemma, we cane.g. use Dubislav’s convention to ‘derive’ an imperative !A from a set ofimperative !B1, ..., !Bn if the set {B1, ..., Bn} classically implies A, then anassumption of closure of the set of norms, perhaps together with an assump-tion of ‘normative consistency’ (exclusion of conflicts), yields a deontic logic(a logic of propositions about what is obligatory according to the norms) thatis isomorphic to such a logic of norms. The ‘logic of normative systems’ pro-posed by Stenius serves as a typical example: here deontic logic is restrictedto stating what prescriptions exist in a normative system that is closed un-der given rules, i.e. the deontic proposition OA is true if and only if theprescription OA exists in the system of of norms S. From such a viewpoint,deontic logic must then appear as a ‘dull isomorphism’ or ‘ersatz theory’.However, Ross’s paradox, in which from the imperative ‘post the letter’ theimperative ‘post the letter or burn it’ is ‘derived’, and Weinberger’s variant,the ‘window paradox’, in which from the imperative ‘close the window andplay the piano’ the imperative ‘play the piano’ is ‘derived’, have drawn into

117

CONCLUSION 118

doubt the adequacy of Dubislav’s convention: in both cases we are forced toacknowledge that the (possibly forbidden) burning of the letter or the (pos-sibly forbidden) playing the piano without closing the window first, satisfiesan imperative, though only a derived one. Since Poincare first suggested thepossibility of imperative inferences, various versions of ‘normological skepti-cism’ have denied the very concept of a logic of imperatives or norms. In thefirst chapter I have laid out my own argument why the search for a logic ofimperatives is futile and should not be continued: there appear to be no log-ical ordinary language arguments in which a premiss and the conclusion arein the imperative mood. So imperative logic turns out to be what Dubislavcalled an Unding or chimaera: a non-thing that exists only as a concept, butno real objects (ordinary language arguments) fall under the concept.

By contrast, ordinary language discourses abound about what one oughtto do, may do, should not do in the face of requests, demands or ordersthat are explicitly given, known and accepted, but are sometimes difficult tofollow and occasionally conflict with each other. Deontic logic may clarifywhat agents should or need not do, first by examining the demands, seeif they are dependent or independent, separable or inseparable, conditionalor unconditional, and then by telling the agents what is, according to theinterpretation used, necessary to satisfy the demands, as far as they stillcan be in given circumstances. When there are conflicts, logic may aid indetermining underlying rankings or priorities, which can then be used to solvethe conflict. Even if a conflict is not resolvable, logic may formalize methodsto localize it and tell the agent what other obligations remain. Deontic logicseen in this way is a logic ‘at work’, it goes beyond merely explaining its modalexpressions, and also it is not primarily a tool to construct normative systemsor give advice to rational law givers, but rather it provides and reconstructsmethods which are and can be used by rational subjects drawing conclusionsabout what they ought to do when faced with multiple demands, in situationsthat the authorities might or might not have foreseen, and even when thedemands are such that a single, rational authority would not have madethem. It is, in this sense, a positivistic deontic logic. Such a logic should notbe seen as an an ersatz theory for some as yet undiscovered logic of norms.Rather, by employing a logical semantics for deontic logic that interprets itsoperators in terms of explicitly given norms, it overcomes the weaknessesof both, proposed logics of norms (that formalize norms, but cannot agreeon logical principles) and of traditional deontic logic (that provides logicalprinciples, but neither formalizes norms nor represents them semantically).

CONCLUSION 119

The reconstructions demonstrate that all the work put into the construc-tion of existing deontic logic systems has not been in vain. Underlying in-tuitions have been captured in axiomatic systems that are still sound andcomplete with respect to a semantics that explicitly represents norms. Inparticular, SDL is sound and complete for a semantics that defines a deonticO-operator in terms of what is necessary to satisfy a set of imperatives, pro-vided that they do not conflict, and Hansson- or Lewis-type standard dyadicdeontic systems are sound and complete with respect to a semantics thatdefines a dyadic deontic O-operator in terms of what is necessary to satisfyall those imperatives that still can be satisfied in possibly contrary-to-dutycircumstances, provided that all conflicts and dilemmas are taken care of bya total priority relation. But aside of these standard systems, new systemshave appeared that allow us to e.g. meaningfully talk about obligations evenif a priority relation does not exist or is not total and therefore conflictsmust be taken into account, and new definitions are possible for operatorsthat e.g. single out obligations pertaining to particular imperatives, be theyoverridden by stronger ones or not, or define what is necessary and sufficientto satisfy at least one imperative. This multitude of possible definitions doesnot, in my view, mean that deontic logic has become a ‘logic a la carte’. Itrather responds to the different circumstances in which deliberation is re-quired, and to different ways we talk about obligations, e.g. when we treatthem as being prima facie or ‘all things considered’.

Inevitably, with the reconstruction of existing deontic logic systems thework has just begun. It started with a semantics that represents uncon-ditional imperatives, but as I demonstrated in [80], the representation ofconditional imperatives presents difficult choices for the definition of an ap-propriate O-operator. Permission is a neglected concept in deontic logic, wemust find a way to represent explicitly given licenses which override pro-hibitions that may apply otherwise (it appears senseless to explicitly per-mit something that has not previously been, and never will be, prohibited).Norms about the creation of new imperatives or licenses must be taken intoaccount. Some of these problems and other challenges have been outlined inHansen & Pigozzi & van der Torre [81]. However, I think that the recon-structions have shown that modeling natural language imperatives and thereasoning of the subjects they address is a fruitful approach to deontic logic.So I am confident that the imperatival tradition of deontic logic, to which mywork contributes, will continue to be useful in developing intuitive solutionsto these problems.

Acknowledgements

My work on imperative-based semantics for deontic logic would not have beenpossible without the fruitful discussions at the biannual ∆EON workshopswhere the small deontic logic community meets. In particular I want tothank Lennart Aqvist, Lou Goble, Jeff Horty, David Makinson and Leon vander Torre for their constant encouragement, helpful suggestions and alwayswell-placed criticism. All errors that remain are mine.

120

Bibliography

[1] Achenwall, G., Prolegomena Iuris Naturalis, In Usum Auditorum, 3rded., Gottingen: Sumtibus Victorini Bossigeli, 1767.

[2] Alchourron, C. E., “Logic of Norms and Logic of Normative Proposi-tions”, Logique & Analyse, 12, 1969, 242–268.

[3] Alchourron, C. E., “The Intuitive Background of Normative Legal Dis-course and its Formalization”, Journal of Philosophicl Logic, 1, 1972,447–463.

[4] Alchourron, C. E., “Conditionality and the Representation of LegalNorms”, in: Martino, A. A. and Socci Natali, F. (eds.), AutomatedAnalysis of Legal Texts, Amsterdam: North Holland, 1986, 173–186.

[5] Alchourron, C. E. and Bulygin, E., “The Expressive Conception ofNorms”, in [94] 95–124.

[6] Alchourron, C. E. and Bulygin, E., “Unvollstandigkeit, Wider-spruchlichkeit und Unbestimmtheit der Normenordnungen”, in: Conte,A. G., Hilpinen, R. and von Wright, G. H. (eds.), Deontische Logik undSemantik, Wiesbaden: Athenaion, 1977, 20–32.

[7] Alchourron, C. E. and Bulygin, E., “Pragmatic Foundations for a Logicof Norms”, Rechtstheorie, 15, 1984, 453–464.

[8] Alchourron, C. E. and Bulygin, E., “On the Logic of Normative Sys-tems”, in: Stachowiak, H. (ed.), Handbuch pragmatischen Denkens,Hamburg: Meiner, 1993, 273–293.

[9] Alchourron, C. E. and Makinson, D., “Hierarchies of Regulations andTheir Logic”, in [94] 125–148.

121

BIBLIOGRAPHY 122

[10] Alchourron, C. E. and Martino, A. A., “Logic Without Truth”, RatioJuris, 3, 1990, 46–67.

[11] Allen, L. E., “Deontic Logic”, Modern Uses of Logic in Law, 2, 1960,13–27.

[12] Anderson, A. R., “The Formal Analysis of Normative Concepts”, YaleUniversity, Technical Report, 1956, reprinted in Rescher, N. (ed.), TheLogic of Decision and Action, Pittsburgh: University Press, 1967, 147-213.

[13] Anderson, A. R., “The Logic of Norms”, Logique & Analyse, 2, 1958,84–91.

[14] Anderson, A. R., “A Reduction of Deontic Logic to Alethic ModalLogic”, Mind, 67, 1958, 100–103.

[15] Anderson, A. R., “On the Logic of Commitment”, Philosophical Stud-ies, 19, 1959, 23–27.

[16] Anderson, A. R., “The Formal Analysis of Normative Concepts”, in:Rescher, N. (ed.), The Logic of Decision and Action, Pittsburgh: Uni-versity Press, 1967, 147–213, first appeared 1956.

[17] Anderson, A. R., “Some Nasty Problems in the Formal Logic of Ethics”,Nous, 1, 1967, 345–360.

[18] Anderson, A. R. and Moore, O. K., “The Formal Analysis of NormativeConcepts”, American Sociological Review, 11, 1957, 9–17.

[19] Anscombe, G. E. M., Intention, Oxford: Blackwell, 1957.

[20] Aqvist, L., “Some Theorems about a ‘Tree’ System of Deontic TenseLogic”, in [94] 187–221.

[21] Aqvist, L., “Interpretations of Deontic Logic”, Mind, 73, 1964, 246–253.

[22] Aqvist, L., “Deontic Logic”, in: Gabbay, D. and Guenthner, F. (eds.),Handbook of Philosophical Logic. Volume II: Extensions of ClassicalLogic, Dordrecht: D. Reidel, 1984, 605–714, reprinted in [59] pp. 147–264.

BIBLIOGRAPHY 123

[23] Aqvist, L., “Some Results on Dyadic Deontic Logic and the Logic ofPreference”, Synthese, 66, 1986, 95–110.

[24] Ayer, A. J., Language, Truth and Logic, New York: Dover Publ., 1938.

[25] Bailhache, P., “Several Possible Systems of Deontic Weak and StrongNorms”, Logique & Analyse, 21, 1980, 89–100.

[26] Bailhache, P., Essai de logique deontique, Paris: LibrairiePhilosophique J. Vrin, 1991.

[27] Barcan Marcus, R., “Iterated Deontic Modalities”, Mind, 75, 1966,580–582.

[28] Becker, O., Untersuchungen uber den Modalkalkul, Meisenheim/Glan:Anton Hain, 1952.

[29] Bentham, J., Of Laws in General, London: Athlone Press, 1970, editedby H. L. A. Hart. First appeared 1782.

[30] Bochman, A., “Credulous Nonmonotonic Inference”, in: Dean, T.(ed.), Proceedings of the 16th International Joint Conference on Arti-ficial Intelligence, IJCAI ’99, Stockholm, August 1999, San Francisco:Morgan Kaufmann, 1999, 30 – 35.

[31] Bochman, A., A Logical Theory of Nonmonotonic Inference and BeliefChange, Berlin: Springer, 2001.

[32] Bochman, A., “Brave Nonmonotonic Inference and Its Kinds”, Annalsof Mathematics and Artificial Intelligence, 39, 2003, 101–121.

[33] Bohnert, H. G., “The Semiotic Status of Commands”, Philosophy ofScience, 12, 1945, 302–315.

[34] Boole, G., The Mathematical Analysis of Logic, Cambridge: Macmil-lan, Barclay, and Machmillan, 1847.

[35] Brewka, G., “Preferred Subtheories: An Extended Logical Frameworkfor Default Reasoning”, in: Sridharan, N. S. (ed.), Proceedings ofthe Eleventh International Joint Conference on Artificial IntelligenceIJCAI-89, Detroit, Michigan, USA, August 20-25, 1989, San Mateo,Calif.: Kaufmann, 1989, 1043–1048.

BIBLIOGRAPHY 124

[36] Brewka, G., “Bevorzugte Teiltheorien: Wissensrevision in einemAnsatz zum Default-Schließen”, Kognitionswissenschaft, 1, 1990, 27–35.

[37] Brewka, G., “Reasoning about Priorities in Default Logic”, in: Hayes-Roth, B. and Korf, R. E. (eds.), Proceedings of the 12th National Con-ference on Artificial Intelligence, Seattle, WA, July 31st - August 4th,1994, vol. 2, Menlo Park: AAAI Press, 1994, 940–945.

[38] Brewka, G. and Eiter, T., “Preferred Answer Sets for Extended LogicPrograms”, Artificial Intelligence, 109, 1999, 297–356.

[39] Bulygin, E., “Existence of Norms”, 1999, in [149] 237-244.

[40] Carmo, J. and Jones, A. J. I., “Deontic Logic and Contrary-to-duties”,in [59] 265-343.

[41] Castaneda, H.-N., “The Paradoxes of Deontic Logic: The SimplestSolution to All of Them in One Fell Swoop”, in [94] 37–85.

[42] Castaneda, H.-N., “Obligation and Modal Logic”, Logique & Analyse,3, 1960, 40–49.

[43] Castaneda, H.-N., “There Are Command sh-Inferences”, Analysis, 32,1971, 13–19.

[44] Chellas, B. F., The Logical Form of Imperatives, Stanford: Perry LanePress, 1969.

[45] Chellas, B. F., Modal Logic, Cambridge/New York: Cambridge Uni-versity Press, 1980.

[46] Chisholm, R. M., “Contrary-to-Duty Imperatives and Deontic Logic”,Analysis, 24, 1963, 33–36.

[47] Danielsson, S., “Taking Ross’s Paradox Seriously. A Note on the Orig-inal Problems of Deontic Logic”, Theoria, 71, 2005, 20–28.

[48] Downing, P., “Opposite Conditionals and Deontic Logic”, Mind, 63,1959, 491–502.

[49] Dubislav, W., Die Definition, 3rd ed., Leipzig: Meiner, 1931.

BIBLIOGRAPHY 125

[50] Dubislav, W., “Zur Unbegrundbarkeit der Forderungssatze”, Theoria,3, 1938, 330–342.

[51] van Eck, J., “A System of Temporally Relative Modal and DeonticPredicate Logic and Its Philosophical Applications”, Logique & Anal-yse, 25, 1982, 249–290, 339–381.

[52] Englis, K., “Die Norm ist kein Urteil”, Archiv fur Rechts- und Sozial-philosophie, 50, 1964, 305–316.

[53] Føllesdal, D. and Hilpinen, R., “Deontic Logic: An Introduction”, in[93] 1–35.

[54] Foot, P., “Moral Realism and Moral Dilemma”, Journal of Philosophy,80, 1983, 379–389.

[55] Forget, L., Risch, V. and Siegel, P., “Preferential Logics are X-logics”,Journal of Logic and Computation, 11, 2001, 71–83.

[56] Forrester, J. W., “Gentle Murder, or the Adverbial Samaritan”, Jour-nal of Philosophy, 81, 1984, 193–197.

[57] van Fraassen, B., “Values and the Heart’s Command”, Journal of Phi-losophy, 70, 1973, 5–19.

[58] Frey, G., “Idee einer Wissenschaftslogik. Grundzuge einer Logik imper-ativer Satze”, Philosophia Naturalis, 4, 1957, 434–491.

[59] Gabbay, D. M. and Guenthner, F. (eds.), Handbook of PhilosophicalLogic, vol. 8, 2nd ed., Dordrecht: Kluwer, 2002.

[60] Geach, P. T., “Imperative and Deontic Logic”, Analysis, 18, 1958,49–56.

[61] Geach, P. T., “Imperative Inference”, Analysis, 23, 1963, 37–42.

[62] Geach, P. T., “Whatever Happened to Deontic Logic?”, Philosophia(Israel), 11, 1982, 1–13.

[63] Gentzen, G., “Untersuchungen uber das logische Schließen”, Mathema-tische Zeitschrift, 39, 1934, 176–210; 405–431.

BIBLIOGRAPHY 126

[64] Goble, L., “The Iteration of Deontic Modalities”, Logique & Analyse,9, 1966, 197–209.

[65] Goble, L., “A Logic of Good, Should, and Would”, Journal of Philo-sophical Logic, 19, 1990, 169–199.

[66] Goble, L., “Preference Semantics for Deontic Logics: Part I – SimpleModels”, Logique & Analyse, 2003, 383–418.

[67] Goble, L., “Preference Semantics for Deontic Logics: Part II – Multi-plex Models”, Logique & Analyse, 2004, 335–363.

[68] Goble, L., “A Proposal for Dealing with Deontic Dilemmas”, in: Lo-muscio, A. and Nute, D. (eds.), Deontic Logic in Computer Science(Proceedings of the 7th International Workshop on Deontic Logic inComputer Science, DEON 2004, Madeira, May 2004), LNAI 3065,Berlin: Springer, 2004, 75 – 113.

[69] Goble, L., “A Logic for Deontic Dilemmas”, Journal of Applied Logic,3, 2005, 461–483.

[70] Goble, L. and Lokhorst, G.-J. C., “Mally’s Deontik”, GrazerPhilosophische Studien, 67, 2004, 37–57.

[71] Gombay, A., “What is Imperative Inference?”, Analysis, 27, 1967, 145–152.

[72] Greenspan, P., “Conditional Oughts and Hypothetical Imperatives”,Journal of Philosophy, 72, 1975, 259–276.

[73] Grelling, K., “Zur Logik der Sollsatze”, Unity of Science Forum, 1939,44–47.

[74] Hamblin, C. L., Imperatives, Oxford: Blackwell, 1987.

[75] Hansen, J., “Sets, Sentences, and Some Logics about Imperatives”,Fundamenta Informaticae, 48, 2001, 205–226.

[76] Hansen, J., “Problems and Results for Logics about Imperatives”,Journal of Applied Logic, 2, 2004, 39–61.

BIBLIOGRAPHY 127

[77] Hansen, J., “Conflicting Imperatives and Dyadic Deontic Logic”, Jour-nal of Applied Logic, 3, 2005, 484–511.

[78] Hansen, J., “Deontic Logics for Prioritized Imperatives”, Artificial In-telligence and Law, 14, 2006, 1–34.

[79] Hansen, J., “The Paradoxes of Deontic Logic: Alive and Kicking”,Theoria, 72, 2006, 221–232.

[80] Hansen, J., “Prioritized Conditional Imperatives: Problems and a NewProposal”, Autonomous Agents and Multi-Agent Systems, 17, 2008,11–35.

[81] Hansen, J., Pigozzi, G. and van der Torre, L., “Ten Philosophical Prob-lems in Deontic Logic”, in: Boella, G., van der Torre, L. and Verhagen,H. (eds.), Normative Multi-Agent Systems, no. 07122 in Dagstuhl Sem-inar Proceedings, Internationales Begegnungs- und Forschungszentrumfur Informatik (IBFI), Schloss Dagstuhl, Germany, 2007.

[82] Hanson, W. H., “Semantics for Deontic Logic”, Logique & Analyse, 31,1965, 177–190.

[83] Hanson, W. H., “A Logic of Commands”, Logique & Analyse, 9, 1966,329–343.

[84] Hansson, B., “An Analysis of Some Deontic Logics”, Nous, 3, 1969,373–398, reprinted in [93] 121–147.

[85] Hansson, S. O., “Preference-based Deontic Logic (PDL)”, Journal ofPhilosophical Logic, 19, 1990, 75–93.

[86] Hare, R. M., “Imperative Sentences”, Mind, 58, 1949, 21–39.

[87] Hare, R. M., The Language of Morals, Oxford: Oxford UniversityPress, 1952.

[88] Hare, R. M., “Some Alleged Differences Between Imperatives and In-dicatives”, Mind, 76, 1967, 309–326.

[89] Harris, Z. S., “Transformational Theory”, Language, 41, 1965, 363–401.

BIBLIOGRAPHY 128

[90] Harrison, J., “Deontic Logic and Imperative Logic”, in: Geach, P. T.(ed.), Logic and Ethics, Dordrecht: Kluwer, 1991, 79–129.

[91] Hempel, C. G., “Review of Alf Ross: ‘Imperatives and Logic”’, Journalof Symbolic Logic, 6, 1941, 105–106.

[92] Hilpinen, R., “On Deontic Logic, Pragmatics and Modality”, in: Sta-chowiak, H. (ed.), Pragmatik: Handbuch des pragmatischen Denkens,vol. 4, Hamburg: Felix Meiner.

[93] Hilpinen, R. (ed.), Deontic Logic: Introductory and Systematic Read-ings, Dordrecht: Reidel, 1971.

[94] Hilpinen, R. (ed.), New Studies in Deontic Logic, Dordrecht: Reidel,1981.

[95] Hintikka, J., “Some Main Problems of Deontic Logic”, in [93] 59–104.

[96] Hintikka, J., “The Ross Paradox as Evidence for the Reality of Se-mantical Games”, in: Saarinen, E. (ed.), Game-Theoretical Semantics,Dordrecht: Reidel, 1977, 329–345.

[97] Hintikka, J. K., Quantifiers in Deontic Logic, Helsingfors: SocietasScientiarum Fennica, Commentationes Humanarum Litterarum 23:4,1957.

[98] Hodges, W., Logic, Harmondsworth: Penguin, 1977.

[99] Hofstadter, A. and McKinsey, J. C. C., “On the Logic of Imperatives”,Philosophy of Science, 6, 1938, 446–457.

[100] Hollander, P., Rechtsnorm, Logik und Wahrheitswerte. Versucheiner kritischen Losung des Jorgensenschen Dilemmas, Baden-Baden:Nomos, 1993.

[101] Horty, J. F., “Moral Dilemmas and Nonmonotonic Logic”, Journal ofPhilosophical Logic, 23, 1994, 35–65.

[102] Horty, J. F., “Nonmonotonic Foundations for Deontic Logic”, in: Nute,D. (ed.), Defeasible Deontic Logic, Dordrecht: Kluwer, 1997, 17–44.

BIBLIOGRAPHY 129

[103] Horty, J. F., Agency and Deontic Logic, Oxford: Oxford UniversityPress, 2000.

[104] Horty, J. F., “Reasoning with Moral Conflicts”, Nous, 37, 2003, 557–605.

[105] Horty, J. F., “Defaults with Priorities”, Journal of Philosophical Logic,36, 2007, 367–413.

[106] Hruschka, J., Das deontologische Sechseck bei Gottfried Achenwall imJahre 1767, Gottingen: Vandenhoek & Ruprecht, 1986.

[107] Hume, D., A Treatise of Human Nature, Oxford: Oxford UniversityPress, 1888, edited by L. A. Selby-Bigge. First appeared 1739.

[108] Husserl, E., Logische Untersuchungen, vol. 1, Halle: Max Niemeyer,1900.

[109] Jacquette, D., “Moral Dilemmas, Disjunctive Obligations, and Kant’sPrinciple that ‘Ought’ implies ‘Can”’, Synthese, 88, 1991, 43–55.

[110] Jørgensen, J., “Imperatives and Logic”, Erkenntnis, 7, 1938, 288–296.

[111] Kalinowski, G., Le Probleme de la Verite en Morale et en Droit, Lyon:Emmanuel Vitte, 1967.

[112] Kalinowski, G., Einfuhrung in die Normenlogik, Frankfurt/M.:Athenaum, 1973.

[113] Kalinowski, G., “Uber die deontischen Funktoren”, in: Lenk, H. (ed.),Normenlogik. Grundprobleme der deontischen Logik, Pullach: Doku-mentation, 1974, 39–63.

[114] Kalinowski, G., “Uber die Bedeutung der Deontik fur Ethik und Recht-sphilosophie”, in: Conte, A. G., Hilpinen, R. and von Wright, G. H.(eds.), Deontische Logik und Semantik, Wiesbaden: Athenaion, 1977,101–129.

[115] Kalinowski, G., “Zur Semantik der Rechtssprache”, in: Krawietz, W.,Opa lek, K., Peczenik, A. and Schramm, A. (eds.), Argumentation undHermeneutik in der Jurisprudenz, vol. 1 of Rechtstheorie, Beihefte,Berlin: Duncker & Humblot, 1979, 239–252.

BIBLIOGRAPHY 130

[116] Kamp, G., Logik und Deontik, Paderborn: Mentis, 2001.

[117] Kamp, H., “Free Choice Permission”, Proceedings of the AristotelianSociety, 74, 1973/74, 57–74.

[118] Kamp, H., “Semantics versus Pragmatics”, in: Guenthner, F. andSchmidt, S. J. (eds.), Formal Semantics and Pragmatics for NaturalLanguages, Dordrecht: Reidel, 1978, 255–287.

[119] Kanger, S., “New Foundations for Ethical Theory: Part 1”, duplic., 42p., 1957, reprinted in [93] 36–58.

[120] Keene, G. B., “Can Commands Have Logical Consequences?”, Ameri-can Philosophical Quarterly, 3, 1966, 57–63.

[121] Kelsen, H., Allgemeine Theorie der Normen, Wien: Manz, 1979.

[122] Kenny, A. J., “Practical Inference”, Analysis, 26, 1966, 65–75.

[123] Keuth, H., “Deontische Logik und Logik der Normen”, in: Lenk, H.(ed.), Normenlogik. Grundprobleme der deontischen Logik, Pullach:Dokumentation, 1974, 64–86.

[124] Knuuttila, S., “The Emergence of Deontic Logic in the FourteenthCentury”, in [94] 225–248.

[125] Kratzer, A., “Conditional Necessity and Possibility”, in: Bauerle, R.,Egli, U. and von Stechow, A. (eds.), Semantics from Different Pointsof View, Berlin: Springer, 1979, 116–147.

[126] Kraus, S., Lehmann, D. and Magidor, M., “Nonmonotonic Reasoning,Preferential Models and Cumulative Logics”, Artificial Intelligence, 44,1990, 167–207.

[127] Kripke, S. A., “Semantical Analysis of Modal Logic I: Normal ModalPropositional Calculi”, Zeitschrift fur mathematische Logik und Grund-lagen der Mathematik, 9, 1963, 67–96.

[128] von Kutschera, F., Einfuhrung in die Logik der Normen, Werte undEntscheidungen, Freiburg/Munchen: Karl Alber, 1973.

BIBLIOGRAPHY 131

[129] Ledig, G., “Zur Klarung einiger Grundbegriffe. Imperativ, Rat, Bitte,Beschluß, Versprechen”, Revue Internationale de la Theorie du Droit,3, 1928/29, 260–270.

[130] Ledig, G., “Zur Logik des Sollens”, Der Gerichtssaal, 100, 1931, 368–385.

[131] Leibniz, G. W., “Elementa Juris Naturalis [2nd half 1671, holographicconcept B (Hannover)]”, in: Preussische Akademie der Wissenschaften(ed.), Gottfried Wilhelm Leibniz: Samtliche Schriften und Briefe, vol. 1of 6. Reihe: Philosophische Schriften, Darmstadt: Otto Reichl, 1930,465–485.

[132] Lemmon, E. J., “Moral Dilemmas”, Philosophical Review, 71, 1962,139–158.

[133] Lemmon, E. J., “Deontic Logic and the Logic of Imperatives”, Logique& Analyse, 8, 1965, 39–71.

[134] Lemmon, E. J., Beginning Logic, 2nd ed., London: Chapman and Hall,1987.

[135] Leonard, H. S., “Interrogatives, Imperatives, Truth, Falsity and Lies”,Philosophy of Science, 26, 1959, 172–186.

[136] Lewis, D., Counterfactuals, Oxford: Basil Blackwell, 1973.

[137] Lewis, D., “Semantic Analyses for Dyadic Deontic Logic”, in: Stenlund,S. (ed.), Logical Theory and Semantic Analysis, Dordrecht: Reidel,1974, 1 – 14.

[138] Lewis, D., “Ordering Semantics and Premise Semantics for Counter-factuals”, Journal of Philosophical Logic, 10, 1981, 217–234.

[139] MacKay, A. F., “Inferential Validity and Imperative Inference Rules”,Analysis, 29, 1969, 145–156.

[140] Makinson, D., “General Patterns in Nonmonotonic Reasoning”, in:Gabbay, D. M., Hogger, C. and Robinson, J. (eds.), Handbook of Logicin Artificial Intelligence and Logic Programming, vol. 3: NonmonotonicReasoning and Uncertain Reasoning, Oxford: Clarendon Press, 1994,35 – 110.

BIBLIOGRAPHY 132

[141] Makinson, D., “On a Fundamental Problem of Deontic Logic”, in: Mc-Namara, P. and Prakken, H. (eds.), Norms, Logics and InformationSystems, Amsterdam: IOS, 1999, 29–53.

[142] Makinson, D. and van der Torre, L., “Input/Output Logics”, Journalof Philosophical Logic, 29, 2000, 383–408.

[143] Makinson, D. and van der Torre, L., “Constraints for Input/OutputLogics”, Journal of Philosophical Logic, 30, 2001, 155–185.

[144] Makinson, D. and van der Torre, L., “What is Input/Output Logic”,in: Lowe, B., Malzkom, W. and Rasch, T. (eds.), Foundations of theFormal Sciences II : Applications of Mathematical Logic in Philosophyand Linguistics (Papers of a conference held in Bonn, November 10-13,2000), Trends in Logic, vol. 17, Dordrecht: Kluwer, 2003, 163–174.

[145] Mally, E., Grundgesetze des Sollens. Elemente der Logik des Willens,Graz: Leuschner & Lubensky, 1926.

[146] Marek, V. W. and Truszczynski, M., Nonmonotonic Logic. Context-Dependent Reasoning, Berlin: Springer, 1993.

[147] Mates, B., Elementary Logic, 2nd ed., Oxford: Oxford University Press,1972.

[148] Mazzarese, T., “‘Norm Propositions’: Epistemic and SemanticQueries”, Rechtstheorie, 22, 1991, 39–70.

[149] Meggle, G. (ed.), Actions, Norms, Values. Discussions with Georg Hen-rik von Wright, Berlin: de Gruyter, 1999.

[150] Menger, K., “A Logic of the Doubtful. On Optative and Impera-tive Logic”, Reports of a Mathematical Colloquium, 2, 1939, 53–64,reprinted in [151] 91–102.

[151] Menger, K., Selected Papers in Logic and Foundations, Didactics, Eco-nomics, Dordrecht: Reidel, 1979.

[152] Meyer, J.-J. C., “A Different Approach to Deontic Logic: Deontic LogicViewed as a Variant of Dynamic Logic”, Notre Dame Journal of FormalLogic, 29, 1988, 109–136.

BIBLIOGRAPHY 133

[153] Moritz, M., “Der praktische Syllogismus und das juridische Denken”,Theoria, 20, 1954, 78–127.

[154] Mott, P. L., “On Chishom’s Paradox”, Journal of Philosophical Logic,2, 1973, 197–211.

[155] Moutafakis, N. J., Imperatives and their Logic, New Delhi: Sterling,1975.

[156] Niiniluoto, I., “Truth and Legal Norms”, Archiv fur Rechts- und Sozial-philosophie Beiheft, 25, 1985, 168–190.

[157] Niiniluoto, I., “Hypothetical Imperatives and Conditional Obliga-tions”, Synthese, 66, 1986, 111–133.

[158] Niiniluoto, I., “Norm Propositions Defended”, Ratio Juris, 4, 1991,367–373.

[159] Nortmann, U., Deontische Logik ohne Paradoxien. Semantik und Logikdes Normativen, Munchen: Philosophia Verlag, 1989.

[160] Nowell-Smith, P. H. and Lemmon, E. J., “Escapism: the Logical Basisof Ethics”, Mind, 69, 1960, 289–300.

[161] Nute, D. and Yu, X., “Defeasible Deontic Logic. Introduction”, in:Nute, D. (ed.), Defeasible Deontic Logic, Dordrecht: Kluwer AcademicPublishers, 1997, 1–16.

[162] Opa lek, K., “On the Logical-Semantic Structure of Directives”, Logique& Analyse, 13, 1970, 169–196.

[163] Opa lek, K., Theorie der Direktiven und Normen, Wien: Springer, 1986.

[164] Opa lek, K. and Wolenski, J., “Is, Ought, and Logic”, Archiv fur Rechts-und Sozialphilosophie, 73, 1987, 373–385.

[165] Opa lek, K. and Wolenski, J., “Normative Systems, Permission andDeontic Logic”, Ratio Juris, 4, 1991, 334–348.

[166] Peczenik, A., “Doctrinal Study of Law and Science”, OsterreichischeZeitschrift fur offentliches Recht, 17, 1967, 128–141.

BIBLIOGRAPHY 134

[167] Peczenik, A., “Norms and Reality”, Theoria, 34, 1968, 117–133.

[168] Philipp, P., “Logik deskriptiver normativer Begriffe”, 1989/90, In [170]pp. 241–290. First published as reports of the Karl-Marx-UniversitatLeipzig, Untersuchungen zur Logik und zur Methodologie 6:65–88, 1989,and 7:51–82, 1990.

[169] Philipp, P., “Normative Logic Without Norms”, 1991, In [170] pp. 291–301. Edited version of a presentation to the Meeting of the CentralDivision of the American Philosophical Association, Chicago, 1991.

[170] Philipp, P., Logisch-philosophische Untersuchungen. Edited by IngolfMax and Richard Raatzsch., Berlin: de Gruyter, 1998.

[171] Philipps, L., “Braucht die Rechtswissenschaft eine deontische Logik?”,in: G., J. and Maihofer, W. (eds.), Rechtstheorie. Beitrage zur Grund-lagendiskkussion, Frankfurt: Klostermann, 1971, 352–368.

[172] Poincare, H., Dernieres Pensees, Paris: Ernest Flammarion, 1913.

[173] Poole, D., “A Logical Framework for Default Reasoning”, ArtificialIntelligence, 36, 1988, 27–47.

[174] Powers, L., “Some Deontic Logicians”, Nous, 1, 1967, 380–400.

[175] Prakken, H., Logical Tools for Modelling Legal Argument, Dordrecht:Kluwer, 1997.

[176] Prakken, H. and Sergot, M., “Contrary-to-duty obligations.”, StudiaLogica, 52, 1996, 91–115.

[177] Prior, A. N., “The Paradoxes of Derived Obligation”, Mind, 63, 1954,64–65.

[178] Prior, A. N., Formal Logic, Oxford: Clarendon Press, 1955.

[179] Prior, A. N., “Escapism: the Logical Basis of Ethics”, in: Melden, A. I.(ed.), Essays in Moral Philosophy, Seattle: University of WashingtonPress, 1966, 135–146.

[180] Reichenbach, H., Elements of Symbolic Logic, New York: Collier-Macmillan, 1947.

BIBLIOGRAPHY 135

[181] Reiter, R., “A Logic for Default Reasoning”, Artificial Intelligence, 13,1980, 81–132.

[182] Rescher, N., “An Axiom System for Deontic Logic”, Philosophical Stud-ies, 9, 1958, 24–30.

[183] Rescher, N., Hypothetical Reasoning, Amsterdam: North-Holland,1964.

[184] Rescher, N., The Logic of Commands, London: Routledge & KeganPaul, 1966.

[185] Rescher, N., “Assertion Logic”, in: Rescher, N. (ed.), Topics in Philo-sophical Logic, chap. XIV, Dordrecht: Reidel, 1968.

[186] Rodig, J., “Kritik des Normlogischen Schließens”, Theory and Decision,2, 1971, 79–93.

[187] Rodig, J., “Uber die Notwendigkeit einer besonderen Logik der Nor-men”, Jahrbuch fur Rechtssoziologie und Rechtstheorie, 2, 1972, 163–185.

[188] Ross, A., “Imperatives and Logic”, Theoria, 7, 1941, 53–71, Reprintedwith only editorial changes in Philosophy of Science 11:30–46, 1944.

[189] Ross, W. D., The Right and the Good, Oxford: Clarendon Press, 1930.

[190] Sartor, G., “Inconsistency and Legal Reasoning”, in: Breuker, J. A.,De Mulder, R. V. and Hage, J. C. (eds.), Legal Knowledge Based Sys-tems: Model-based Legal Reasoning (Proceedings of the Fourth Inter-national Conference on Legal Knowledge Based Systems JURIX 1991,Lelystad, December 1991), Lelystad: Koninklijke Vermande, 1991, 92–112.

[191] Sayre-McCord, G., “Deontic Logic and the Priority of Moral Theory”,Nous, 20, 1986, 179–197.

[192] Schotch, P. K. and Jennings, R. E., “Non-Kripkean Deontic Logic”, in[94] 149–162.

[193] Sellars, W., “Imperatives, Intentions and the Logic of “Ought””,Methodos, 8, 1956, 227–268.

BIBLIOGRAPHY 136

[194] Siegel, P. and Forget, L., “A Representation Theorem for PreferentialLogics”, in: Aiello, L. C., Doyle, J. and Shapiro, S. C. (eds.), Principlesof Knowledge Representation and Reasoning (Proceedings of the 5thInternational Conference, KR ’96), San Francisco: Morgan Kaufmann,1996, 453–460.

[195] Sigwart, C., Logic, vol. I. The Judgment, Concept, and Inference, 2nded., London: Swan Sonnenschein & Co., 1895.

[196] Simon, H. A., “The Logic of Rational Decision”, The British Journalfor the Philosophy of Science, 16, 1965, 169–186.

[197] Sinowjew, A. A., Komplexe Logik, Berlin: VEB Deutscher Verlag derWissenschaften, 1970.

[198] Smiley, T. J., “The Logical Basis of Ethics”, Acta Philosophica Fennica,16, 1963, 237–246.

[199] Solt, K., “Deontic Alternative Worlds and the Truth-Value of ‘OA’”,Logique & Analyse, 27, 1984, 349–351.

[200] Sosa, E., “The Logic of Imperatives”, Theoria, 32, 1966, 224–235.

[201] Sosa, E., “On Practical Inference and the Logic of Imperatives”, Theo-ria, 32, 1966, 211–223.

[202] Sosa, E., “The Semantics of Imperatives”, American PhilosophicalQuarterly, 4, 1967, 57–64.

[203] Spohn, W., “An Analysis of Hansson’s Dyadic Deontic Logic”, Journalof Philosophical Logic, 4, 1975, 237–252.

[204] Stekeler-Weithofer, P., Grundprobleme der Logik, Berlin/New York:Walter de Gruyter, 1986.

[205] Stenius, E., “The Principles of a Logic of Normative Systems”, ActaPhilosophica Fennica, 16, 1963, 247–260.

[206] Stenius, E., “Ross’ Paradox and Well-Formed Codices”, Theoria, 48,1982, 49–77.

BIBLIOGRAPHY 137

[207] Stranzinger, R., “Ein paradoxienfreies deontisches System”, in: Tam-melo, I. and Schreiner, H. (eds.), Strukturierungen und Entscheidungenim Rechtsdenken, Wien: Springer, 1978, 183–193.

[208] Swirydowicz, K., “Normative Consequence Relation and ConsequenceOperations on the Language of Dyadic Deontic Logic”, Theoria, 60,1994, 27–47.

[209] Swirydowicz, K., “A New Approach to Dyadic Deontic Logic and theNormative Consequence Relation”, in: Murawski, R. and Pogonowski,J. (eds.), Euphony and Logos. Essays in Honour of Maria Steffen-Batogand Tadeusz Batog, Amsterdam/Atlanta: Rodopi, 1997, 283–317.

[210] Tarski, A., “Uber einige fundamentale Begriffe der Metamathematik”,Comptes Rendus des Seances de la Societe des Sciences et des Lettresde Varsovie, 23, 1930, 22–29, published in English under the title “OnSome Fundamental Concepts of Metamathematics” in [212] pp. 30–37.

[211] Tarski, A., “Der Wahrheitsbegriff in den formalisierten Sprachen”, Stu-dia Philosophica, I, 1935, 261–405.

[212] Tarski, A., Logic, Semantics, Metamathematics, Oxford: Oxford Uni-versity Press, 1956.

[213] van der Torre, L., Reasoning About Obligations, Amsterdam: ThesisPubl., 1997.

[214] van der Torre, L. and Tan, Y.-H., “Two-Phase Deontic Logic”, Logique& Analyse, 43, 2000, 411–456.

[215] Urmson, J. O., “Saints and Heroes”, in: Melden, A. I. (ed.), Essaysin Moral Philosophy, Seattle: University of Washington Press, 1958,198–216.

[216] Wagner, H. and Haag, K., Die moderne Logik in der Rechtswis-senschaft, Bad Homburg: Gehlen, 1970.

[217] Walter, R., “Jorgensen’s Dilemma and How to Face It”, Ratio Juris,9, 1996, 168–171.

BIBLIOGRAPHY 138

[218] Walter, R., “Some Thoughts on Peczenik’s Replies to ‘Jorgensen’sDilemma and How to Face It’ (with Two Letters by A. Peczenik)”,Ratio Juris, 10, 1997, 392–396.

[219] Wansing, H., “Deontic Modalities: Another View of Parking on High-ways”, Erkenntnis, 49, 1998, 185–199.

[220] Watts, I., Logick: or, the Right Use of Reason in the Inquiry afterTruth, with a Variety of Rules to Guard against Error in the Affairsof Religion and Human Life, as well as in the Sciences, London: JohnClark, Richard Hett et al., 1725, republished at Whitefish, Montana:Kessinger Publishing, 2004.

[221] Wedeking, G. A., “Are There Command Arguments”, Analysis, 30,1970, 161–166.

[222] Weinberger, O., “Uber die Negation von Sollsatzen”, Theoria, 23, 1957,102–132.

[223] Weinberger, O., Die Sollsatzproblematik in der modernen Logik:Konnen Sollsatze (Imperative) als wahr bezeichnet werden?, Prague:Nakladatelstvı Ceskoslovenske Akademie Ved, 1958, page numbers re-fer to the reprint in [227] pp. 59–186.

[224] Weinberger, O., “Bemerkungen zur Grundlegung der Theorie des ju-ristischen Denkens”, Jahrbuch fur Rechtssoziologie und Rechtstheorie,2, 1972, 134–161.

[225] Weinberger, O., “Der Begriff der Nicht-Erfullung und die Normen-logik”, Ratio, 14, 1972, 15–32.

[226] Weinberger, O., “Ideen zur logischen Normensemantik”, in: Haller,R. (ed.), Jenseits von Sein und Nichtsein. Beitrage zur Meinong-Forschung, Graz: Akademische Druck- und Verlagsanstalt, 1974, 295–311, Reprinted in [227] pp. 259–277.

[227] Weinberger, O., Studien zur Normlogik und Rechtsinformatik, Berlin:Schweitzer, 1974.

[228] Weinberger, O., “Ex falso quodlibet in der deskriptiven und praskrip-tiven Sprache”, Rechtstheorie, 6, 1975, 17–32.

BIBLIOGRAPHY 139

[229] Weinberger, O., Normentheorie als Grundlage der Jurisprudenz undEthik, Berlin: Duncker&Humblot, 1981.

[230] Weinberger, O., “Der normenlogische Skeptizismus”, Rechtstheorie, 17,1986, 13–81, reprinted in [233] pp. 431–499.

[231] Weinberger, O., Rechtslogik, 2nd ed., Berlin: Duncker&Humblot, 1989.

[232] Weinberger, O., “The Logic of Norms Founded on Descriptive Lan-guage”, Ratio Juris, 4, 1991, 284–307.

[233] Weinberger, O., Moral und Vernunft, Wien: Bohlau, 1992.

[234] Weinberger, O., Alternative Handlungstheorie, Wien: Bohlau, 1996.

[235] Weinberger, O., “Logical Analysis in the Realm of Law”, 1999, in [149]291-304.

[236] Weingartner, P., “Reasons from Science for Limiting Classical Logic”,in: Weingartner, P. (ed.), Springer, Berlin, 2003, 233–248.

[237] Weingartner, P. and Schurz, G., “Paradoxes Solved by Simple Rele-vance Criteria”, Logique & Analyse, 29, 1986, 3–40.

[238] Williams, B. A. O., “Imperative Inference”, Analysis, 23, 1963, 30–36.

[239] Wittgenstein, L., Philosophical Investigations, Oxford: Blackwell,1953.

[240] Wolenski, J., “Deontic Logic and Possible Worlds Semantics: A His-torical Sketch”, Studia Logica, 49, 1990, 273–282.

[241] Wolenski, J., “Remarks on Mally’s Deontik”, in: Hieke, A. (ed.), ErnstMally – Versuch einer Neubewertung, St. Augustin: Akademia, 1998,73–80.

[242] von Wright, G. H., “Deontic Logic”, Mind, 60, 1951, 1–15.

[243] von Wright, G. H., An Essay in Modal Logic, Amsterdam: North Hol-land, 1951.

[244] von Wright, G. H., “A Note on Deontic Logic and Derived Obligation”,Mind, 65, 1956, 507–509.

BIBLIOGRAPHY 140

[245] von Wright, G. H., Logical Studies, London: Routledge and KeganPaul, 1957.

[246] von Wright, G. H., Norm and Action, London: Routledge & KeganPaul, 1963.

[247] von Wright, G. H., “A New System of Deontic Logic”, Danish Yearbookof Philosophy, 1, 1964, 173–182, reprinted in [93] 105–115.

[248] von Wright, G. H., “A Correction to a New System of Deontic Logic”,Danish Yearbook of Philosophy, 2, 1965, 103–107, reprinted in [93] 115–120.

[249] von Wright, G. H., An Essay in Deontic Logic and the General Theoryof Action, Amsterdam: North Holland, 1968.

[250] von Wright, G. H., “Normenlogik”, in: Lenk, H. (ed.), Normenlogik.Grundprobleme der deontischen Logik, Pullach b. Munchen: VerlagDokumentation, 1974, 25 – 38.

[251] von Wright, G. H., “Problems and Prospects of Deontic Logic”, in:Agazzi, E. (ed.), Modern Logic: A Survey, Dordrecht: D. Reidel, 1980,399–423.

[252] von Wright, G. H., “Norms, Truth and Logic”, in: von Wright, G. H.(ed.), Practical Reason: Philosophical Papers vol. I, Oxford: Blackwell,1983, 130–209.

[253] von Wright, G. H., “Bedingungsnormen, ein Prufstein fur die Normen-logik”, in: Krawietz, W., Schelsky, H., Weinberger, O. and Winkler,G. (eds.), Theorie der Normen, Berlin: Duncker & Humblot, 1984,447–456.

[254] von Wright, G. H., “Is and Ought”, in: Bulygin, E., Gardies, J.-L. andNiiniluoto, I. (eds.), Man, Law and Modern Forms of Life, Dordrecht:D. Reidel, 1985, 263–281.

[255] von Wright, G. H., “Is there a Logic of Norms?”, Ratio Juris, 4, 1991,265–283.

[256] von Wright, G. H., “A Pilgrim’s Progress”, in: von Wright, G. H. (ed.),The Tree of Knowledge and Other Essays, Leiden: Brill, 1993, 103–113.

BIBLIOGRAPHY 141

[257] Yoshino, H., “Uber die Notwendigkeit einer besonderen Normenlogikals Methode der juristischen Logik”, in: Klug, U. (ed.), Gesetzgebungs-theorie, Juristische Logik, Zivil- und Prozessrecht. Gedachtnisschriftfur Jurgen Rodig, Berlin: Springer, 1978, 141–161.

[258] Ziemba, Z., “Deontic Syllogistics”, Studia Logica, 28, 1971, 139–159.

[259] Ziemba, Z., “Deontic Logic”, 1976, Appendix in [260], pp. 360–430.

[260] Ziembinski, Z., Practical Logic, Dordrecht: Reidel, 1976.

Appendix: Research Papers

“Sets, Sentences, and Some Logics about Imperatives”, Fundamenta Infor-maticae, 48, 2001, 205–226.

“Problems and Results for Logics about Imperatives”, Journal of AppliedLogic, 2, 2004, 39-61.

“Conflicting Imperatives and Dyadic Deontic Logic”, Journal of AppliedLogic, 3, 2005, 484–511.

“Deontic Logics for Prioritized Imperatives”, Artificial Intelligence and Law,14, 2006, 1–34.

“Prioritized Conditional Imperatives: Problems and a New Proposal”, Au-tonomous Agents and Multi-Agent Systems, 2007, demnachst. Online verfug-bar unter http://www.springerlink.com/content/p253015562l43212.

142

http://www.springerlink.com/content/p253015562l43212

APPENDIX: RESEARCH PAPERS 143

Fundamenta Informaticae 48 (2001) 205–226 205

IOS Press

Sets, Sentences, and Some Logics about Imperatives�

Jorg Hansen�

Institut fur Philosophie

Universitat Leipzig

Leipzig, Germany

[email protected]

Abstract. Though deontic logic is regarded as the logic of normative reasoning, norms – as entitieslacking truth values – are usually represented neither in its language nor its semantics. Limitingourselves to unconditional imperatives, we propose a concept for their semantic representation andshow that existing systems of monadic and dyadic deontic logic can be reconstructed accordingly.

Keywords: deontic logic, logic of imperatives

1. Norms and Deontic Logic

Deontic logic, so the definition runs, is the logical study ofthe normative use of language. Its subjectmatter is normative concepts, notably those ofobligation, prohibition, andpermission. To this effect,the systems of deontic logic employ modal operators� , � , and � , where �� is read as ‘it ought to bethat � ’ or ‘it is obligatory that � is done’, depending on language and the nature of� . While normativeconcepts are its subject, it is quite difficult to see how deontic logic relates tonorms. Imperative orpermissive expressions such as “John, leave the room!” and “Mary, you may enter now” do not describe,but demand or allow a behaviour on the part of of John and Mary.Being non-descriptive, they cannotmeaningfully be termed true or false. Lacking truth values,these expressions cannot – in the usual sense– be premise or conclusion in an inference, be termed consistent or contradictory, or be compounded by�The ideas of this paper were motivated by discussions at the� EON ’98 conference. The analysis of dyadic deontic logics

considerably gained from Wlodek Rabinowicz’ critical remarks on an earlier version of this paper, and the paper entirelyprofited from Leon van der Torre’s constant encouragement and well-placed criticism. I am obliged to an anonymous refereefor many helpful comments.Address for correspondence: Institut fur Philosophie, Universitat Leipzig, Leipzig, Germany


206 J. Hansen / Some Logics about Imperatives

truth-functional operators. Hence it seems there cannot bea logic of norms – this is Jørgensen’s dilemma[28].

Though norms are neither true nor false, one may state thataccording to the norms, something oughtto be (be done) or is permitted: the statements “John ought toleave the room”, “Mary is permitted to en-ter”, are then true or false descriptions of the normative situation. Such statements are sometimes callednormative statements, as distinguished from norms. To express principles such as

(Conj) ��

with Boolean operators having truth-functional meaning atall places, deontic logic has resorted to in-terpreting its formulas�� , � � , � � as representing such normative statements.1 A possible logic ofnormative statements may then reflect logical properties ofunderlying norms – thus logic may have a“wider reach than truth”, as von Wright [54] famously stated.

Norms, for the reasons given, are absent in thelanguage of deontic logic. But since the truth ofnormative statements depends on a normative situation, it seems that norms may be represented in alogicalsemantics that models such truth or falsity. Standard deontic semantics evaluates deontic formulaswith respect to sets of worlds, in which some are ‘ideal’ or ‘better’ than others. Deviating from thisstandard approach, a semantics that seeks to represent norms has been proposed by several authors,namely Erik Stenius, Stig Kanger, Carlos Alchourron and Eugenio Bulygin, and Bas van Fraassen.

Stenius [47] uses a set� of prescriptively interpreted norm sentences, each norm sentence having theform �� , and � being a Boolean formula. The following principle restricts� :

(r-St) if � �� and ��"!��$#��%# ��!�� , then ��&!'� �)(+*-, �.�The truth of descriptively interpreted sentences�� is then defined as

(td-St) �� is true if and only if (iff) ��/!'�Obviously, � is a normal modal operator. Note, however, that Stenius’ closure principle yields Ross’paradox [44], where the imperative ‘Post the letter or burn it!’ is ‘derived’ from the imperative ‘Postthe letter!’: even the forbidden will satisfy a norm sentence in any set� that conforms to (r-St). Also,it is quite a matter of dispute whether norms ‘contain’ indicatives2, and Stenius does in fact define� as‘sentence radical’, so the formulation of his principle canbe criticized, cf. Makinson [34]. – Kanger[29] and Alchourron/Bulygin [2] instead use a set of descriptive sentences, where Kanger thinks of thesesentences as representing a ‘welfare program for the universe of discourse’, and to Alchourron/Bulyginthe sentences in their set0 are the ‘sentences commanded by’ an authority. Their truth definition for ��then reads

(td-AB) �� is true iff �/!'1 (2� 0��1 (2� 0�� is the consequential closure of0 . Again, � is normal. Ross’ paradox can be argued not toarise: �3!/0 yields �

� �54+�� for any ��# but � does not necessarily succeed in satisfying a normsince 0 is not closed under consequence. Yet Kanger and Alchourron/Bulygin make it appear as ifnorms can somehow be replaced by descriptive sentences in normative reasoning – an approach termed‘reductionist’ by some philosophers (cf. Moutafakis [37],Hamblin [17]). But sentences are true or false,

1Cf.Føllesdal/Hilpinen [16]. This already holds for Mally’sdeontic logic [35], cf. p. 12, where he writes that to interpret‘! 6 ’as the theoretical statement ‘it is a fact that6 ought to be’ is the standpoint of (his) theory that describesobligating norms andtheir laws.2Some believe imperatives ‘contain’ indicatives as their ‘propositional core’ (Sosa [45], also cf. Alchourron/Bulygin [3]),whereas others deny any straight-forward grammatical relation and more cautiously speak of a ‘thematically parallel sentence’(Opałek [39] p. 35).


J. Hansen / Some Logics about Imperatives 207

norms are not, so semantics that claims to be about norms should describe the gap, and explain how itcan be bridged. And certainly norms have their special properties: the implication that any feature onemay want to semantically model – time of promulgation, authority, subject, hierarchical position, etc. –must relate to a sentence which is but one of these features, not the norm itself, seems hardly acceptable.

Van Fraassen [50] has a set of ‘imperatives in force’, where to each imperative�

there correspondsa class of possible outcomes

��in which the imperative is satisfied. The truth of�� is evaluated with

respect to single imperatives, where at first the truth definition reads (� �� is the set of outcomes where� is true, and� the set of imperatives in force):

(td-vF1) �� is true iff there is an� !�� such that

�� .��Van Fraassen then introduces the notion of a ‘score’ relative to an outcome and the set of imperatives� , where the score of the outcome is the subset of imperatives it satisfies. His final truth definition reads

(td-vF2) �� is true iff there is a�!��.�� such that for all � !�� where �� is the score of the outcome among the set of imperatives� . Van Fraassen’s suggestivepaper extends this analysis to conditional imperatives, where these relate to sentences that describe theopportunity for satisfying the imperative. The idea of a ‘score’ has been further examined by Horty [26],also cf. [25] p. 284, and related research on the subject of conditional imperatives was carried out byMakinson [34].

According to Makinson [34], the “fundamental problem” of deontic logic is to reconstruct it suchthat it is in accord with the position that norms are devoid oftruth values. I propose a semantic plat-form, related to the above semantics, on which such reconstructions can be carried out. The paper isconfined to unconditional obligating norms (imperatives, commands), and will not treat explicit permis-sions, derogations, general norms, or other types. The basic semantics is given in section 2. In section3 we provide reconstructions for systems of monadic deonticlogic. In section 4 we consider the ‘reduc-tion’ of deontic logic to alethic modal logic by Anderson [7]and Kanger [29], reconstructing it such thatthe Andersonian-Kangerian constants have definite meanings. Since the late Fifties, deontic logics haveemployed dyadic operators to formalize conditional obligations. In section 5 we propose reconstructionsfor well-known systems of dyadic deontic logics. The results may be viewed as justifying and explainingthe view that deontic logic is a logic of reasoning about, if not of, norms.

2. Imperative Structures

2.1. The Basic Concept

What is needed to model imperatives in a logical semantics? Imperatives have a source, and a recipient,and some authority must be assumed on the part of the source, of which the recipient is a subject (cf.Rescher [43] ch. 2). The decisive component of an imperativeis held to be its ‘theme of demand’: astate the authority wants to see realized, or an action the subject ought to perform. If this state is realizedor the action performed, then the imperative is satisfied. Wecan identify this state or action by using adescriptive sentence which corresponds to the imperative and is true just when the imperative is satisfied.This sentence will usually be rather complex, e.g. it must spell out whether chance fulfilment or anattempt suffices, and special attention must be paid to seemingly ‘ordinary’ connectives and quantifiers.Hence transforming the exact words of the imperative into indicative mood is just an initial approach.That there nevertheless is such a correspondence between imperatives and indicatives is even accepted



by the ‘anti-reductionists’ (cf. Weinberger [60] p. 69, Moutafakis [37] T3 pp. 155-156, Hamblin [17]pp. 112, 139/140, 152-153), so we may formulate the following postulate:

(W) To each imperative there corresponds a descriptive sentence that is true if the imperative is satisfied,and false if it is not-satisfied (violated).

Let an imperative structure�

therefore be a pair� � #�� , where� is a (possibly empty) set of objects;they are meant to be imperatives. We make no assumptions about the nature of� : it may be the setof imperatives from a specific source, directed towards a certain subject, belonging to a system, beingvalid or in force at a certain time, or derivable by some here unspecified procedure from some otherset, e.g. as individual obligations from more general norms. The function � � � �� associates asentence in the language�� of a basic logicBL with each imperative in� . These are meant to be thedescriptive sentences that ‘correspond’ to imperatives. In the reconstructions,BL will be a propositionallogic defined below. ButBL may be any logic that accommodates concepts oftruth, semanticvalidityof a sentence� (we write � �� ), and satisfiability, so we could assumeBL to be a logic of actpropositions (cf. Alchourron/Bulygin [3]) or a logic thatrepresents time and intentionality. This is ourbasic platform to develop and reconstruct specific deontic logics. However, this platform is open to, andrequires, extensions and restrictions to model further qualities of imperatives.

2.2. Semantic Restrictions

The norm-theoretical assumptions of our basic concept are very poor: all we suppose is that it makessense to treat imperatives as members of a set, to which sentences that describe the circumstances inwhich an imperative is satisfied can be related. Further assumptions can be made explicit by addingsemantic restrictions to the basic concept. Consider e.g.:

(r-1) For all� !�� : � ��

(r-2) � � � � is �� -satisfiable

(r-3) For all� !�� : � ��

(r-4) If � � � �"� �� then there exists an� !�� such that� � � � �

Someone taking the view that imperatives express a will, andthat a will cannot be directed towardsthings that are logically impossible, and so imperatives that demand the impossible cannot exist (cf.Kelsen [31] p. 171), may want to restrict the semantics of imperative structures by (r-1). – Kelsen[30] p. 136 argued that conflicting norms cannot coexist within the same system. Then (r-2) might beadequate, demanding thatall imperatives must be satisfiablecollectively.3 – It has often been assumedthat the nature of imperatives is to direct, to serve as a means tochangepresent facts or prevent unwantedchanges taking place (cf. Jørgensen [28]). (r-3) might thenserve to exclude the case that an imperativedemands what is true on logical grounds alone (cf. Weinberger [59] p. 121, Alchourron/Bulygin [4] p.284). – Alchourron/Bulygin [4] also believe that one cannot command without implicitly commandingall logical consequences of the commands’ contents, and then (r-4) might be accepted. We have alreadynoted that Ross’ paradox is connected with such closure conditions.

The effect of employing semantic restrictions is that imperative structures that do not conform tothese are excluded from analysis. For a general exclusion the grounds must be very good indeed, and

3 �� is the set of values of� to any member of�� .



though the above may appear reasonable, we hesitate to integrate them into the basic concept. Still,restrictions might be useful if the behaviour of specific imperative structures that satisfy these is to beexamined. In this sense, restrictions are a primary tool to reconstruct specific deontic logics on the basisof our concept.

2.3. Truth Definitions

We do not predetermine how deontic operators should be defined in a deontic logic that makes use of theabove semantics. If we consider deontic sentences�� , �� as giving information about a specific set ofimperatives (or norms), then the truth definitions will depend on the questions we want to ask about thisset. For�� consider e.g.:

(td-1) �� is true iff there is an� !�� such that� � � ��

(td-2) �� is true iff there is an� !�� such that� �� 3�

(td-3) �� is true iff there is an� !�� such that� �� 3�

(td-4) �� is true iff � � � � � ��According to (td-1)�� is true iff � is the descriptive sentence correlated to some imperative.An � suchthat �� is defined true by (td-1) might be considered an answer to the question ‘What sentence (literally,grammatically) corresponds to a given imperative?’.4 – If the primary interest is not in the wording,but the �� -logical meaning we assume of the imperative-correlated sentences, we may instead employ(td-2). Each� of a sentence�� defined true by this definition may then be considered an answer to thequestion ‘What describes a state logically necessary and sufficient to satisfy one imperative?’. – (td-3) issimilar to van Fraassen’s truth definition in [50]. Each� of a sentence�� defined true by this definitionanswers the question ‘What is necessary to satisfy one imperative?’. – A subject may be interested notso much insingle imperatives, but what must be achieved to haveall imperatives satisfied. Then, (td-4), similar to the truth definition of Alchourron/Bulygin in [2], defines each� of a sentence�� to bean answer to the question ‘What is logically necessary to satisfy all imperatives?’. We are at liberty tomix the above, and any other deontic operators we choose to define, and can then state truths such as

��

��

�� .

2.4. Logical Relations Between Imperatives?

The expression ‘it ought to be that ...’ is ambiguous: it may be used to describe what ought to be accordingto some norm, but also to prescribe, tocreate an obligation to this effect. Jørgensen’s dilemma withstand-ing, authors have wondered whether logical relations can beconstructed not just between descriptive, butalso between prescriptive oughts (cf.Aqvist [10], Alchourron/Martino [5], Alchourron/Bulygin [4]).

In a strict sense, a logicof imperatives is not possible here: To write ‘� �� ’ or ‘

� �+� � � ’ ismeaningless if the operators ‘� ’ or ‘ � ’ are to be truth-functional, since we have not assumed truthvalues of these objects – this is our tribute to Jørgensen’s dilemma. But while the truth of a sentence

�� depends on the existence of some imperative in the set� , the other side of the coin is that�� canbe made true by putting new imperatives into this set. This property may be used to define relations

4Except for imperatives with identical corresponding sentences, the set of (td-1)-true -sentences will mirror the set of imper-atives. So (td-1) comes close to an operator that allows us to directly talk about imperatives (cf. van Fraassen [50] p. 18 for theneed for such operators).



between imperativeson topof a logic about imperatives:5 Let� � � #�� be any imperative structure,

and let� �� be shorthand for� �� #�� # � � � # so

� �� is the imperative structure we obtainfrom adding a new imperative

��to � and expanding� such that� � �� . Let

� � be the name ofthis new imperative.

� � might be called an ‘imperative-candidate’ with respect to�

. Suppose we have alogic about imperatives that defines deontic operators with respect to an imperativestructure

�. Since

the semantics of may employ semantic restrictions, we call an imperative structure that conforms withthese a -structure. We now define (where�5! � �� ,

�is a -structure, and�� denotes the set of

-sentences defined true with respect to�

):� � is -covered by

�iff

� �� is a -structure, and�� is -rejected by�

iff� �� is not a -structure� � is -independent from

�iff

� �� is a -structure, and�� If

� � is -covered by�

, then – from the point of someone using as a tool for the analysis of a set ofimperatives – it does not matter whether this imperative-candidate is added to the set or not, since the setof -truths does not change.6 On the other hand, someone who believes that imperative structures shouldconform to the semantic restrictions of could not tolerate the addition of an imperative-candidate

� � ifit is -rejected by

�, since thereby these restrictions would be violated. Finally if

� � is -independentfrom

�, then whether it should be added to

�or not seems not disputable on -logical grounds alone.

We cannot further explore this topic here. But it seems important that the above semantics may notonly be used to construct descriptively interpreted deontic operators, but also to explain (meta-)logicalrelations between prescriptions.

Ourbasic logicwill be propositional logic given in the usual fashion: Thealphabethas a setPropofproposition letters ‘�$� ’ # ‘ � � ’ #�� , logical signs ‘� ’,‘ � ’, ‘ 4 ’, ‘ � ’, ‘ � ’, and brackets ‘

�’, ‘ � ’. All proposition

letters arePL-sentences, and if � and � arePL-sentences, so are� �� (negation),� � �� (conjunction),� � 4�� (disjunction), � � � �� (conditional), and� � � �� (biconditional), and nothing else. Outerbrackets and ‘corner quotes’ for ‘mixed’ expressions are mostly omitted. Thelanguage�� is the set ofPL-sentences. – Avaluation � �� !� , ##" assigns a truth value to each� ! �� . B is the set ofall such valuations.Truth of a PL-sentence� at a valuation (we write +� �� is defined recursively,starting with

�� $� iff � �� %"�� iff � is not true

and as usual for the remaining connectives.�.�� is the set of all !'& at which � is true. Validity ofa sentence� (we write � ��'�� , tautologies, andcontradictionsare defined as usual (( designates anarbitrary tautology and) an arbitrary contradiction).� is contingentiff it is neither a tautology nor acontradiction. A set ofPL-sentences*+ +�%� �%# � � #�� is PL-satisfiableiff there is a such that for any��,�!-* , � ��, , and * �� -impliesa sentence�/! �"�� iff for every that satisfies* , � � (we then

5The idea of constructing normative logics as built on top of logics of normative statements appears in Alchourron/Bulygin[3] p. 463, whereas in Alchourron/Bulygin [4] p. 285 the relation is vice versa. The idea of defining the normative status ofa considered norm by testing what happens to a system if this norm is added is the basis of von Wright’s logic of normativeentailment (cf. [57], [58]).6The term ‘coverage’ is borrowed from Rescher [43] ch. 6. The notion is akin to that of ‘acceptance’ used in default logic, cf.van der Torre [48].



write * � �� ). – We suppose a sound and completeaxiomatic systemfor PL.� �� means that� is

provable in this system.

3. Monadic Deontic Logic

To reconstruct monadic deontic logics, we employ two operators � and � on sentences of our basic logicPL. We intend to give the following meaning to the operator� :

�� is true iff � is necessary to satisfy the imperatives.

Since we have not introduced permissive norms, to give�� the usual reading as ‘it is permitted that� ’is somewhat ambiguous. What we intend to express is that the state described by� does notopposethesatisfaction of any imperative, and so define:

�� is true iff the imperatives and� are co-satisfiable.

Both definitions refer to a ‘satisfaction’ of imperatives; in the second the terms of ‘ordinary’ satisfiabilityand ‘satisfaction of an imperative’ even appear mixed. Thisseems more that a semantic coincidence ifwe accept postulate (W), for then we know a set of imperativesto be ‘satisfiable’ iff the set of correlatedsentences is satisfiable, and can rephrase the definitions accordingly. – Both definitions also refer to‘the imperatives’. However, the question is what to do if there are none. Instead of rejoicing as anormal person would, most deontic logicians churlishly exclude this possibility by demanding that atleast something must be ‘obligatory’. We want to cover the case where the imperative set is empty, andthen nothing naturally appears to be ‘necessary to satisfy the imperatives’ or as ‘opposing an imperative’ssatisfaction’. So the truth definitions then have�� false, and�� true, for all sentences� .7

Thelanguage of analysiswill be �� , where thealphabetis like that of �� , except that additionallywe have ‘� ’ and ‘ � ’. Then �� is the smallest set such that

(a) If �5! � �� then �� and ��/! ��(b) If � # �&! �� # then so are�� # � �-��.# � �-4��.# � � �3��.# and

� � �3��The truth definitionsare relative to an imperative structure

� � � #�� and are just like those forPL-sentences, except that their first clause is replaced by thefollowing two clauses:� � �� iff �� and � � � � � �� iff �� or � � � � ��%� is �� -satisfiable

We use the followingrestrictions:

(R1) � � � � is �� -satisfiable

(R2) � � ��(R1) excludes that the imperatives are, for reasons of the logic of their correlated sentences, not or notall satisfiable. Otherwise we may have the result that everything is ‘obligatory’ – everything is necessaryto satisfy the imperatives though nothing can – and nothing is ‘permitted’ – but it is not particular statesbut the imperatives themselves that oppose their satisfaction. (R2) may then be used to rule out the casethat the imperative set is empty.

Consider the following axiom-schemes and rules:

7Beyond intuition, to have such a definition may be important in a more dynamic setting where we successively removeimperatives invariably satisfied or violated from a given set. With �� we then have a formula true when an agent is ‘done’.



(Def) �� (Conj) �

� � �� -� ��

(D) ��((N) ��((Ext) If

� �� 3� then� � ��

Let be any of the names in the left column of table 1:

T A B L E 1

Name System Semantic Restrictions� � �(Def), (Conj), (Ext) None� � (Def), (Conj), (Ext), (D) (R1)

� � � �(Def), (Conj), (Ext), (N) (R2)

� � � (Def), (Conj), (Ext), (D), (N) (R1), (R2)

Theaxiomatic system is the smallest set ofDL-sentences that contains all� �� -instances ofPL-theorems as well as allPL-instances of the axiom-schemes indicated to the right of the system name inthe center column and is closed undermodus ponensand any indicated rule. If�&! , we write

� ��and call � provableand �� refutablein . If

� � � and�

satisfies the restrictions for indicated inthe right column of Table 1 and in the same row as the system name, we call � -satisfiable. If

� � �for all such

�, we call � -valid and write � �� .

DL is von Wright’s original deontic logic in [52],DL�

is DL without the claim that obligations mustbe consistent,SDL is the so-called standard deontic logic, andSDL

�the system that demands something

be obligatory, but not that obligations are consistent (thename is adopted from al-Hibri [1]). In allsystems the following derived rule holds:

(Mon) If� �� 3� then

��

Theorem 3.1. All systemsDL�

, DL, SDL�

, andSDLare sound and complete.

Proof: Soundnessis trivial with respect to truth definitions and restrictions. To prove (weak)complete-ness, for any

� � -sentence� such that�� we construct an -structure�

such that� � �� : We

assume� � � so �� is not refutable in . We form a disjunctive normal form of�� and eliminateall negation signs in front of deontic operators using (Def). The result is a disjunction of conjunctions,where each conjunct is a sentence�� or �� . One disjunct must then be not refutable in . Let �be that disjunct. Let� be the set of�� -sentences� such that�� appears in� , and � be the set of

�� -sentences� such that�� appears in� . Let

� � � � ��( if for all �&!��# � � is �� -satisfiable� otherwise

and � � � � � be an identity function on� . This is our desired�

. We show that� � �� by

demonstrating that any conjunct of� is true: Let �� be a conjunction member of� . Then � !�� #



so � � � and since� � � � � , � � � � � �� , so �� is true. Let �� be a conjunction member of� . Now suppose�� is false, i.e. � � � and � � � � � � � is not �� -satisfiable. So�+� �%� is not�� -satisfiable, and there are�%� � #�� # � � � � such that

� �� . By (Mon), (Conj)and (Def) we have

� � �� '� � � � �� . So � , containing the conjuncts�� #�� # �� # �� ,is – contrary to our assumption – refutable in .

� � � �� will be �� -satisfiable (R1) if contains (D): If � is not �� -satisfiable, then there are�%� ��#�� # � � �� such that

� �� and therefore we have� � � � �� by

(D), (Def), (Ext) and (Conj), so� is refutable in again. � will be non-empty (R2) if contains (N):Suppose�' � . Then � � and for one� ! � , �%� is not �� -satisfiable, so

� �� . But then� � � �� by (N), (Def) and (Ext), so again� is refutable in .��

Remark 3.2. Suppose that in the truth definitions we drop the clause regarding empty� :� � �� iff � � � �"� �� iff � � � � ��%� is �� -satisfiable

ThenSDL�

is sound and complete with respect to semantics that employsthese definitions, andSDL issound and complete for such semantics if restricted by (R1).(Validity of ��( is trivial with respect to thenew definitions. To adjust the above completeness proof, define �� .)

4. Anderson’s and Kanger’s Reduction

We are at liberty tosort given imperatives, according to some quality of rank or importance, and alsoaccording to what the world is like, whether it leaves them satisfied or violated. The semantics ofPLmakes use of a valuation to logically model truth and falsity of sentences describing the world. Wedefine thesatisfaction set �� to be the set of all imperatives of� of which the sentences correlatedby � are true at , and likewise theviolation set �� as the set of all imperatives of� of which thesentences correlated by� are false at :

�� !�� !��

If the imperative structure and the valuation are clear fromthe context, we will just write� and � . Since�� accepts theprinciple of bivalence we have�� and �� .

Anderson’s [7] and Kanger’s [29] idea was to define deontic operators by a combination of alethicoperators and one constant. Unhappily, Anderson [7] and Anderson/Moore [9] explained their constant‘S’ as representing a ‘sanction’ or ‘bad thing’, and defined the statement�� as meaning that not doing�necessitates the sanction. Lemmon and Nowell-Smith [32] objected, since “in this wicked world it is notalways true that when (say) a robbery is committed someone suffers a sanction” and accused Andersonof the naturalistic fallacy. Interestingly, in answering these accusations Anderson [8] argued that if onefails to fulfill ones obligations, thatis the ‘bad thing’, no ‘sanction’ being presupposed. But then we cangive Anderson’s constant a definite meaning: the bad thing isthat � � � , since then an imperative hasbeen violated. – Kanger [29] introduced a constant ‘Q’ standing for ‘what morality prescribes’. Thoughthis seems just as dubious at first, Kanger likened ‘morality’ to a ‘welfare program for the universe ofdiscourse’, represented by a set� of propositions, and defined Q to be true iff all propositionsof thisprogram are true.�� is then defined true iff the truth of Q necessitates the truth of � . Now we have a set



of sentences that might be understood as a ‘welfare program’: the set of sentences� � � � that correspondto the imperatives. We therefore interpret Kanger’s constant as meaning that all sentences in� � � � aretrue, hence by (W) every imperative is satisfied, so� � .

Here we will havefour constants, of which only two can be defined. The constants of Andersonand Kanger neither alone nor combined suffice for logics properly parallel to

� � �and

� � , or axiomsexpressing the principle of bivalence which the notions of satisfaction and violation inherited from ourbasic logic. Thelanguage of analysiswill be � �� , where thealphabetis like that of �� , except thatadditionally we have operators ‘� ’ and ‘ � ’ and constants ‘� ’, ‘ � ’, ‘ � ’, and ‘� ’. The definition of thesetof sentencesdiffers from that of�� only in the respect that now also� , � , � , and � are sentences, andif � is an � -sentence, then also� � and �� are. Note that� will represent Anderson’s constant, and� is our interpretation of Kanger’s Q. Thetruth definitionsare relative to a valuation and an imperativestructure

� � � #�� , and are as for�� -sentences, except for these additional clauses:� #�� iff for all :

� #�� (logical necessity)� #�� iff for some :� #�� (logical possibility)� #�� iff � � (every imperative is satisfied)� #�� iff � � �� (some imperative is satisfied)� #�� iff � � (every imperative is violated)� #�� iff � � �� (some imperative is violated)


(Df1) � � ��(Df2) � � ��(Biv1)

��

(Biv2)��

(AK-D) ��(AK-N) �� 4��(Nec) If

� �� then� ��


T A B L E 2

Name System Semantic Restrictions

� �(Df1), (Df2), (Biv1), (Biv2), (Nec) None

� (Df1), (Df2), (Biv1), (Biv2), (Nec), (AK-D) (R1)

�� (Df1), (Df2), (Biv1), (Biv2), (Nec), (AK-N) (R2)

�� (Df1), (Df2), (Biv1), (Biv2), (Nec), (AK-D), (AK-N) (R1), (R2)

We define theaxiomatic system to be the smallest set ofAK-sentences that contains allAK-instances ofPL-theorems and the modal logicS5 as well as all axioms indicated to the right of thesystem name in the center column, and is closed undermodus ponensand any indicated rule. If� ! we write

� � � and call � provable in . -satisfiability and -validity of AK-sentences� are thendefined as before.



5. Dyadic Deontic Logic

The aim of dyadic deontic logic, that formalizes “If1 is the case, it ought to be that� is the case”as �

� � � 1 � , was to cope with deficiencies of monadic deontic logic with respect to the representationof conditional obligations. First, Prior’sparadox of derived obligation[41] showed�

� 1 � �� to beinadequate for “contrary-to-duty” imperatives expressing a secondary duty in case a primary norm isviolated: From � ��1 we derive �

� 1 � �� with (Mon) for any � (“be hanged for a sheep as for alamb”). Then, Chisholm [14] showed that if we formalize contrary-to-duty obligations as1 � �� ,the set� � 1 #��1 � �� # �

� 1 � ��.# ��1 , expressing a primary obligation, a contrary-to-duty im-perative, an “ordinary” conditional imperative, and the fact that the primary obligation was violated, isinconsistent in standard deontic logic, and otherwise formalized mutually dependent, though intuitivelythey should be neither.

To solve these problems, von Wright [53] introduced dyadic deontic operators, meaning to reflect aconditional structure of the operators’ scope. To reconstruct dyadic deontic logics, we add an auxiliarysign ‘/’ to the alphabet and define thelanguageof dyadic deontic logic� � �� to be like �� # exceptthat in the definition of sentences clause (a) now reads

(a) If � # 1 ! � �� , then �� 1 � and �

� � � 1 � ! �� Semantics for dyadic deontic logics is usually based on a preference relation� (‘at least as good as’)

on the set of all Boolean valuationsB (Hansson [19]) or a set of possible worldsW (Lewis [33], Aqvist[12]). According to the Hansson-type definition,�

� � � 1 � is true iff the best1 -worlds are� -worlds,whereas the Lewis-type truth definition demands that there is a

� 1 �� -world such that no� 1-� �� -

world is at least as good.9 Hansson motivated his semantics by the observation that while one shouldalways try to make the real world an ideal world, in circumstances in which somebody neverthelessperforms a forbidden act ideal worlds may be excluded, but some achievable worlds may still be betterthan others. One should then ‘make the best out of sad circumstances’, i.e. realize the best worlds left. Inour semantic framework, the ‘best’ state obviously is that all imperatives are satisfied. However, supposethat in circumstances1 some imperatives cannot be satisfied. It then seems reasonable to demand thatall other imperativesstill satisfiable in circumstances1 be satisfied, and to ‘permit’ only what is co-satisfiable in1 with the imperatives left. The truth of dyadic-deontic sentences�

� � � 1 � and �� 1 �

will be defined with respect to the subset of imperatives in� not yet violated in situation1 – eachsentence� such that�

� � � 1 � is true answers the question: “What is necessary to satisfy all imperativesthat are still satisfiable, given circumstances1 ?” Note that we do not presuppose any special conditionalstructure of the imperatives.10

For thesemantics, let � be a function that removes those sentences from a set* which (individually)imply the truth of a sentence1 :

* � 15 *��'�%�5!�* � � �� 1

Let� � � #�� be any imperative structure.� � � � � ��1 is then the set of just those sentences in� � � �

that are individually satisfiable with1 . We give the followingtruth conditions:9Lewis-type truth definitions have become more standardly employed with Horty ([25], [27]) to define deontic operators withregard to (general, dominance, maximin) orderings instit-models.10So in the present setting, the solution to Chisholm’s paradox is to let �� 6 �� 6 �� In the originalparadox, the case that the man can still go (� ), but has already told he won’t (6 ), is not accounted for. If the priority is to go,we can change the above set into� � �� 6 �� 6 ��



� � � � � �� 1 � iff � � � � � ��1 � �� and � � � � � ��1 ��1 � �� 1 � iff � � � � � ��1 �� or � � � � � ��1 �� 1 ��%� is �� -satisfiable

Before we take a closer look at� # note that the truth definitions are strict parallels of thosein section3. Concerning their first clause, I believe that when all imperative correlated sentences are rejected by� � � � � ��1 , i.e. all imperatives are violated in circumstances1 , then there is nothing left to be done. Tocall something obligatory or not permitted in such a situation seems as pointless as when there are noimperatives at all: E.g. if I’m to see to it that the two witnesses appear at the wedding, and one won’tcome, I might just as well tell the other one it’s off. However, we shall also consider truth conditionswhere this clause is dropped:� � * � � � �

� � � 1 � iff � � � � � ��1 ��1 � �� * � � � �

� � � 1 � iff � � � � � ��1 ��1 ��%� is �� -satisfiable

We will then see that whereas conditions (a) and (b) produce Lewis-type dyadic logics, semantics thatmakes use of conditions (a*) and (b*) will relate to Hansson-type systems.

Obviously, � is a contraction function, and the truth of�� 1 � depends on whether� is implied

by the corresponding revision of� � � � with 1 . Contraction functions serve to remove sentences fromtheory bases if a certain sentence is desired not to be in the theory (cf. [6], [20]). The problem isthat the process is ambiguous: Consider a basis that contains the individual sentences� , �� Then for acontraction with

� � �� it suffices to remove just one of the two, so there is a choice. ‘Absolutely safe’full meet contraction removes both, while partial meet contraction employs a choice function to pick thesentences to be removed, leaving us with several possibly resulting contracted theory bases. Now� is‘naive’ in the sense that it avoids ambiguity, but at the price of producing no contraction in the classicalsense: If we

(-contract* with a non-tautological sentence, this sentence may still be a consequence of

the contraction (see the above case). So ‘naive’ contraction is really the opposite of ‘absolutely safe’ fullmeet contraction. Why use this weird function instead of some other available concept?

My position is that we should not exclude dilemmas created byimperatives that conflict for somesituation. The logical evidence of there being a dilemma is the truth of both�

� � � 1 � , �� 1 � . A

situation where��1 was demanded initially, but1 is now true, is a state of violation, not a dilemma.Such demands are disregarded by� � � � � ��1 � In a dilemma, we also know that invariably some demandswill be violated, but which ones is still up to us. Then saying‘You should do� ’ and also ‘You shoulddo �� ’ may reflect a tenable position (this van Fraassen [50] argued for in detail, also cf. von Wright[55], [56], Hamblin [17] ch. 5). So some logic should be able to state the dilemma, and we should notbeforehand smooth out the situation by removing one of the still satisfiable demands, or change them intoa demand for, say,

� � 4 �� . Also, in a dyadic setting the existence of a dilemma for a situation 1 is notentirely devastating, even if� is constructed as a regular operator (supporting agglomeration of contents)so subsequently�

� ) � 1 � is true: There might neither be a dilemma prior to the truth of1 , nor mighta dilemma ever arise again once one of the demands is violated, as it must be. So dilemma-permittinglogics with regular operators may still be of interest.11

We use the followingrestrictions on�

, which parallel (R1), (R2) employed above (p. 211):

11To avoid the truth of � �� while accepting the possible truth of both � 6�� , � � 6�� van Fraassen [50] proposeddefining deontic operators with regard to maximal sets of collectively satisfiable imperatives. This idea was explored indetailby Horty [26], and implies giving up agglomeration for some cases, though not all.



(DR1) If � �� 1 then � � � � � ��1 �� 1 is �� -satisfiable (no predicaments)

(DR2) If � �� 1 then � � � � � ��1 � �� (no outlaws)

(DR1) expresses the assumption that all possible conflicts (dilemmas, predicaments) are resolved in� . If the circumstances can be realized at all, the imperatives that are then ‘left’ must be ‘co-satisfiable’with them: Though in possible circumstances1 e.g. two imperatives

� � # � � !�� with � � � � � &1 � �and � � � � �2 /1/� �� might be satisfiable with1 individually, they cannotboth be satisfied any longerand therefore are not both allowed in the set of imperatives.Note that if (DR1) applies, then for all� � # � � !�� either � �� or � �� .12

(DR2) demands that in all possibly realized circumstances1 there are still some unviolated impera-tives left. I call this restriction ‘No outlaws’, since it might be understood as stating that we cannot getinto a situation, however despicable, that would completely rid us of all duties. (DR2) may be motivatedby the assumption that� contains an infinite supply of imperatives for any occasion,though of courseone ‘tautological imperative’ would suffice.


(DDef) �� 1 ��

� �� 1 �(DConj) �

� �-�� 1 � � �� 1 � � �

� � � 1 � �(CExt) if

� ��15� � � �3�� then� � �

� � � 1 �� 1 �

(ExtC) if� ��15� �

then� � �

� � � 1 ��

(Up) �� 1 � � � � �

� � �3� � 1 �(Down)

�� 1 4 � ��

� 1 � � � �2� �� 1 � �

(DD�

) �� ( � )��

(DN�

) �� ( � )��

(DDR) if � �� 1 then� � �

� ( � 1 �(DNR) if � �� 1 then

� � �� ( � 1 �

(DDef) and (DConj) are the analogues of monadic (Def), (Conj). (CExt) is a contextual ‘extensionality’rule for the consequent in�

� � � 1 �.# 13 and (ExtC) an extensionality rule for circumstances. (Up) trans-fers obligations conditionally from stronger to weaker circumstances, (Down) allows for a correspondingtransfer of obligations from weaker to stronger, yet not contrary-to-duty circumstances.14 Concerning(DD

�

), (DN�

), it is obvious that axiomatic systems that contain both and(DDef), (CExt) will be incon-sistent. So we have to choose between (DD

�

), (DN�

), and the analogues of the monadic (D) and (N)axioms must then be restricted as in (DDR), (DNR).15


12I owe this observation to Leon van der Torre. So the contents ofall norms must be chained: The norm giver must, implicitlyor explicitly (e.g. by use of ‘unless-clauses’), indicate which demands should be given priority in case each is still satisfiableindividually, but not all collectively. This does not mean that in a finite code all but the strongest norms are redundant,sincesome norms may be contrary-to-duty.13Lewis [33], van Fraassen [49] employ non-contextual extensionality and an axiom to the same effect.14(Up) and (Down) are derivable in the systemsCU of Lewis [33], Spohn [46], andAqvist [11], [12], [13]. (Up) was firstemployed by Rescher [42], and (Down) was first proposed by Føllesdal/Hilpinen [16], p. 31.15Hansson [19], Spohn [46], andAqvist [11] give priority to the analogue of monadic (N) and restrict the analogue of monadic(D) to noncontradictory (possible) antecedents, whereas vanFraassen [49] and Lewis ([33] cf. p. 5) give priority to the analogueof (D), Lewis restricting his analogue of (N) similarly.



T A B L E 3

Name System Restrictions

DDL�

(DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DD�

) None

DDL (DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DD�

), (DDR) (DR1)

DSDL�

(DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DD�

), (DNR) (DR2)

DSDL (DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DD�

), (DDR), (DNR) (DR1), (DR2)

DSDL3�

(DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DN�

), (DNR) None

DSDL3 (DDef), (DConj), (CExt), (ExtC), (Up), (Down), (DN�

), (DNR), (DDR) (DR1)

We define the axiomatic system to be the smallest set ofDDL-sentences that contains allDDL-instances ofPL-theorems as well as allPL-instances of the axioms indicated to the right of the systemname in the center column, and is closed undermodus ponens and any indicated rule. If�/! we write� � � and call � provable in . For systemsDDL

�, DDL, DSDL

�, DSDL, validity and satisfiability are

defined with respect to truth conditions (a) and (b) and the indicated restrictions, and similar for systemsDSDL3

�, DSDL3 with respect to truth conditions (a*) and (b*) (the horizontal line indicates the change

in truth definitions).Neglecting differences of language,DSDL is the synoptical systemDFL of Aqvist [11], [12], [13],16

andDSDL3�

is the system of von Kutschera [51]17. DSDL3 is the system of Hansson [19] as axiomatizedby Spohn [46]. Analogues of the remaining systems seem to be missing in the literature. We give sometheorems common to all:

(CMon) if� ��15� � � �3�� then

� � �� 1 ��

� � � 1 �(T1) �

� � � 1 � � �� 3� � 1 ��

� � � 1 � � � �(T2)

�� 1 � � �

� � � 1 � � � �� 1 �

(T3)�

�� 1 � � �

� � � � � �� 1 4 � �

(T4) �� ) � 1 � � �

� ��1 � 1 4 � �(T5) �

� ) � 1 � � �� ) � 1 � � �

(T1) is the ‘down’ axiom employed e.g. by Rescher [42] andAqvist [12],18 and (T2) is an equivalent ofLewis’ [33] ‘down’ axiom (A8). In the presence of (Up) and (CExt), (T1) and (T2) are derivable from(Down), while for a derivation of (Down) from (T1) the use of (DNR), and for a derivation of (Down)from (T2) the use of (DDR) seems crucial. (T3) is Lewis’ [33] ‘upward’ axiom (A6), derivable here from(Up) and (CExt). (T4) and (T5) both derive from (Up) and will be useful in the proofs below.

Theorem 5.1. The systems� � � � # � � �"# � � � � � # � � � � are sound.

Proof: The validity of (DDef), (DConj), (CExt), and (ExtC) is trivial, and so is (DD�

) since � � � � � ( � . For the validity of (Up), first observe that* � � � � � �� * � �� * � �� Assume the truthof �

� � � 1 � � �.# so � � � � � � � 1 � � � � � and also� � � � � ��1 � � � We have � � � � � � � 1 � � � �16DFL contains the systems of Danielsson [15], van Fraassen [49], and Lewis [33].17We add (ExtC), which von Kutschera erroneously believed to bederivable.18Von Wright’s axiom (A2) in [53] is equivalent to the conjunction of (T1) and (Up), and so are von Kutschera’s (A17) in [51],and Spohn’s (P5). For a discussion of versions of (Up) and (Down)cf. [40].



��1 � � � �� # so � � � � � � � 1 � � � �-� 1 � �� # and since� � � � � � � 1 � � � � � � � � � ��1 ,from monotony of�� we obtain� � � � � ��1 �� 1 � �� 3� �

For (Down), consider its equivalent�

�� 1 � � �

� 1 � � � � � �� 154 � � � : Again observe first

that * � � � �-4�� * � ��'� * � �� Suppose�� 1 � and �

� 1 � � � are true. Either� � � � � ��15 � �Suppose� � � � � � � �� 1 is �� -satisfiable, so for all� � � �2! � � � � � � � # � � � � is satisfiable with1 ,but we assumed the contrary for all

� !�� So for �� 1 � � � to be true,� � � � � � � must be empty, but then

� � � � � � � 1 4 � �2 � so �� 1 4 � � holds trivially. Or � � � � � ��1 �� 1 �� is �� -satisfiable. If

� � � � � � � �� then � � � � � � � 1�4 � � � � � � � ��1 # and again trivially� � � � � � � 1�4 � �#� ��1 4 � � �%� is �� -satisfiable. Suppose� � � � � � � �� 1 is �� -satisfiable, so for all� � � � ! � � � � � � � # � � � �is satisfiable with1 # so � � � � � � � � � � � � � ��1 # so once more� � � � � � � 1 4 � �' � � � � � ��1 and� � � � � � � 1 4 � � �-� 1-4 � �-� � is �� -satisfiable. So in all three possible cases in which�

� � � 1 �and �

� 1 � � � are true,�� 1 4 � � is true.

For (DDR), if � �� 1 then due to (DR1)� � � � � ��1%�'� 1 �'��( is satisfiable, so the seconddisjunct of the truth definition holds for�

� ( � 1 � . Concerning (DNR), if � �� 1 then due to (DR2)� � � � � ��1 � � , so the first conjunct of the truth definition holds for�

� ( � 1 � , and the second holdstrivially since ( is always a consequence.

��

Theorem 5.2. The systemsDDL�

, DDL, DSDL�

, andDSDL are complete.

Proof: To prove (weak)completeness, we have to show that� � � � � � � for any� � � -sentence

� and any system � � � � # � � �"# � � � � � # � � � �"� We assume� �� so �� is not refutable in .We build a disjunctive normal form of�� and eliminate all negation signs in front of deontic operatorsby using (DDef). The result is a disjunction of conjunctions, where each conjunct is a sentence�

� � � 1 �or �

� � � 1 � . One disjunct must then be not refutable in . Let � be that disjunct. Let�� be the

�� -sentences that contain only proposition letters occurring in � . Let � � � �� be a set of 2��

mutuallynon-equivalent representatives of�

�� , where(

is the number of proposition letters in� . By writing PL-sentences we now mean their unique representatives in� � � �� . We construct a set

�with the following

restrictions:

(a) Any (with CExt, ExtC from� � � �� equivalently replaced) conjunct of� is in�

,

(b) for all ��# 1/! � � � �� .# either �� 1 � ! � or �

� �� 1 ��! � ,

(c)�

is -consistent.

It then suffices to prove that there is an imperative structure for that satisfies all� ! � . The prooffollows Hansson [19] and Spohn [46], i.e. we first identify the deontic basis of a sentence1 , identifysupersets� � � such that the deontic basis of

�is implied by that of1 (which reflects partly ordinary,

not contrary-to-duty, conditionality of1 ), and then see to it that the deontic basis of�

is implied by� � � � � � � � � � (so the same holds for1 ). However, unlike Hansson and Spohn, we must allow forsentences1 with an empty deontic basis (in case�

� ) � 1 ��! � ), or no deontic basis at all (in case�� ) � 1 ��! � ). So let �� be the (finite) set of sentences� such that�

� � � 1 ��! � , and �� be thecorresponding set of sentences� such that�

� � � 1 � ! �. Observe that�� .# and�� For all 1 such that�� we define thedeontic basis �� as the conjunction of all

sentences in�� : � �+ �� We have:

��- �� iff �� ) � 1 �2! � iff �� is not defined��+ �� iff �� ) � 1 �2! � iff ��



For all 1 with a defined basis,�� ! �� by (DConj), and�� 1 � by (CExt). Furthermore, from(CMon) and (b), (c), for all such1 we have:

(L1) � � � � � �.�� iff �� 1 � ! �

(L2) � � � �� .�� iff �� 1 � ! �

For each sentence� , the sentences� such that�� ! � may be ordered by extensional inclusion.

We define� � �� to be the set of all sentences maximal within that order, and prove some observations

regarding� � �� :

� � �� ! � � �� "� �

� � � �� ! � and for all 1 � � : if �� 1 � ! � then �.�� 1 ��

(L3) If �� 1 ��! � then there is a� ! � � �� such that� 1 � � ��

Proof : If 1 �! � � �� then there must be a� �� 1 # such that�

� � � � �� ! � and ��1 � � � � � � #and if

� � �! � � ��.# then likewise for some sentence� � � � � . There is only a finite number of

sentences in� � �� .# so for some such

� , there cannot be a� , � � such that�

� � � � , � ��! � ,� � ,�� , � � � and

� , � � � � , � So� ,$! � � ��.# and � 1 � � � � , � �

(L4) For all sentences� # � � �� Proof: Apparent from (L3) and the fact that�

� � � )�� ! � .

(L5) For any� such that� ! � � �� for some� , we have� � � ��

Proof : Suppose� � ! � � � �.� Then �� .# �

� � � � �.# �� ! � , so we obtain�

� � � � 4�� .# �� 4

� � � ! � from (Down) , and so�2 � � 4�� .(L6) Let � ! � � ��.#� � ! � � �� for some� # �� Then either�� or �

� �� ! � �Proof : Suppose�

� � � � � �! � . Then from �� ! � and (Down) we have�

� � � ��4 � � � ! � ,so �� 4�� # ��

(L7) If � � � �� , then there is a sentence� � �� such that�.�� and� � ��

Proof : Suppose� #�� ! � � ��.� We have�� .# �

� � � � � ! � and by (Down) also�� 4�� !�

. So � �� by definition of� � ��.� We have�

� � � �� ! � and by (CMon)�� ! � , so

with (Down) we obtain�� 4�� .� But then �� #�� )�

(L8) Let � ! � � �� for some� . Then �� )�� and if � � � �� then � � � �� )�� .

Proof : �� .# �

� ) � � � )�� "! � , so �� )�� ! � with (CMon), we have�

� � � � 4�� )�� ! �from (Down) and by definition of

� � ��.#�� 4�� )�� # so �� )�� Suppose�� )�� -� � � Then �

� ) � � � )�� .# �� )�� ! � , so by use of (Down) we obtain�

� ) � � � ! �contrary to the assumption� � � ��

(L9) For all sentences� such that� � � �� and � � � �� : � � � � �.�� Proof : From � �� and �

� � � � �� ! � we have�� ! � by use of (Up).

So �� and �.��

� � � � � � From �� .# �

� � �� ! �we have�

� � �� "! � by use of (T2). So�� and since also� � � � � �.�� weobtain ��



Definition 5.3. For the desired imperative structure� � � #�� , let

–� � � ! �� and � ! � � �� for some�/! �� ,

– � �� ! � ,– � be some proposition letter not in�

�� .

� is the set given by the following clauses, and� is again an identity function on� : 19

(i) For each� ! � ,� �� 2!�� and � �� ! �

(ii)� �� )�� !�� and

� �� )�� 2!��Lemma 5.4. (‘Coincidence Lemma’)For all � # 1 ! � � � �� iff �

� � � 1 � ! �

Proof: For each1 , either � �- �� # or ��+� �� and �� , or ��+ �� .Suppose�� . So �

� ) � 1 � ! � . We have� � 1 � � � )�� from (L6) and (L8). From (L8) itfollows that 1 logically implies � for each�"! � , and also that� � logically implies ��1 for each� ! � .So no sentence from clause (i) can be in� � ��1 � Concerning the sentences from clause (ii), by the samereasoning1 logically implies �� and also the negation of the consequent. So neither are thesetwosentences in� � ��1 , so � � ��15 �� # and both sides of the iff-clause prove false for any�5! � � � �� .

Suppose��%� � and �� Due to (L7) there is a� � 1 � ! � such that1 logically implies � � 1 � ,and ��1 � �� for otherwise� �� from (L9) and� �+ �� contrary to our assumption. Itfollows that �� 1 � �!�� 1 and �� 1 �� ! � � ��1 # so � � ��1 � ��1 � ��1-� � �� # so from(L9) � � ��1 �-��1 � �� . Now for any other� ! � , �� 1 � we have that if

� � � � � � or �� are in� � ��1 # then � �� +� �� : Assume��1 � � �� Then �� 1 � � � �� since � 1 � � �� 1 � � #so �� 1 � � � �� # so ��

� � �� from (L6), and since (L9) gives us�� we have� �� Assume ��1 � � � � �'� � � So �

� 1 � � � ! �, and since� 1 � � �� 1 � � we have

�� 1 � � � � ! � , so �� 1 � � from (L6). If also �� # then �

� � � 1 ��! � , and from�� 1 � � � 1 � � ! � and (Down) we have�

� � � � � 1 � � ! � and �� 1 � � � �� from (L6), so�� 1 �.# but weassumed otherwise. So�� # so again��

� � �� # � � � � � �� So for all sentences� ! � � ��1 deriving from clause (i) we have� �� # and for the sentences from clause (ii) thisholds trivially since� ��

� �� . So with (L9) and (L1) we have� � ��1 � ��1 � �� iff�� .�� iff �

� � � 1 � ! � �Suppose� �+ �� Then �

� ) � 1 �2! � � We have��1 � �� )�� # for otherwise� 1 � � �� )�� and �

� ) � 1 � ! � by (T5), but we assumed� �- � � So both sentences from clause (ii) are in� � ��1 �Assume

� � 1 �� )�� Then for all � ! � , � � � � ��1 � �� So � 1 � � �� # so � � ��1�� 1 � ��) � Assume that there are� ��#��%#�� ! � � 1 �.#�� ,�� )��.� For each such��, # � 1 �� , � # for otherwise�� 1 � due to (T4). So� 1 � ��, �� # so for each such��, # � �� , ! � � ��1 � For any��! � , either �

� 1 � � �.# then due to (L3) there is a� , ! � � 1 � such that �� ,�� # or �� 1 � � �.� So

� � ��1 �-��1 � �� and again� � ��1 �-��1 � ��) � So both sides of the iff-clause prove true forany �5! � � � �� . ��

19By these clauses� is constructed inconsistent: The sentences from clause (ii) jointly imply that some �� is true, which isdenied by the sum of all sentences from clause (i). It is the ‘job’ of the contraction function

�to remove sentences to make� � �� consistent, if so required.



Lemma 5.5. (‘Verification Lemma’)If contains (DDR), then (DR1) holds for

�. If contains (DNR), then (DR2) holds for

�.

Proof: For each1 ! � �� there is a1 � ! � � �� such that��1 � � � 1 � � and for all � ! � � �

�� .#�.�� 1 � � � iff �.�� 1 � � � � � For (DR1) suppose� 1 �'� � � Then � 1 � � � � � We have

�� ( � 1 � �! � from (DDR), so � � � � � � If � � � � then � � ��1 � � , so also� � ��1 � # and we

assumed1 to be satisfiable. If� � � � � then � � ��1 � � ��1 � � �� iff � � � � � �� # � � � � �� #so � � ��1 �#� � 1 � is satisfiable.� � � ! � � �

�� .# so �� 1 � � �� # so also� � ��1 �� 1 and, since� � ��1 � � � ��1 � , � � ��1 � � 1 is satisfiable. For (DR2) suppose again� 1 � � � # so � 1 � � � � � Wehave �

� ( � 1 � �"! � from (DNR), so � � � � � � If � � � � � then� � � 1 � � � � �� 2! � � ��1 � and since� � � 1 � �� !'� � �

�� .# � � � 1 � �� !�� 1 � If � � � � then � 1 � �� )�� soboth imperatives from clause (ii) are in� � ��1 � and at least one must also be in� � ��1 � ��

This completes the proof of Theorem 5.2.��

Remark 5.6. If is DDL or DSDL, we can more easily construct�

by use of the ‘system of spheres’characteristic of these logics. Let� � � � �

�� be the smallest set such that

(i)

(ii)

if � � � �� then (5!'�if 1 !�� and � � �� then

� 1 � �� !'�� is then defined as��15� ��-� 1 !'� . The obvious fact here that for all

� # � � ! � , either � �� or � ��

, thus reflects a property of deontic bases resulting from (DDR) that, when presented inthe usual ‘possible worlds’ style, goes through without comment.

Theorem 5.7. The systemsDSDL3�

, DSDL3 are sound and complete.

Proof: Regardingsoundness, the validity of (DN�

), (DNR) is now trivial with respect to the changedtruth definitions, and so is the validity of (DDR) for semantics restricted by (DR1). (Up) and (Down) areproved as before (for Down there is now just one case to consider). To prove (weak)completeness, theabove proof may be adapted in the following way: First observe that for all 1 # � � is now well defined,since � �%� � due to (DNR), (DN

�

). Now possibly� � �� (just where in the previous construction� � �� )�� ), so (L4) only holds for1 such that�� # and (L8) cannot be stated since (L7) does

not define� � )�� any more. All other observations still hold. In the definition of�

, clause (ii) now has� �� ! � , � �� ! � . In the proof of the coincidence lemma, case� � �� is now excludedand case� � � � done as before. In case� � � # if ��1 � � then � � � � 1 � � ��1 � �� ) holdstrivially, so suppose� 1 �$� � � Then both sentences of clause (ii) are in� � � � 1 � since ��1 � neitherexcludes � � � nor � �� . Assume first

� � 1 � � # then for all ��! �we have �� 1 � � # so

� 1 � � �� , and � � � � 1 � ��1 � �� ) � The case� � 1 �� is then proved as before. In the proof

of the verification lemma, for (DR1) case� � � is redundant and the remainder proved identically.Note that also (DR2) holds trivially since any nonempty��1 � has a nonempty intersection with� � � or�� # so one of the sentences from clause (ii) is in� � � � 1 � . ��

6. Future Study

We motivated a semantics that models imperatives, and demonstrated that existing systems of deon-tic logic reconstruct nicely with respect to this semantics. In particular we could observe which truth



definitions and restrictions on imperative structures are necessary to obtain these reconstructions, whilestandard possible worlds semantics tends to conceal that and how its restrictions on relations betweenworlds are in fact restrictions on the presupposed normative system. We can only hint at how otherconcepts may be accomodated:

� We may define some subset ofI as ‘minimal requirements’. We can then define operators express-ing what is necessary to satisfy the minimal requirements, but also operators of supererogation asmeaning that while something is not minimally required, it still satisfies an imperative, or opera-tors of ‘offense’ as meaning that some not minimally required imperative is violated (cf. [36] fora possible worlds setting of supererogation).

� We may have hierarchies of imperative structures, and defineoperators in relation to satisfactionsets that fulfil e.g. the following criterion: Any imperative of the highest authority is satisfied, anyimperative of the second highest authority is satisfied thatcan be if the imperatives of the highestauthority are, etc. (cf. [18] for such command hierarchies).

� We might use a second function to associate (numerical) values with each imperative, and define�� as true iff � is necessary for achieving that the sum of all values of imperatives in the satis-faction set is the sum of all values of imperatives in� , the highest achievable sum, or surpasses afixed value. It may then be obligatory to violate one imperative when it serves to satisfy others ofhigher value.

� Regarding conditional imperatives, we may follow van Fraassen’s proposal in [50] and employa function � that associates a second�� -sentence with each imperative, describing theopportu-nity for its satisfaction or violation. For all valuations we then have the set� � of ‘actualized’imperatives

� !�� such that � �� . We can define a monadic operator�� as true at iff� � �� , and a dyadic operator�

� � � 1 � as true iff �� is true in all �!�� 1 � . This operatorwill be indefeasible, i.e. allow strengthening of the antecedent. We may then combine such dyadicoperators with contrary-to-duty operators as defined above(cf. [48] for a similar enterprise).

Some puzzling detail remains: In section 2, we have separated imperatives and their ‘propositionalcontent’, linking them by a function� that associates a descriptive sentence with each imperative. But inthe reconstructions this separation was not used – the restrictions and truth definitions could equally wellbe given with respect to a set of sentences instead. This doesnot mean that imperatives are ‘reducible’ toindicatives. It merely shows that the reconstructed logicsdo not require the modelling of other qualities ofimperatives. The real puzzlement comes from the other side of the equation: Quite a number of deonticlogics may be characterized as being not about specifically normative notions at all, but as reasoningabout sets of sentences and their underlying classical (meta-)logic. To modify von Wright’s famousstatement: It may have turned out that deontic logics have a wider reach than normative concepts.

References

[1] al-Hibri, A.: Deontic Logic, Washington: University Press, 1978.

[2] Alchourron, C. E. and Bulygin, E.: “The Expressive Conception of Norms”, in [24], 95–123.



[3] Alchourron, C. E. and Bulygin, E.: “Pragmatic Foundations for a Logic of Norms”,Rechtstheorie, 15, 1984,453–464.

[4] Alchourron, C. E. and Bulygin, E.: “On the Logic of Normative Systems”, in Stachowiak, H. (ed.):Handbuchpragmatischen Denkens, vol. IV, Hamburg: Meiner, 1993, 273–294.

[5] Alchourron, C. E. and Martino, A. A.: “Logic Without Truth”, Ratio Juris, 3, 1990, 46–67.

[6] Alchourron, C. E., Gardenfors, P., and Makinson, D.: “On the Logic of Theory Change: Partial Meet Con-traction and Revision Functions”,Journal of Symbolic Logic, 50, 1985, 510–530.

[7] Anderson, A. R.:The Formal Analysis of Normative Concepts, in Rescher, N. (ed.):The Logic of Decisionand Action, Pittsburgh: University Press, 1967, 147–213.

[8] Anderson, A. R.: “Some Nasty Problems in the Formal Logic of Ethics”, Nous,1, 1967, 345–359.

[9] Anderson, A. R. and Moore, O. K.: “The Formal Analysis of Normative Concepts”,American SociologicalReview, 22, 1957, 9–17.

[10] Aqvist, L.: “Interpretations of Deontic Logic”,Mind, 73, 1964, 246–253.

[11] Aqvist, L.: “Deontic Logic”, in Gabbay, D. and Guenthner, F. (eds.): Handbook of Philosophical Logic, vol.II, Dordrecht: Reidel, 1984, 605–714.

[12] Aqvist, L.: “Some Results on Dyadic Deontic Logic and the Logic of Preference”,Synthese, 66, 1986,95–110.

[13] Aqvist, L.: Introduction to Deontic Logic and the Theory of Normative Systems, Naples: Bibliopolis, 1987.

[14] Chisholm, R. M.: “Contrary-to-Duty Imperatives and Deontic Logic”, Analysis, 24, 1963, 33–36.

[15] Danielsson, S.:Preference and Obligation, Uppsala: Filosofiska foreningen, 1968.

[16] Føllesdal, D. and Hilpinen, R.: “Deontic Logic: An Introduction”, in [23], 1–35.

[17] Hamblin, C. L.:Imperatives, Oxford: Blackwell, 1987.

[18] Hanson, W. H.: “A Logic of Commands”,Logique et Analyse, 9, 1966, 329–343.

[19] Hansson, B.: “An Analysis of Some Deontic Logics”,Nous, 3, 1969, 373–398. Reprinted in [23], 121–147.

[20] Hansson, S. O.: “New Operators for Theory Change”,Theoria, 55, 1989, 114–131.

[21] Hare, R. M.: “Imperative Sentences”,Mind, 58, 1949, 21–39.

[22] Hare, R. M.:The Language of Morals, 2nd ed., Oxford: University Press, 1964.

[23] Hilpinen, R. (ed.):Deontic Logic: Introductary and Systematic Readings, Dordrecht: Reidel, 1971.

[24] Hilpinen, R. (ed.):New Studies in Deontic Logic, Dordrecht: Reidel, 1981.

[25] Horty, J. F.: “Agency and Obligation”,Synthese, 108, 1996, 269–307.

[26] Horty, J. F.: “Nonmonotonic Foundations for Deontic Logic”, in [38], 17–44.

[27] Horty, J. F.:Agency and Deontic Logic, New York: Oxford University Press, 2001.

[28] Jørgensen, J.: “Imperatives and Logic”,Erkenntnis, 7, 1937/38, 288–296.

[29] Kanger, S.: “New Foundations for Ethical Theory”, in [23], 36–58.

[30] Kelsen, H.:Reine Rechtslehre, Leipzig/Wien: Deuticke, 1934.

[31] Kelsen, H.:Allgemeine Theorie der Normen, Wien: Manz, 1979.



[32] Lemmon, E. J. and Nowell-Smith, P. H.: “Escapism: The Logical Basis of Ethics”,Mind, 69, 1960, 289–300.

[33] Lewis, D.: “Semantic Analyses for Dyadic Deontic Logic”, in Stenlund, S. (ed.):Logical Theory and Seman-tic Analysis, Dordrecht: Reidel, 1974, 1–14.

[34] Makinson, D.: “On a Fundamental Problem of Deontic Logic”,in McNamara, P. and Prakken, H. (eds.):Norms, Logics and Information Systems, Amsterdam: IOS, 1999, 29–53.

[35] Mally, E.: Grundgesetze des Sollens, Graz: Leuschner und Lubensky, 1926.

[36] McNamara, P.: “Doing Well Enough: Toward a Logic for Common Sense Morality”,Studia Logica, 57,1996, 167–192.

[37] Moutafakis, N. J.,Imperatives and their Logics, New Delhi: Sterling, 1975.

[38] Nute, D.:Defeasible Deontic Logic, Dordrecht: Kluwer, 1997.

[39] Opałek, K.:Theorie der Direktiven und Normen, Wien: Springer, 1986.

[40] Prakken, H. and Sergot, M.: “Dyadic Deontic Logic and Contrary-to-Duty Obligations”, in [38], 223–262.

[41] Prior, A. N.: “The Paradoxes of Derived Obligation”,Mind, 63, 1954, 64–65.

[42] Rescher, N.: “An Axiom System for Deontic Logic”,Philosophical Studies, 9, 1958, 24–30.

[43] Rescher, N.:The Logic of Commands, London: Routledge & Kegan, 1966.

[44] Ross, A.: “Imperatives and Logic”,Theoria, 7, 53–71.

[45] Sosa, E.: “The Logic of Imperatives”,Theoria, 32, 1966, 224–235.

[46] Spohn, W.: “An Analysis of Hansson’s Dyadic Deontic Logic”,Journal of Philosophical Logic, 3, 1975,237–252.

[47] Stenius, E.: “The Principles of a Logic of Normative Systems”,Acta Philosophica Fennica, 16, 1963, 247–260.

[48] van der Torre, L.:Reasoning about Obligations, Amsterdam: Thesis Publishers, 1997.

[49] van Fraassen, B.: “The Logic of Conditional Obligation”, Journal of Philosophical Logic, 1, 1972, 417–438.

[50] van Fraassen, B.: “Values and the Heart’s Command”,Journal of Philosophy, 70, 1973, 5–19.

[51] von Kutschera, F.: “Normative Praferenzen und bedingte Gebote”, in Lenk, H. (ed.):Normenlogik, Pullach:UTB, 1974, 137–165.

[52] von Wright, G. H.: “Deontic Logic”,Mind, 60, 1951, 1–15.

[53] von Wright, G. H.: “A Note on Deontic Logic and Derived Obligation”, Mind, 65, 1956, 507–509.

[54] von Wright, G.H.:Logical Studies, London: Routledge and Kegan, 1957.

[55] von Wright, G.H.: “A New System of Deontic Logic”,Danish Yearbook of Philosophy, 1, 1964, 173–182.Reprinted in [23], 105–120.

[56] von Wright, G.H.: “A Correction to a New System of Deontic Logic”, Danish Yearbook of Philosophy, 2,1965, 103–107. Reprinted in [23], 105–120.

[57] von Wright, G. H.: “Norms, Truth, and Logic”, in von Wright,G. H.: Practical Reason, Oxford: Blackwell,1983, 130–209.

[58] von Wright, G. H.: “Is There a Logic of Norms?”,Ratio Juris, 4, 1991, 265–283.

[59] Weinberger, O.: “Uber die Negation von Sollsatzen”,Theoria, 23, 1957, 103–132.

[60] Weinberger, O.:Normentheorie als Grundlage der Jurisprudenz und Ethik, Berlin: Duncker & Humblot,1981.


Journal of Applied Logic 2 (2004) 39–61

www.elsevier.com/locate/jal

Problems and results for logics about imperatives

Jörg Hansen

Institut für Philosophie, Universität Leipzig, Leipzig, Germany

Abstract

Deviating from standard possible-worlds semantics, authors belonging to what might be calledthe ‘imperative tradition’ of deontic logic have proposed a semantics that directly representsnorms(or imperatives). The paper examines possible definitions of (monadic) deontic operators in such asemantics and some properties of the resulting logical systems. 2004 Elsevier B.V. All rights reserved.

Keywords:Logic of imperatives; Deontic logic

1. Imperative semantics and basic operators

Deontic logic, i.e., the logical analysis of obligation, permission and prohibition, is usu-ally modelled by a possible-worlds semantics, in which among the set of worlds a deonticalternative or preference relation selects some as ‘ideal’ or ‘better’ compared to others.However, when there is an explicit set of given imperatives, a code of norms, or a numberof specified tasks to complete for the agent, the appeal to some prohairetic notion for themeaning of ought seems out of place. Instead, what is obligatory then depends on whetherit serves to satisfy the imperatives, avoid norm violations, or complete all the tasks.

In what might be termed the imperative tradition of deontic logic, a number of authorshave deviated from the standard approach by giving semantics that relates the meaningof deontic operators to an explicitly given set of norms or imperatives.1 The general idea

E-mail address:[email protected] In [8] I mention semantics of Kanger [14], Stenius [20], van Fraassen [21], Alchourrón and Bulygin [1].

A more comprehensive list should also include Smiley [19], and Niiniluoto [17]. Smiley, in a concept later en-dorsed by Ruth Barcan Marcus [15] and motivated by an Andersonian [2] definition of a constant representingthe satisfaction ofall normative demands, uses a ‘normative code’ consisting of propositions to defineOA asmeaning that there is a finite number of propositionsp1, . . . ,pn in this code such that(p1 ∧ · · · ∧ pn → A) is alogical truth. Niiniluoto represents commands by a tuple containing a propositionp, where the truth ofOA thendepends on whetherA is logically implied by one suchp.

1570-8683/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jal.2004.01.003


40 J. Hansen / Journal of Applied Logic 2 (2004) 39–61

behind imperative semantics is that to each command or imperative there is a descriptivesentence that describes what must hold iff this imperative is satisfied. If a set of impera-tives is under consideration, the set of corresponding descriptive sentences is then used ofin the definition of deontic operators. The proper representation of this set is controversial:Directly representing imperatives by a set ofdescriptive sentences, as Kanger [14] and Al-chourrón and Bulygin [1] have done, makes it appear as if norms can somehow be reducedto factual statements. Others like van Fraassen [21], Niiniluoto [17] and myself [8], havemore cautiously represented imperatives by a set of objects thatrefer to states of affairs orpropositions, thereby following the doctrine that norms bear no truth values. For reasons ofreadability I skip this intermediate step here, and use a setI ⊆ L[BL] of descriptive sen-tences that—in the language of a basic logicBL—mean the sentences associated with someimperatives in the above sense. Deontic operators are then considered as giving informationabout the properties of this set of imperative-associated descriptive sentences. The follow-ing definitions of operatorsO1−5, differing in strength, will be employed throughout:

I |= O1A iff A ∈ I ,

I |= O2A iff ∃B ∈ I :|=BL B ↔ A,

I |= O3A iff ∃B ∈ I :|=BL B → A,

I |= O4A iff ∃B1, . . . ,Bn ∈ I :|=BL (B1 ∧ · · · ∧ Bn) → A,

I |= O5A iff I |=BL A.

O1A is true iff A is one of the sentences inI , i.e.,A (literally) corresponds to what an im-perative demands.O2A is true iff the truth ofA is BL-logically a necessary and sufficientcondition for the fulfillment of what one imperative demands.2 Similar to Niiniluoto’s [17]operator,O3A is true iff A is BL-implied by what some imperative demands. Similar toSmiley’s [19] operator,O4A is true iff a finite subset of imperative-associated sentencesBL-impliesA. Finally and similar to Alchourrón–Bulygin’s [1] operator,O5A is true iff Ais BL-implied by the set of all such sentences. Note that while all ofO1 − O4 presupposethe existence of some sentence inI , O5A can hold for (BL-valid) sentencesA even whenI contains no object.

Discussing the general framework, I showed in [8] how a number of well-known sys-tems of monadic and dyadic deontic logic canbe reconstructed with respect to imperativesemantics. The present paper examines some details and problems connected with monadicdeontic logics facilitated by the above semantics. Section 2 gives logical systems for eachof the above operators. Section 3 then examines systems that include several of these op-erators. In Section 4 we take a look at possible semantic restrictions on the content ofimperatives, and see what changes must be applied to logical systems to correspond tothem. Section 5 examines more closely the definitions van Fraassen [21] proposed forO-operators, and provides a monotonic system to resemble his reasoning, and Section 6addresses a ‘sceptical’ definition of ought proposed by Horty [12] to deal with normativeconflicts. Finally, in Section 7 I give systems and semantics for extended languages thatpermit iterated and nested deontic operators of the types presented above.

2 In the terminology of Brown [4]O2 is a “type 2” operator, whereas operatorsO3−5 would be “type 1”, theirscope being a necessary condition for norm satisfaction only.


J. Hansen / Journal of Applied Logic 2 (2004) 39–61 41

2. Logics DL1−51 for deontic operators O1−5

Let our basic logicbe propositional logicPL. The alphabet of the languageL[PL]

has a set of proposition lettersProp= p1,p2, . . . , truth-functional operators ‘¬’, ‘ ∧’, ‘ ∨’,‘→’, ‘ ↔’ and brackets ‘(’, ‘)’. The set of sentences is defined as usual.

∧,∨

in frontof sets of sentences means their conjunction and disjunction, and, e.g.,

∧ni=1 Ai further

abbreviates∧

{Ai, . . . ,An}. In the semantics, valuation functionsv : Prop→ {1,0} definethe truth of sentencesA ∈ L[PL] as usual,‖A‖ meaning the set of valuationsv that makeA true.⊤ means an arbitrary tautology, and⊥ an arbitrary contradiction. We suppose asound and complete axiomatic system forPL.

In this section, we examine deontic logicsDLi1 for operatorsO i,1 � i � 5, defined

above. The upper index in the system name (and the names of most axiom schemes) indi-cates the type of operator used, and the subscript indicates that the scope ofO i is L[PL],not a deontic language (in Section 7 we give up this restriction). The alphabet oflanguagesL[DLi

1], 1� i � 5, is like L[PL] except for one additional operatorO i . L[DLi1] is then

the smallest set such that

(a) if A ∈ L[PL] thenA ∈ L[DLi1],

(b) if A ∈ L[PL] thenO iA ∈ L[DLi1], and

(c) if A, B ∈ L[DLi1], so are¬A, (A ∧ B), (A ∨ B), (A → B), and(A ↔ B).

Note that we permit ‘mixed’ expressions like ‘p1 → O1p2’. The axiomatic systemsDLi1

are then defined by the following clauses (we write⊢DLi1A for A ∈ DLi

1):

(a) All L[DLi1]-instances ofPL-tautologies are inDLi

1.

(b) DLi1 is closed undermodus ponensand the following rule:

(Exti1) If ⊢PL A ↔ B then⊢DLi1O iA ↔ O iB, 2� i � 5.

(c) For allA, B ∈ L[PL] (⊤ being an arbitrary tautology):(Mi

1) ⊢DLi1O i(A ∧ B) → (O iA ∧ O iB), 3� i � 5,

(Ci1) ⊢DLi

1(O iA ∧ O iB) → O i(A ∧ B), 4� i � 5,

(Ni1) ⊢DLi

1O i⊤, i = 5.

As usual, a set of sentencesΓ ⊆ L[DLi1] is DLi

1-inconsistentiff there areA1, . . . ,An

in Γ, n � 1, such that⊢DLi1(A1 ∧ · · · ∧ An) → ⊥, andΓ is DLi

1-consistentotherwise.

A sentenceA ∈ L[DLi1] is DLi

1-derivablefrom a setΓ ⊆ L[DLi1] (we writeΓ ⊢DLi

1A) iff

Γ ∪{¬A} is DLi1-inconsistent. If¬A is derivable from the empty set,A is calledrefutable.

For the semantics, let v be as before, andI be a set ofPL-sentences. The truth-definitions ofDLi

1-sentences are relative tov andI , where the truth of a proposition letteris defined as before, and the truth ofO iA is defined with respect toI as given in Section 1,truth-definitions for Boolean operators being as usual.VerDLi

1(I, v) denotes the subset of

DLi1-sentences defined true forI , v, and we writeI , v |= A for A ∈ VerDLi

1(I, v). A is



DLi1-valid (we write |=DLi

1A) iff A ∈ L[DLi

1] is true for arbitraryI , v, andΓ ⊆ L[DLi1]

DLi1-entailsA (we writeΓ |=DLi

1A) iff I , v |= A for all I , v such thatI , v |= B for all

B ∈ Γ. If there is a pairI , v such thatI , v |= B for all B ∈ Γ ⊆ L[DLi1], we call Γ

DLi1-satisfiable. Note that Boolean valuationsv are introduced just to give non-deontic

formulae their usual interpretation, and the set of sentencesI is not made relative to onesuch valuation. If imperatives are meant to arise in specific circumstances only, this is amatter of proper representation of (conditional) imperatives, for which here no means areprovided. So for examplep1 → O1p2 states that either¬p1 is true, or thatp2 is whatsome imperative demands, regardless of the situation.

Theorem 2.1. Each systemDLi1,1 � i � 5, is sound and complete.

Proof. Soundnessis trivial with respect to the truth definitions employed. Forcomplete-ness, we prove equivalently that ifΓ is DLi

1-consistent thenΓ is DLi1-satisfiable:

Let A1, A2, . . . be a fixed enumeration ofL[DLi1]. Let ∆ =

⋃

n ∆n, where∆0 = Γ,

and

∆n+1 =

{

∆n ∪ {An+1}, if this is consistent,∆n ∪ {¬An+1}, otherwise.

Clearly each∆n must beDLi1-consistent with eitherAn+1 or ¬An+1, hence (i)∆ is DLi

1-consistent, (ii) for allA ∈ L[DLi

1] eitherA ∈ ∆ or ¬A ∈ ∆. Now we define:

v =

{

1, if p ∈ ∆,

0, otherwise,I = {A | O iA ∈ ∆}.

We proveI, v |= A iff A ∈ ∆ by induction onA. We only give the case forA = O iB, theothers are trivial: SupposeO iB ∈ ∆. ThenB ∈ I and for all operatorsO i , I, v |= O iB.

SupposeO iB /∈ ∆, so¬O iB ∈ ∆:

i = 1: AssumeI, v |= O1B, soB ∈ I, soO1B ∈ ∆, but then∆ is DL11-inconsistent.

i = 2: AssumeI, v |= O2B, so there is aC ∈ I such that|=PL B ↔ C. SoO2C ∈ ∆.

But from(Ext21) andPL-completeness we have⊢DL2

1O2B ↔ O2C, so∆ is DL2

1-inconsistent.

i = 3: AssumeI, v |= O3B, so there is aC ∈ I such that|=PL C → B, so O3C ∈ ∆.

But from(Ext31), (M3

1), andPL-completeness we have⊢DL31O3C → O3B, so∆

is DL31-inconsistent.

i = 4: AssumeI, v |= O4B, so there is a non-empty finite set{C1, . . . ,Cn} ⊆ I such that|=PL (C1 ∧ · · · ∧ Cn) → B. So {O4C1, . . . ,O

4Cn} ⊂ ∆. But from (Ext41), (C4

1),(M4

1), andPL-completeness we obtain⊢DL41(O4C1 ∧ · · · ∧ O4Cn) → O4B, so

∆ is DL41-inconsistent.

i = 5: AssumeI, v |= O5B, soI |=PL B. If I �= ∅, then by strong completeness ofPLthere is a non-empty finite set{C1, . . . ,Cn} ⊆ I such that|=PL (C1 ∧· · ·∧Cn) →

B, so the r.a.a. is done as in casei = 4. If I = ∅ then⊢DL51O5B by (Ext51), (N4

1),

so∆ is DL51-inconsistent. �



3. Combined logics DLOp1 for operators O1−5

The definitions of operatorsO1−5 all refer to a setI . Instead of having languagesL[DLi

1] that make use of just one of these operators, it makes sense to permit mixed state-ments such as ‘O4p1 ∧ ¬O3p1’ which expresses that thoughp1 is not implied by what asingle imperative demands, it is implied by a finite subset of all imperative demands. So letOp be a subset of operator indices{1,2,3,4,5}, card(Op) > 1, and letL[DLOp

1 ] be likeL[DLi

1] except that the alphabet contains all ofO i, i ∈ Op, and clause b) in the definitionof sentences now reads

b) if A ∈ L[PL] thenO iA ∈ L[DLOp

1 ], for all i ∈ Op.

Let the description of systemsDLOp

1 follow that of DLi1, where the clause a) is now

relative to the languageL[DLOp

1 ], and clauses b), c) hold for alli ∈ Op satisfying theadditional requirements. We have the following additional axiom schemes:

(OiOj ) ⊢DLOp

1O iA → OjA, {i, j } ⊆ Op, i < j ,

(O3O2) ⊢DLOp

1O3⊥ → O2⊥, {2,3} ⊆ Op,

(O4O3) ⊢DLOp

1O4⊤ → O3⊤, {3,4} ⊆ Op,

(O5O3) if �PL A then⊢DLOp

1O5A → O3⊤, {3,5} ⊆ Op,

(O5O4) if �PL A then⊢DLOp

1O5A → O4A, {4,5} ⊆ Op.

Theorem 3.1. All systemsDLOp

1 are sound and weakly complete.

Proof. Soundnessis again trivial with respect toO i -truth definitions. For weakcomplete-ness, we have to prove that if�DLOp

1A then⊢DLOp

1A. We assume�DLF

1A so¬A is not

refutable inDLOp

1 . One disjunctδ in a disjunctive normal form of¬A is then not refutable

in DLOp

1 . From the non-deontic conjuncts ofδ we obtain a valuationv that satisfies them.As for the deontic conjuncts, for alli ∈ Op let Oi = {B ∈ L[PL] | O iB is a conjunct ofδ}, andQi = {B ∈ L[PL] | ¬O iB is a conjunct ofδ}. Let π1,π2 be proposition letters thatdo not occur inδ. We define

I = O1 1 ∈ Op (a)

∪{

(B ∧ (π1 ∨ ¬π1)) | B ∈ O2}

2 ∈ Op (b)

∪{

(B ∧ π1) | B ∈ O3}

3 ∈ Op (c)

∪{

π2, (π2 → (B ∧ π1)) | B ∈ O4}

4 ∈ Op (d)

∪{

π2, (π2 → (B ∧ π1)) | B ∈ O5 and�PL B}

5 ∈ Op (e).

Obviously, for allB ∈ Oi , I, v � O iB. We prove that for eachB ∈ Qi , I, v � ¬O iB:

B ∈ Q1: If B ∈ I thenB /∈ O1 by PL-consistency ofδ. But all otherB ∈ I contain propo-sition letterπ1 not occurring in anyB ∈ Q1.



B ∈ Q2: For all C ∈ I by clause (a) or (b),�PL (B ↔ C) by PL-consistency ofδ andaxiom scheme(O1O2). All other C ∈ I derive some contingent formulaπ2 or(π2 → π1). SinceB contains no occurrence ofπ1,π2, if �PL B → C then�PL¬B. The C ∈ I generated by clauses (d), (e) all are contingently true or false,so if there is aC ∈ I such that�PL ¬C it must be due to clause (c). But thenO3CDLOp

1 -derivesO2C by (O3O2) andO2C DLOp

1 -derivesO2B by (Ext21), so

δ is DLOp

1 -refutable.B ∈ Q3: For allC ∈ I by clause (a), (b), or (c),�PL (C → B) by PL-consistency ofδ and

axiom schemes(O1O3), (O2O3), (M31), (Ext31). So only by virtue of clauses (d),

(e) can there be aC ∈ I such that�PL (C → B). But no singleC generated bythese clausesPL-impliesB unless�PL B. But then there is some conjunctO4C

in δ, or some conjunctO5D with non-tautologicalD, from whichDLOp

1 -derives

O3C by use of(O5O3) or (O4O3), (M41), (Ext41). Soδ is DLOp

1 -refutable.B ∈ Q4: If O4B is true, thenI �= ∅, Oi �= ∅ for some i ∈ Op. By the construction

�PL∧

I → B iff �PL∧

(⋃4

i=1 Oi ∪ {C ∈ O5 |�PL C}) → B. For anyC in⋃4

i=1 Oi ∪ {C ∈ O5 |�PL C} we deriveO4C by (OiO4), i ∈ Op, andO5B using

(M41), (Ext41), soδ is DLOp

1 -refutable.

B ∈ Q5: If �PL B thenδ is DLOp

1 -refutable by(N51), so againI �= ∅, Oi �= ∅ for some

i ∈ Op. The proof relies on axiom schemes(OiO5), i ∈ Op, and is done as incaseQ4. �

Theorem 3.2. No DLOp

1 , Op �= {4,5}, is strongly complete.

Proof. The semantics ofDLOp

1 for Op �= {4,5} is not compact, i.e., there are finitely

satisfiable setsΓ ⊆ L[DLOp

1 ] such thatΓ is not satisfiable: Suppose{i, j } ⊆ Op. We givenon-satisfiable setsΓ and anIf for each finiteΓf ⊆ Γ such thatIf , v satisfiesΓf , v beingarbitrary.

i = 1, j = 2: Let Γ = {¬O1A ||=PL A ↔ p1} ∪ {O2p1}. For any finite subsetΓf of Γ

there is a proposition letterπ that does not occur in anyA ∈ Γf . Each setΓf issatisfied byIf = {((π ∨ ¬π) ∧ p1)}.

i = 1,2, j = 3: Let Γ = {¬O iA ||=PL A → p1} ∪ {O3p1}. Let Γf be any finite subset,andπ be as before. Each setΓf is satisfied byIf = {(π ∧ p1)}.

i = 1,2,3, j = 4,5: LetΓ = {¬O iA �PL A} ∪ {Ojp1}. Let Γf be any finite subset, andπ be as before. Each setΓf is satisfied byIf = {(π → p1),π}. �

Theorem 3.3. DL{4,5}1 is (strongly) complete.

Proof. We adapt the completeness proof ofDL41 in Theorem 2.1:∆ is now defined with

respect to an enumeration ofDL{4,5}1 -sentences andDL{4,5}

1 -consistency.I , v is constructedas forDL4

1, in particularI = {A | O4A ∈ ∆}. The proof thatI , v |= O4A iff O4A ∈ ∆ is asbefore. To prove thatI, v |= O5A iff O5A ∈ ∆, assume firstO5A ∈ ∆. EitherO4A ∈ ∆,



but thenA ∈ I , andI |=PL A is trivial. Or O4A /∈ ∆, then¬O4A ∈ ∆, but thenA mustbe tautological, for otherwise by use of(O5O4) the finite subset{¬O4A,O5A} ⊆ ∆ isDL{4,5}

1 -inconsistent. So|=PL A andI, v |= O5A is true. For the other direction assumeO5A /∈ ∆, so¬O5A ∈ ∆. For r.a.a. assumeI, v |= O5A, soI |=PL A. If I �= ∅ thenI, v |=

O4A, so by completeness ofPL O4A ∈ ∆, but then due to(O4O5) the set{¬O5A,O4A}

is DL{4,5}

1 -inconsistent. IfI = ∅ thenA is tautological, so by using(N51), (Ext5

1){¬O5A}

is DL{4,5}1 -inconsistent. �

4. DLi1-logics and semantic restrictions

So far, a set of sentences has been used to model what a set of imperatives demands,there being neither restrictions on the size ofI , nor restrictions on the logical type ofsentences inI . But concerning the size ofI , it may be argued that a deontic logic shouldbe applied only if there is at least one demand to be considered, since reasoning about whatone ought to do when nothing is explicitly obligatory seems a borderline case. Concerningthe elements ofI , it may be argued that imperatives that demand the logically impossibleor logically necessary are not ‘proper’, and one may also want to use a ‘rationality restraint’to the effect that what the imperatives demand should be (jointly) satisfiable. Considerthefollowing restrictions:

[R-0] I �= ∅ (Non-Triviality),[R-1] ∀A ∈ I :�BL ¬A (Excluded Impossibility),[R-2] I �BL ⊥ (Collective Satisfiability),[R-3] ∀A ∈ I :�BL A (Excluded Necessity).

Consider the logicsDLi1 defined in Section 2 and the following additional axiom

schemes:

(XIi1) if ⊢PL ¬A then⊢DLi

1¬O iA,

(Di1) if ⊢PL ¬(A1 ∧ · · · ∧ An) then⊢DLi

1¬(O iA1 ∧ · · · ∧ O iAn),

(XNi1) if ⊢PL A then⊢DLi

1¬O iA.

Warranted by the following theorems, the table below lists, for languagesL[DLi1],

1 � i � 5, axiom systems sound and complete with respect toDLi1-semantics conforming

to any of the above restrictionsindividually (square brackets indicate weak completenessonly):

L[DLi1] i = 1 i = 2 i = 3 i = 4 i = 5

[R-0] [DLi1] DLi

1 + (Ni1) DL5

1

[R-1] DLi1 + (XIi1) DLi

1

[R-2] DLi1 + (Di

1)

[R-3] DLi1 + (XNi

1) [DLi1] DL5

1



Theorem 4.1. The following systems are sound and(strongly) complete:

1. DLi1 + (Ni

1) for DLi1-semantics restricted by[R-0], i = 3,4.

2. DL51 for DL5

1-semantics restricted by[R-0].3. DLi

1 + (XIi1) for DLi

1-semantics restricted by[R-1], i = 1,2,3.4. DLi

1 for DLi1-semantics restricted by[R-1], i = 4,5.

5. DLi1 + (Di

1) for DLi1-semantics restricted by[R-2], i = 1,2,3,4,5.

6. DLi1 + (XNi

1) for DLi1-semantics restricted by[R-3], i = 1,2.

7. DL51 for DL5

1-semantics restricted by[R-3].

Proof. Soundness is again obvious. The completeness proofs are forthcoming adaptationsof that for theDLi

1-systems. When there is a specific axiom, it obviously ensures that theconstructedI conforms to the given restriction. Consider cases 2,4,7 where there are nospecific axioms:

Case2: O5⊤ ∈ ∆ by (N51), so by the construction⊤ ∈ I �= ∅.

Case4: We construct∆ as before, but now defineI = {A | O iA ∈ ∆ and�PL ¬A}. Sup-poseO iA ∈ ∆ and A /∈ I , so |=PL ¬A. Then |=PL A → p1, |=PL A → ¬p1,

so by (Exti1) we haveO ip,O i¬p1 ∈ ∆, so {p1,¬p1} ⊆ I, so I |=PL A. Theremainder is as before.

Case7: Let ∆ be as before, but nowI = {A | O5A ∈ ∆ and�PL A}. SupposeO5A ∈ ∆

andA /∈ I , so|=PL A. But then triviallyI |=PL A, and the remainder of the proofis as before. �

Theorem 4.2. DLi1-semantics,i = 1,2, is not compact if restricted by[R-0]. DLi

1-semantics,i = 3,4, is not compact if restricted by[R-3].

Proof. To show thatDLi1-semantics,i = 1,2, is not compact if restricted by[R-0], let

Γ = {¬O iA | A ∈ L[DLi1]}. To satisfy all ofΓ, I must be empty, which is excluded by

[R-0], soΓ is not satisfiable by[R-0]-restricted semantics. But any finite subsetΓf ⊆ Γ

is satisfied byIf , v, whereIf = {π}, andπ is a proposition letter that does not occur inanyA ∈ Γf , v being arbitrary.

For the proof thatDLi1-semantics,i = 3,4, is not compact if restricted by[R-3], let

Γ = {¬O iA �PL A} ∪ {O i⊤}. To satisfy all ofΓ, I must contain tautologies only, whichis excluded by[R-3], yet it also cannot be empty due to the expected truth ofO i⊤, soΓ

is not satisfiable by[R-3]-restricted semantics. But anyΓf is again satisfied byIf , v, asdescribed above.�

Theorem 4.3. The following systems are sound and weakly complete:

1. DLi1 for DLi

1-semantics restricted by[R-0], i = 1,2.2. DLi

1 for DLi1-semantics restricted by[R-3], i = 3,4.



Proof. Soundness is again obvious. For weak completeness, suppose{A} is DLi1-

consistent. To demonstrate that there areI, v such thatI, v � A, we do a construction as inthe proof of Theorem 3.1, i.e., form a disjunctive normal form ofA, from the non-deonticconjuncts of a non-refutable disjunct obtain a valuationv that satisfies these, and from thedeontic conjuncts construct setsOi andQi as before. Again, letπ be a proposition letternot occurring inA.

Case1: Let

I =

{

Oi, if Oi �= ∅,

{π}, otherwise.

The construction ensures thatI �= ∅ (R-0), and for allB ∈ Oi , I, v � O iB. If i = 1 andB ∈ Q1, thenB /∈ O1 by PL-consistency of{A}, andB �= π sinceπ does not occur inA,soB /∈ I andI, v � ¬O1B. If i = 2 andB ∈ Q2, then noC ∈ O2 is PL-equivalent toB dueto (Ext21) andPL-consistency of{A}, and no formula not containingπ is PL-equivalentto π . So there is noC ∈ I s.t.�PL (B ↔ C), andI, v � ¬O2B.

Case2: Let

I =

Oi − {A ∈ L[PL] |�PL A}, if this is non-empty,{π}, otherwise and ifQi ∩ {A ∈ L[PL] |�PL A} = ∅,

∅, otherwise.

ObviouslyI contains no tautological element (R-3). SupposeB ∈ Oi . If there is a non-tautologicalC ∈ Oi , for all B ∈ Oi , tautological or non-tautological,I, v � O iB. If thereare tautologicalB ∈ Oi (only), there cannot be a tautologicalC ∈ Qi due to(Exti1) andPL-consistency of{A}, soπ ∈ I and againI, v � O iB. For eachB ∈ Qi , I, v � ¬O iB:

B ∈ Q3: If there is a tautologicalC ∈ Q3, O3 = ∅ due to(Ext31), (M31) andPL-consistency

of {A}, and I = ∅ by the above construction, soI, v � ¬O3B. Otherwise, noC ∈ O3 PL-impliesB again due to(Ext31), (M3

1) andPL-consistency of{A}, andif π does thenB is tautological, but this was excluded.

B ∈ Q4: As before, ifC ∈ Q4 is tautological thenO4 = ∅ andI = ∅, so I, v � ¬O4B.Otherwise,B is neither implied by any{C1, . . . ,Cn} ⊆ O4 due to(Ext41), (M4

1),(C4

1) andPL-consistency of{A}, nor byI = {π} since a tautologicalB was ex-cluded. �

If the semantics employs more than one of the above restrictions, then generally thereisa sound and (strongly) complete axiomatic system iff each of the semantics with just oneof these restrictions has such a system, and the axiomatic system is obtained by includingall axioms and rules of the systems for the singularly restricted semantics. However, thisrule fails in one case:

Theorem 4.4. DL51-semantics is not compact if restricted by[R-0], [R-3].

Proof. To show incompactness, letΓ = {¬O5A �PL A}. To satisfy all ofΓ, I cannotcontain anything but tautologies, which is excluded by[R-3], yet it also cannot be emptydue to [R-0], so Γ is not satisfiable by[R-0][R-3]-restricted semantics. But any finite



subsetΓf is again satisfied byIf , v, whereIf = {π}, π being as before, andv beingarbitrary. �

Theorem 4.5. DL51 is sound and weakly complete for[R-0][R-3]-restrictedDL5

1-sem-antics.

Proof. Soundness is again obvious. For completeness we assume that{A} is DL51-

consistent and construct anI, v such thatI, v � A as in the proof of Theorem 4.3, nowletting

I =

{

Oi − {A ∈ L[PL] |�PL A}, if this is non-empty,{π}, otherwise.

ObviouslyI satisfies[R-0] and[R-3]. For allB ∈ O5, if B is non-tautological thenB ∈ I

andI, v � O5B, and trivially so ifB is tautological. SupposeB ∈ Q5: B cannot be tauto-logical due to(N4

1) andPL-consistency of{A}, so alsoB cannot bePL-implied by {π},so if I �PL B by completeness ofPL there must be some{C1, . . . ,Cn} ⊆ O5 such that⊢PL ((C1∧· · ·∧Cn) → B), but this is excluded by(Ext4

1), (M41), (C

41) andPL-consistency

of {A}. �

5. Van Fraassen’s imperative logic

Van Fraassen [21] discusses three truth definitions ofO-operators with respect to givenimperatives. For all these operators, axiom scheme(C) does not hold, for van Fraassen isconcerned with possibly conflicting imperatives, and argues that when there is a demandfor A, and a demand for¬A, we should admit the truth ofOA ∧ O¬A, but the derivationof O(A ∧ ¬A) should be blocked. In van Fraassen’s semantics, to each imperativei thereis a class of possible outcomesi+ in which i is fulfilled. His first definition reads:

[Df-F1] OA is true iff, for some imperativei that is in force,i+ is part of the set ofpossible outcomes in whichA is true.

In the terms used here, van Fraassen’s definition may be reformulated as

[Df-F1∗] I |= OA iff ∃B ∈ I :|=PL B → A.

Since van Fraassen also presupposes that no single imperative may be impossible tofulfill, van Fraassen’s first logic coincides withDL3

1+ (XI 31) above. However, van Fraassen

thinks definition[Df-F1] too ‘simple minded’, for two reasons: For one, imperatives areconditional, and according to van Fraassen a conditional imperative can be fulfilled orviolated only if its condition is the case. His second logic thereforeintroduces conditionalimperatives and a corresponding dyadicO-operator. I leave this definition aside, since weare not concerned with conditional imperatives here. The other argument against[Df-F1]is that it should be allowed to draw conclusions from imperatives that donot conflict,e.g., from two imperatives that demandp1 ∨ p2 and¬p2 respectively conclude thatp1



is obligatory. But since there is no single imperative that demandsp1, [Df-F1] does notallow the derivation ofOp1. Van Fraassen therefore introduces the notion of ascoreof anoutcomev, i.e., the set of imperatives thatv fulfills, and defines:

[Df-F3] OA is true iff there is a possible state of affairsv in ‖A‖ whose score is notincluded in the score of anyv′ in ‖¬A‖.

Again, this definition may be reformulated in the terms used here:

[Df-F3∗] I |= OF A iff ∃v ∈ ‖A‖: ∀v′ ∈ ‖¬A‖: {B ∈ I | v |= B} � {B ∈ I | v′ |= B}.

Let the set ofmaximallyPL-consistent subsetsI −⊥ of I be the set of subsetsI ′ ⊆ I suchthat (i) I ′ � ⊥, and (ii) there is noI ′′ such thatI ′ ⊂ I ′′ ⊆ I andI ′′ � ⊥. It can then beproven that[Df-F3∗] is equivalent to[Df-F3∗∗] (cf. Horty [12, p. 30, Theorem 2]):

[Df-F3∗∗] I |= OF A iff ∃I ′ ∈ I − ⊥: I ′ ⊢PL A.

Let the languageL[DLF1 ] be like any of theL[DLi

1], except thatOF replacesO i inall DLi

1-sentences. Let the axiomatic systemDLF1 be like DL3

1 + (XI31) + (N3

1), exceptthatOF replacesO3. Let DLF

1 -semantics be defined likeDL31-semantics, except that truth

definition[Df-F3∗∗] replaces that forO3. We then have the following results:

Theorem 5.1. DLF1 is sound and weakly complete.

Proof. Concerningsoundness, the validity of(Ext1F1 ), (MF

1 ), (XIF1 ) and(NF

1 ) is imme-diate. Concerning (weak)completeness, we have to prove that if�DLF

1A then⊢DLF

1A. We

assume�DLF1

A so ¬A is not refutable inDLF1 . As before (cf. Theorem 3.1), letδ be a

non-refutable disjunct in a disjunctive normal form of¬A. From its non-deontic conjunctswe again obtain a valuationv that satisfies these. Concerning the deontic conjuncts ofδ,let OF be the set ofPL-sentencesB such thatOFB is a conjunct ofδ, andQF be the setof PL-sentencesB such that¬OFB is a conjunct ofδ. For any setΓ ⊆ L[PL] let

minPLΓ = {A ∈ Γ | ∀B ∈ Γ : if ⊢PL B → A then ⊢PL B ↔ A}.

Let n = card(minPLOF ). Let {π1, . . . , πn} ⊂ Prop be an arbitrary set ofn propositionletters not occurring inδ, andσ be a function that mapsminPLOF onto the set{π1,¬π1 ∧

π2, . . . ,¬π1 ∧ · · ·¬πn−1 ∧ πn} of n mutually exclusivePL-sentences. We then define:

I ={

B ∧ σ(B) | B ∈ minPLOF}

.

Note thatI − ⊥ = {{i} | i ∈ I }, sinceσ(B) is PL-inconsistent with anyσ(C), B,C ∈

minPLOF andB �= C. No B ∈ OF is a contradiction due to(XIF1 ), and for each consistent

B ∈ OF the definition ofminPLOF ensures that there is anI ′ ∈ I − ⊥ such thatI ⊢PL B,soOF B is true. SupposeB ∈ QF but I, v � OFB, so there is anI ′ ∈ I − ⊥: I ′ ⊢PL B.Then there is aC ∈ minPLOF such thatI ′ = {C ∧ σ(C)} and⊢PL C ∧ σ(C) → B. Since



σ(C) contains no sentence letters occurring in eitherB or C, ⊢PL C → B. But then¬OFC

DLF1 -derives from¬OFB by (MF

1 ), (ExtF1 ), soδ is DLF1 -inconsistent. �

Theorem 5.2. There is no strongly complete axiomatic systemDLF1 .

Proof. Let

Γ ={

OF p1,OF p2,¬OF (p1 ∧ p2),¬OF¬p1,¬OF ¬(p1 ∧ p2),

¬OF¬(p1 ∧ p2 ∧ p3), . . .}

.

Γ is finitely satisfiable: Letπ be a proposition letter that does not occur in some sentence ofa finiteΓf ⊂ Γ . ThenI = {p1 ∧ π,p2 ∧ ¬π} satisfiesΓf . However,Γ is not satisfiable:Suppose there is a setI that satisfiesΓ . OF p1,O

F p2 ∈ Γ , so there arePL-consistentfinite setsΓ1,Γ2 ⊆ I that derivep1 andp2 respectively.Γ1 ∪ Γ2 is inconsistent, sinceotherwise there is someI ′ ∈ I − ⊥ such thatΓ1 ∪ Γ2 ⊆ I ′ andI ′ ⊢ p1 ∧ p2, so I wouldnot satisfy¬OF (p1 ∧ p2). Now �PL (

∧

Γ1 ∧ p1 ∧ · · · ∧ pn) → ⊥ for any n � 1, forotherwiseΓ1 ⊢PL ¬(p1 ∧ · · · ∧ pn) would contradict¬OF¬(p1 ∧ · · · ∧ pn). The sameholds forΓ2. Let n be the highest index of proposition letters occurring inΓ2. SinceΓ2 ∪

{p1,p2, . . . , pn} is PL-consistent, i.e.,(p1 ∧· · ·∧pn) is the first disjunct in the disjunctivenormal form of

∧

Γ2, we have⊢PL (∧

Γ2 ∧ p1 ∧ · · · ∧ pn) ↔ (p1 ∧ · · · ∧ pn). So weobtain�PL (

∧

Γ1 ∧∧

Γ2 ∧ p1 ∧ · · · ∧ pn) → ⊥. But Γ1 ∪ Γ2 wasPL-inconsistent. �

After explaining that semantics based on[Df-F3] satisfies the ‘basic criteria’ (the axiomschemes ofDL3

1 + (XI31), van Fraassen notes that in the semantics we have the following

additional truth: If there are two imperativesi1, i2 in force, and if there is av that belongsto i+1 and also toi+2 , then if [all such]v ∈ ‖A‖, OF A is true. He then asks:

“But can this happy circumstance be reflected in the logic of the ought-statements alone?Or can it be expressed only in a language in which we can talk directly about the im-peratives as well? This is an important question because it is the question whether theinferential structure of the ‘ought’ language game can be stated in so simple a mannerthat it can be grasped in and by itself.”

If what van Fraassen means by ‘the logic of the ought-statements alone’ is a monotonicaxiomatic system, then van Fraassen’s first question deserves a negative answer: There isa logic of ought-statementsDLF

1 that is weakly complete, but the only additional truth ifcompared to[Df-F1] is OF⊤.3 In particular,DLF

1 does not provide the inferences vanFraassen desires. And Theorem 5.2 shows that there is no strongly complete axiomatic

3 Van Fraassen seems to have overlooked the additional truth ofOF ⊤. To avoid its truth, theO5-type defini-tion [Df-F3∗∗] may be changed into a moreO4-type definition:

I |= OF A iff ∃I ′ ∈ I − ⊥: ∃B1, . . . ,Bn ∈ I ′ |=BL B1 ∧ · · · ∧ Bn → A.

It can then be shown thatDL31 + (XI3

1 ) is sound and weakly complete with respect to the changed definition, andthe counterexample in Theorem 5.2 again disproves compactness.



system in terms of theO-operator used. But, regarding his additional question, it seems tome that the “ought language game” extends to operators likeO1 or O2 that more “directlytalk about imperatives”. So perhaps a more powerful language can provide the missinginferences. It is immediate that the counterexample used in Theorem 3.2 to disprove com-pactness ofDL{2,4}

1 also disproves compactness of any semantics that employs operatorsof type O1 or O2 in addition to van Fraassen’s operatorOF . So for an improved char-acterization of van Fraassen’s third semantics it must suffice to give a weakly completesystem:

Let L[DL{2,F }

1 ] be thelanguagethat is likeL[DL{2,3}

1 ] except thatOF replacesO3 in

all DL{2,3}

1 -sentences. For thesemantics, the truth ofO2, OF is defined with respect to asetI as above, and for proposition letters and Boolean connectives it is defined as usual.Let theaxiomatic systemDL{2,F }

1 contain all instances ofL[DL{2,F }

1 ] in PL-theorems, beclosed undermodus ponensand(Ext2

1), (ExtF1 ), contain allL[PL]-instances in the axiomschemes(MF

1 ), (XIF1 ), (NF

1 ) of DLF1 , and in the following schemes:

(F-1) If �PL ¬∧n

i=1 Ai then⊢DL{2,F }

1

∧ni=1 O2Ai → OF

∧ni=1 Ai ,

(F-2) ⊢DL{2,F } (∧n

i=1 O2Ai ∧ OF B ∧ ¬OF (B ∧∧n

i=1 Ai)) → OF (B ∧ ¬∧n

i=1 Ai).

According to (F-1), agglomeration of contents ofO2-obligations is admissible if theseare jointly PL-consistent.4 (F-2) then states that ifO2-contents may not be added to thecontent of anOF -obligation, then there is an obligation to the contrary that goes with thatcontent. So (F-1) and (F-2) now properly define the ‘simple cases’, where agglomerationof contents is admissible.

Theorem 5.3. DL{2,F }

1 is sound and weakly complete.

Proof. For soundnessof the new axiom schemes, validity of (F-1) is immediate from thefact that if there areB1, . . . ,Bn ∈ I equivalent toA1, . . . ,An respectively, andA1, . . . ,An

arePL-consistent, then someI ′ ∈ I − ⊥ contains{B1, . . . ,Bn}, soI ′ ⊢PL A1 ∧ · · · ∧ An.For the validity of (F-2), supposeO2A1, . . . ,O

2An. If ¬OF (B ∧∧n

i=1 Ai), then noI ′ ∈

I − ⊥ may deriveB and allAi, so if OFB is true and there is anI ′ that derivesB, it mustbe inconsistent with

∧ni=1 Ai and consequently deriveB ∧ ¬

∧ni=1 Ai .

For (weak)completeness, we have to prove that ifA is DL{2,F }

1 -valid, thenA ∈ DL{2,F }

1 .

We assumeA /∈ DL{2,F }

1 , so¬A is not refutable inDL{2,F }

1 . As now usual, letδ be a non-refutable disjunct in a disjunctive normal form of¬A. From its non-deontic conjuncts we

4 A difficulty in axiomatizing van Fraassen’s semantics was pointed out by Horty [11, p. 50]: If backgroundimperatives are coded into ought-statements, as they are here using operatorO2, then the set{O2A | A ∈ L[BL]}

must deriveOF B for anyBL-consistentB, though for some basic logics like first order logic these are not recur-

sively enumerable. But then due to (F-1) the set ofDL{2,F }

1 -axioms is not decidable andDL{2,F }

1 not recursivelyenumerable, so for such basic logics no axiomatization in the usual sense is provided here.



again obtain a valuationv that satisfies them. As for the deontic conjuncts ofδ, first let

O2 ={

B ∈ L[PL] | O2B is a conjunct ofδ}

,

Q2 ={

B ∈ L[PL] | ¬O2B is a conjunct ofδ}

.

Let L[PL]δ be the set ofPL-sentences that contain only proposition letters occurring insome deontic conjunct ofδ. Let r(L[PL]δ) be a set of 22

nmutually non-equivalent rep-

resentatives ofL[PL]δ, n being the number of proposition letters occurring in the deonticconjuncts ofδ. By writing PL-sentences we now mean their unique representatives inL[PL]∆. We construct a finite set∆ with the following properties:

(a) If OFB,¬OF C are conjuncts ofδ, thenOFB,¬OF C are in∆.(b) For allB ∈ r(L[PL]δ), eitherOFB ∈ ∆ or ¬OF B ∈ ∆.(c) {δ} ∪ ∆ is DL{2,F }

1 -consistent.

In (a), we may have to replaceB,C by r(L[PL]δ)-equivalents using(ExtF1 ).Let:

OF ={

B ∈ L[PL] | OF B ∈ ∆}

,

QF ={

B ∈ L[PL] | ¬OF B ∈ ∆}

.

Lemma 5.1. For all B ∈ O2: (a) if �PL ¬B then there is aC ∈ minPLOF such that⊢PLC → B, and(b) for all C ∈ minPLOF , if {B,C} is PL-consistent, then⊢PL C → B.

Proof. (a) is immediate from (F-1) and the definition ofminPLOF (cf. the proof of Theo-rem 5.1). Concerning (b), eitherOF (B ∧ C) ∈ ∆ or ¬OF (B ∧ C) ∈ ∆ (clause (b) in theconstruction of∆). If OF (B ∧ C) ∈ ∆ then⊢PL C ↔ (B ∧ C) by definition ofminPLOF ,so ⊢PL C → B. If ¬OF (B ∧ C) ∈ ∆ then O2B,OF C,¬OF (B ∧ C) DL{2,F }

1 -derivesOF (¬B ∧ C) by use of (F-2), so OF (¬B ∧ C) ∈ ∆ by clause (c) in the constructionof ∆. But then⊢PL C ↔ (¬B ∧ C) by definition of minPLOF , so {B,C} is not PL-consistent. �

Let σ : minPLOF → L[PL] be the function defined in the proof of Theorem 5.1. Let

I = O2 ∪{

B ∧ σ(B) | B ∈ minPLOF}

.

Lemma 5.2. For all I ′ ∈ I − ⊥, B ∈ L[PL]δ: if I ′ �= ∅ and I ′ ⊢PL B then there is aC ∈ minPLOF such that⊢PL C → B.

Proof. Immediate from Lemma 5.1, the construction ofI , and the fact thatσ(C) containsno proposition letters occurring inB or C. �

It remains to prove thatI satisfies all deontic formulas inδ and∆:



O2: For all B ∈ O2 we haveB ∈ I , soO2B is true.Q2: For all B ∈ Q2, if C ∈ I and �PL B ↔ C, then C /∈ O2, for otherwiseδ is

refutable by(Ext21). All other C ∈ I derive someπi not occurring inδ, but noB ∈ Q2 does.

OF : Unless B ∈ OF is a contradiction, which is excluded by(XI F1 ) and DL {2,F }

1 -consistency of∆, by definition ofminPLOF and the construction ofI there isa I ′ ∈ I − ⊥ such thatI ′ ⊢PL B, soOF B is true.

QF : For all B ∈ QF , there is noI ′ ∈ I − ⊥ such thatI ′ ⊢PL B: For otherwise thereis a C ∈ minPLOF such that⊢PL C → B (Lemma 5.2). ThenOFC ∈ ∆, and¬OFC ∈ ∆ due to(ExtF1 ), (MF

1 ), so∆ is DL {2,F }

1 -inconsistent. �

6. Saving a twin without guilt—the sceptical oughtOS

In the long-standing philosophical dispute on the existence of moral dilemma andconflicting obligations, Barcan Marcus [16] provided the following Buridan’s ass type ex-ample: Identical twins are in danger of being crushed to death by a rock. They are pinneddown such that only one can be pulled free at a time. If nothing is done, the rock willsoon kill them both, but if either twin is removed this will cause the rock to slide andkill the other. A mountain guide is liable for the lives of both twins. What are her obliga-tions?

As the example is set up, it suggests the co-existence of two conflicting obligations,i.e., the guide has to save one twin (T1), and also save the other (T2), T1 contradict-ing T2. In logicsDL4

1 andDL51 we haveO4/5T1 andO4/5T2, and alsoO4/5t (T1 ∧ T2).

But then a contradiction is obligatory,so we cannot also accept the principle ‘ought im-plies can’ expressed by(XI 4/5

1 ). In van Fraassen’s logic we maintainOF T1 andOF T2,and derivation ofOF (T1 ∧ T2) is blocked by the absence of(CF

1 ) and inconsistency ofT1 ∧ T2 (axiomF-1). However, in addition we haveOF ¬T1 andOF¬T2: by saving onetwin the mountain guide will violate her duties towards the other. So why not simply walkaway?

Conee [5], Donagan [6], and Brink [3] have argued that morally there is no conflict:according to Conee, either act is permitted and neither absolutely obligatory, while forDonagan and Brink there only exists an obligation to save one twinor the other, but not twoconflicting obligations to save either. Deontically, the dilemma was examined by Jacquette[13] and Horty [12]. Jaquette points out that van Fraassen’s approach is unsatisfactory,since it does not hold that an obligatory act is also permitted:OF A → PF A is not a truthin DLF

1 , with PF A defined as¬OF¬A. So according to van Fraassen’s notion of ought,either twin must be saved, but can only be saved at the price of guilt (for doing somethingforbidden). As an alternative to van Fraassen’s logic, Horty, in accord with Donagan’s andBrink’s disjunctive proposal, considers a ‘sceptical theory’ of obligations, where somethingcan be obligatoryS only if there is no consistent set of norms that demand the contrary. Withrespect to our semantics, Horty’s definition reads:

[Df-S] I |= OSA iff ∀I ′ ∈ I − ⊥: I ′ ⊢PL A.



SoA is obligatory iff all maximally consistent subsets of imperative-associated sentencesderiveA.5 Let L[DLS

1] be like any of theL[DLi1], except that operatorOS replacesO i .

Let the truth ofDLS1-sentences be defined with respect to[Df-S] and the usual definitions

for proposition letters and Boolean connectives. LetDLS1 be like DL5

1 + (XI51). We then

obtain:

Theorem 6.1. DLS1 is sound and(strongly) complete.

Proof. Immediate. For completeness,∆ is constructed just as in the proof of Theo-rem 2.1. �

The result is welcome at first, for it seems that standard deontic logic, including ag-glomeration(CS

1) and ‘ought implies can’(XIS1), is the proper tool to represent the sceptic

notion of ought. However, the problem is that now all the problems have disappeared:Conflicting imperatives and their impact onwhat is “sceptically obligatory” have becomeimperceivable inDLS

1. To see what goes on behind the curtain, and illustrate the relation ofsceptically defined oughts with van Fraassen’s logic, we must again admit the use of morethan oneO-operator:

Let the languageL[DL{2,F,S}

1 ] be likeL[DL{2,F }

1 ] except that we have the additional

operatorOS . Thetruth of DL{2,F,S}

1 -sentences is defined as forDL{2,F }

1 -sentences with the

additional truth definition[Df-S]. Let thesystemDL{2,F,S}

1 contain all axiom-schemes and

rules ofDL{2,F }

1 andDLS1 , and additionally the following:

(FS-1) (OF A ∧ OSB) → OF (A ∧ B);(FS-2) (

∧ni=1 O2Ai ∧ ¬OF¬(

∧Γ1 →

∨kj=2

∧Γj )) → OS(

∧Γ1 →

∨kj=2

∧Γj );

(FS-3) (∧n

i=1 O2Ai ∧ ¬OF¬∨k

j=1∧

Γj ) → OS∨k

j=1∧

Γj ;

(FS-4) (∧n

i=1 O2Ai ∧ ¬OF∧n

i=1 Ai) → OS¬∧n

i=1 Ai .

In FS-2 andFS-3, Γj is a non-empty subset of{A1, . . . ,An}. Note that (FS-3), (FS-4) arespecial cases of (FS-2). (FS-3) derives(O2A ∧ O2B ∧ ¬OF¬(A ∨ B)) → OS(A ∨ B),

which is a syntactic version of the disjunctive solution to Marcus’s dilemma: the disjunc-tion (A ∨ B) of two possibly conflicting imperative demandsA, B is obligatory, unlessthere is a consistent set of imperatives that demands even¬A ∧ ¬B. Hence one twin or

5 It might seem awkward that ifI contains contradictory imperative demands, e.g., ifI = {p1,¬p1,p2} thenwith [Df-S] the set of truths is the same as forI ′ = {p2}, i.e., the same as for anI ′ without these demands. Ifthe fulfillment of one of the demands is to remain obligatory, an alternative would be to have a “disjunction ofoughts” instead of the “ought of a disjunction”, and define:

[Df-DS] VerDS(I, v) := {A ∈ L[DL51] | ∀I ′ ∈ I − ⊥: A ∈ VerDL5

1(I ′, v)}.

In the above example,(O5p1 ∨O5¬p1) is now true under the new definition,O5(p1 ∨¬p1) remains valid, andO5⊥ remains false. Ignoring operator indices, we haveVerDLS

1(I, v) ⊆ VerDS(I, v) ⊆ VerDL5

1(I, v).



the other must be saved. From (FS-1) and(XIF1 ) we deriveOFA → ¬OS¬A, so what is

obligatory according to van Fraassen’s ought isat least ‘permitted’ in the sceptical sense. Ifguilt is a notion that attaches to a violation of a ‘sceptical ought’ only, then one may save atwin without guilt, but walking away remains forbidden. From the results of Section 5 it isimmediate thatDL{2,F,S}

1 cannot be strongly complete. However we obtain the followingresult:

Theorem 6.2. DL{2,F,S}

1 is sound and weakly complete.

Proof. Concerningsoundness, for the validity of (FS-1) assumeOFA, so someI ′ ∈ I −⊥

derivesA. If OSB is true, then allI ′ ∈ I −⊥ deriveB, soI ′ derivesA∧B andOF (A∧B)

is true. For the validity of (FS-3) and (FS-4) supposeO2Ai is true for all i, 1 � i � n.Concerning (FS-3), assume¬OF ¬

∨kj=1

∧

Γj , so allI ′ ∈ I − ⊥ arePL-consistent with

at least one disjunct, so allI ′ contain someΓj , 1 � j � k, so all I ′ derive∨k

j=1∧

Γj .

Concerning (FS-4), assume¬OS¬∧n

i=1 Ai, so someI ′ ∈ I − ⊥ is PL-consistent with∧n

i=1 Ai , so there is someI ′ ∈ I − ⊥ that contains eachAi . For the validity of (FS-2),assume againO2Ai true for all i, 1 � i � n, and assume¬OF¬(

∧

Γ1 →∨k

j=2∧

Γj ).

ThoseI ′ ∈ I − ⊥ that contain a setΓj also derive∧

Γ1 →∨k

j=2∧

Γj . Consider anI ′

that does not contain anyΓj : SinceI ′ is not consistent with allAi ∈ Γj , it must derive¬

∧

Γj for eachj , so it derives¬∨k

j=2∧

Γj . If I ′ is consistent withΓ1, it derives∧

Γ1 ∧

¬∨k

j=2∧

Γj , but then¬OF¬(∧

Γ1 →∨k

j=2∧

Γj ) is false. SoI ′ must be inconsistent

with Γ1, so it derives¬∧

Γ1 and hence also∧

Γ1 →∨k

j=2∧

Γj .For (weak)completenesswe use the construction employed in the proof of Theorem 5.3

with the following modifications:∆ is now defined with respect toDL{2,F,S}

1 -consistency(clause c in the construction of∆), and additionally we have setsOS andQS :

OS ={

B ∈ r(

L[PL]δ)

| {δ} ∪ ∆ ⊢DL{2,F,S}

1OSB

}

,

QS ={

B ∈ r(

L[PL]δ)

| {δ} ∪ ∆ ⊢DL{2,F,S}

1¬OSB

}

.

I is constructed as in the proof of Theorem 5.3 and additionally contains the followingelements: LetQS = {B1, . . . ,Bm}. For eachBj we addCI ∧ CII ∧ CIII to I , where:

– CI =∧

OS ,– CII = ¬

∨

{∧

Γ | Γ ⊆ O2 andΓ ∪ {CI} ⊢PL Bj },– CIII = ¬π1 ∧ · · · ∧ ¬πn ∧ ¬πn+1 ∧ · · · ∧ πn+j .

In CIII , π1, . . . , πn are the same proposition letters used to defineσ : minPLOF → L[PL]

(cf. the proofs of Theorems 5.1 and 5.3), andπn+1, . . . , πn+m are some other letters notoccurring inδ. We have to prove that the modifiedI still satisfies all conjuncts ofδ. Forthe non-deontic conjuncts and conjuncts of the formO2B, ¬O2B, OF B, the proof is asbefore. Consider conjuncts of the form¬OFB, OSB, ¬OSB, and the setsQF , OS , QS :



QF : For all B ∈ QF , there is noI ′ ∈ I − ⊥ such thatI ′ ⊢PL B: Due to Lemma 5.1no I ′ ∈ I − ⊥ can consist of elements ofOS alone. IfI ′ contains a sentence ofthe formC ∧ σ(C), C ∈ minPLOF , I ′ cannot contain any of the new elementsin I , and the proof is as before. SupposeI ′ contains a sentence of the formCI ∧

CII ∧ CIII andI ′ ⊢PL B. Let C ∈ QS be the sentence used in the construction ofCI ∧CII ∧CIII . First supposeCII is vacuous, i.e.,O2 contains no non-empty subsetΓ such thatΓ ∪ {CI} ⊢PL C. If CI ∧ CII ∧ CIII is the only element ofI ′ then⊢PLCI → B, soOSB is derivable by the use of(MS

1) and(ExtS1), soOF B is derivable

by the use of (NF1 ), (FS-1), so∆∪ δ is DL2,F,S

1 -inconsistent. IfI ′ contains a non-empty subsetΠ ⊆ O2 and⊢PL

∧Π ∧ CI → B, then¬OF (

∧Π ∧ CI) ∈ ∆ due

to (MF1 ) and(ExtF1 ), and¬OF (

∧Π) ∈ ∆ due to (FS-1), but thenOS¬

∧Π is

derivable by the use of (FS-4), soΠ is inconsistent withCI ∧CII ∧CIII and hencecannot be inI ′. Now supposeCII is not vacuous. Then¬OS¬CII is derivable with(CS

1), (MS1) and(ExtS1), andOFCII is derivable by the use of (FS-3), furthermore

due to (FS-1) we haveOF (CI ∧ CII ) ∈ ∆, so again ifCI ∧ CII ∧ CIII is the onlyelement ofI ′ and⊢PL CI ∧ CII → B then∆ is DL2,F,S

1 -inconsistent. IfΠ ⊆ OS

is a non-empty subset ofI ′ such thatΠ ∪ {CI ∧ CII } ⊢PL B then¬OF (∧

Π ∧

CII ) ∈ ∆ due to (FS-1), so¬OF ¬(∧

Π → ¬CII ) ∈ ∆, so with (FS-2) we deriveOS(

∧Π → ¬CII ), but thenΠ is inconsistent withCI ∧ CII and not inI ′.

OS : SupposeB ∈ OS . If I ′ ∈ I − ⊥ containsCI ∧ CII ∧ CIII thenI ′ ⊢PL B by virtueof conjunctCI . OtherwiseI ′ contains a sentence of the formC ∧ σ(C), C ∈

minPLOF (Lemma 5.1). FromOSB, OFC we obtainOFB ∧ C by the use of(FS-1), so OFB ∧ C ∈ ∆, and by the construction ofminPLOF we have⊢PLC ↔ (B ∧ C), so againI ′ ⊢PL B.

QS : If B ∈ QS then the sentenceCI ∧ CII ∧ CIII constructed forB is in I . The con-junctionCI ∧CII ∧CIII is notPL-inconsistent: Suppose it isPL-inconsistent, andassumeO2 = ∅, so CII is vacuous and⊢PL ¬CI , but this is excluded by(CS

1)

and(XIS1). Or O2 �= ∅, but then⊢PL CI → ¬CII , and since⊢PL CI ∧ ¬CII → B

by the construction ofCII , we have⊢PL CI → B, soOSB is DL{2,F,S}-derivableby (CS

1), (MS1), and(ExtS1), andδ refutable. LikewiseCI ∧ CII ∧ CIII does not

derive B, for otherwise⊢PL CI ∧ CII → B, ⊢PL CI ∧ ¬CII → B, and again⊢PL CI → B. So someI ′ ∈ I − ⊥ containsCI ∧ CII ∧ CIII which does not byitself deriveB. Elements of the formC ∧σ(C), C ∈ minPLOF are excluded fromI ′ by CII . If there is some non-emptyΓ ⊆ O2 such thatΓ ∪ {CI ∧ CII } ⊢PL B

then since also⊢PL CI ∧ ¬CII → B we haveΓ ∪ {CI} ⊢PL B, but then consis-tency ofΓ with CI ∧ CII ∧ CIII is ruled out byCII . So oneI ′ ∈ I − ⊥ does notPL-derive B. �

7. Logics for extended languages L[DLi ]

Let us return to the logicsDLi1 treated in Section 2. In the languagesL[DLi

1], thescope of deontic operators is restricted to sentences of propositional logic—formulas likeO4O4p1,¬O5(p2∧O5¬p3), in which deontic operators appear iterated or nested, are not



well-formed. But semantically, this exclusion is artificial:I is a set of descriptive sentences,and sentences likeO4p1 are descriptive, stating that according to what the imperativesdemand,p1 must hold. Natural language gives evidence that imperative expressions may—grammatically well-formed and meaningfully—refer to the existence of other demands orpermissions. Consider, e.g.:

(1) ‘Applicants shall be asked to produce statements in duplicate at their own costs.’(2) ‘Students over 15y may be permitted to leave the grounds during breaks.’

To express a possible appearance of deontic notions within the scope of deontic opera-tors, let the extended languagesL[DLi], 1� i � 5, be the smallest sets such that

(a) if A ∈ L[PL] thenA ∈ L[DLi ],

(b) if A, B ∈ L[DLi], so areO iA,¬A, (A ∧ B), (A ∨ B), (A → B), and(A ↔ B).

Semantically, we again employ a pairI, v, whereI ⊆ L[DLi ]. The truth of propositionletters is as usual, and for the deontic modalities we like to use the truth definitions givenin Section 1. Yet there is a snag: Since anyA ∈ L[DLi ] may appear in the scope of deonticoperators, the basic logicBL appearing in the definitions of operatorsO2−5 cannot bepropositional logic. We would like to make|=BL the entailment relation of the thus definedDLi -semantics itself, but at the present point this creates a circle. We must therefore takea detour: First we define thedeontic degreeδ(A) of anyA ∈ L[DLi] (⋆ is any two-placeBoolean operator):

(i) δ(p) = 0,

(ii) δ(¬A) = δ(A),

(iii ) δ(A ⋆ B) =

{

δ(A), if δ(A) � δ(B),

δ(B), otherwise,

(iv) δ(O iA) = δ(A) + 1.

For eachi, 1 � i � 5, we definesublanguagesL[DLin] = {A ∈ L[DLi ] | δ(A) � n},

n � 0. Obviously,L[DLi0] = L[PL], L[DLi

n] ⊂ L[DLin+1], andL[DLi ] =

⋃

n L[DLin].

For DLin-semantics, the truth of eachA ∈ L[DLi

n] is then defined as usual with respectto a valuationv, and (forn � 1) with respect to a setI ⊆ L[DLi

n−1]. In the DLin-truth

definitions of operatorsO2−5, the entailment relation|=BL is that of DLin−1. The ax-

iomatic systemsDLin, n � 1, are defined likeDLi

1, except that we allow allDLin-instances

of PL-tautologies andDLin−1-instances of the relevant axiom schemes(Mi

n), (Cin), or

(Nin). (Exti1) now reads:

(Extin) If ⊢DLin−1

A ↔ B then⊢DLinO iA ↔ O iB, 2� i � 5.

Theorem 7.1. All systemsDLin, 1 � i � 5, n � 1, are sound and complete with respect to

DLin-semantics.



Proof. The proof proceeds just like for DLi1. Note that strong completeness is proved

inductively for DLin, using strong completeness of DLi

n−1. �

Let the axiomatic systemsDLi be like DLin, except we now allow all DLi -instances in

PL-tautologies and relevant axiom schemes (Mi), (Ci), or (Ni), and (Exti) now reads:

(Exti) If ⊢DLi A ↔ B then ⊢DLi O iA ↔ O iB , 2 � i � 5.

Theorem 7.2. For all A ∈ L[DLi]: ⊢DLi A iff ⊢DLinA, n = δ(A).

Proof. For the left-to-right direction we do an induction on the construction of DLi : Sup-pose ⊢DLi A. Induction basis: Assume A ∈ DLi by instantiation of sentences B1, . . . ,Bk

into a PL-theorem. Then {B1, . . . ,Bk} ⊆ L[DLin], so there is the same instance for DLi

n.Assume A ∈ DLi by instantiation of sentences B1,B2 into any of the DLi -axiom-schemes(Mi), (Ci), (Ni) or (Exti). Then B1,B2 ∈ L[DLi

n−1], so there is the same instance forDLi

n. The induction step for modus ponensis trivial. For the right-to-left direction do adouble induction on the construction of DLi

n: Suppose ⊢DLinA and suppose n = 1. Induc-

tion basis: Assume A ∈ DLi1 by instantiation of sentences B1, . . . ,Bk into a PL-theorem.

Then {B1, . . . ,Bk} ⊆ L[DLi1] ⊆ L[DLi ], so there is the same instance for DLi . Assume

A ∈ DLi1 by instantiation of sentences B1,B2 into any of the DLi

1-axiom-schemes (Mi1),

(Ci1), or (Ni

1). Then B1,B2 ∈ L[PL] ⊆ L[DLi], so there is the same instance for DLi . As-sume A ∈ DLi

1 by an application of (Exti1). So (Exti1) is applied on a PL-theorem, whichis also in DLi . The induction step for modus ponensis again trivial. Suppose n � 1. Theonly non-trivial case is that A ∈ DLi

n by an application of (Extin): Then (Extin) is appliedto a DLi

n−1-theorem, which by the induction hypothesis is in DLi . �

Corollary 7.1. For all 1 � i � 5, n � 1: DLin ⊂ DLi

n+1 andDLi =⋃

n DLin.

Proof. DLi =⋃

n DLin is immediate from Theorem 7.2, and DLi

n ⊂ DLin+1 is proved by

induction on the construction of DLin just as in the right-to-left proof of Theorem 7.2. �

We are now able to give proper truth definitionsfor the deontic modalities of L[DLi ]:

I , v |= p iff v(p) = 1;

I , v |= O1A iff A ∈ I ;

I , v |= O2A iff ∃n ∈ N: ∃B ∈ I ∩ L[DL2n]: |=DL2

nB ↔ A;

I , v |= O3A iff ∃n ∈ N: ∃B ∈ I ∩ L[DL3n]: |=DL3

nB → A;

I , v |= O4A iff ∃n ∈ N: ∃B1, . . . ,Bk ∈ I ∩ L[DL4n]: |=DL4

n(B1 ∧ · · · ∧ Bk) → A;

I , v |= O5A iff ∃n ∈ N: I ∩ L[DL5n] |=DL5

nA.

Theorem 7.3. DLi is sound and complete with respect toDLi -semantics.



Proof. For soundness, the validity ofDLi -instances inPL-theorems is trivial, and so isthe validity of (Mi). The validity of(Ci) is easily established using Corollary 7.1 as wellas the soundness and completeness of eachDLi

n. For theDL4-validity of (Ext4), suppose⊢DL4 A ↔ B, so by Theorem 7.2⊢DL4

nA ↔ B for n = δ(A ↔ B). So |=DL4

nA ↔ B

by completeness ofDL4n. AssumeI, v |= O4A. So there areB1, . . . ,Bk ∈ I s.t. |=DLi

m

B1 ∧ · · · ∧ Bk → A for somem ∈ N. Supposem � n. Then we have⊢DL4m

A ↔ B byCorollary 7.1, so|=DLi

mB1 ∧ · · · ∧ Bk → B by the truth definitions for Boolean operators.

So alsoI, v |= O4B. Supposen � m. Then|=DLinB1 ∧ · · · ∧ Bk → A by Corollary 7.1, so

we also have|=DLinB1 ∧ · · · ∧ Bk → B andI, v |= O4B. The assumptionI, v |= O4B is

done likewise, soI, v |= O4A ↔ O4B, and|=DL4 O4A ↔ O4B sinceI, v were arbitrary.The proof ofDL5-validity of (Ext5) is done similarly.

The completenessof DLi is immediate fori = 1,2,3 due to Theorems 7.1, 7.2, soconsider casei = 4. As in the completeness proof forDL4

1, we construct a maximallyconsistent set∆ such that for allA ∈ L[DL4] either O4A ∈ ∆ or ¬O4A ∈ ∆. In or-der to prove this set satisfiable, we letI = {A ∈ L[DL4] | O4A ∈ ∆}, and definev asbefore (proof of Theorem 2.1). We have to showO4A ∈ ∆ iff I, v |= O4A. The left-to-right direction is again trivial. For the right-to-left direction supposeO4A /∈ ∆, so¬O4A ∈ ∆, andI, v |= O4A. Then there areB1, . . . ,Bk ∈ I s.t.|=DL4

nB1 ∧ · · · ∧ Bk → A

for somen ∈ N. By completeness ofDL4n we have⊢DL4

nB1 ∧ · · · ∧ Bk → A, so by

the use of(Ext4n+1) we obtain⊢DL4

n+1O4B1 ∧ · · · ∧ O4Bk → O4A and from Theo-

rem 7.2 we likewise have⊢DL4 O4B1 ∧ · · · ∧ O4Bk → O4A. SinceB1, . . . ,Bk ∈ I wehave{O4B1, . . . ,O

4Bk,¬O4A} ⊆ ∆, and⊢DL4 (O4B1 ∧ · · ·∧ O4Bk ∧ ¬O4A) → ⊥. Sothe r.a.a. makes∆ inconsistent. Casei = 5 is done similarly, using strong completenessof DL5

n. �

Corollary 7.2. For all A ∈ L[DLi]: |=DLi A iff |=DLinA, n = δ(A).

Proof. Immediate from Theorems 7.2 and 7.3.�

Corollary 7.3. The following equivalences hold:

I, v |= O1A iff A ∈ I ,

I, v |= O2A iff ∃B ∈ I : |=DL2 B ↔ A,

I, v |= O3A iff ∃B ∈ I : |=DL3 B → A,

I, v |= O4A iff ∃B1, . . . ,Bk ∈ I : |=DL4 (B1 ∧ · · · ∧ Bk) → A,

I, v |= O5A iff I |=DL5 A.

Proof. Immediate from the truth definitions forO i -operators and Corollary 7.2. So thetruth definitions ofO i-operators are equivalent to takingDLi as ‘basic logic’. �

Consider the followingDLi -sentences:



(DT) O i(O iA → A);(D4) O iA → O iO iA;(D4c) O iO iA → O iA;(DB) O i(A → O i¬O i¬A);(DE) ¬O i¬A → O i¬O i¬A.

None is a theorem of anyDLi . To refute(D4c) in DL5, consider the setI = {O5p1}. WehaveI |=DL5 O5p1, but notI |=DL5 p1, soI, v |= O5O5p1 is true, andI, v |= O5p1 falsefor anyv. SoI, v |= O5O5p1∧¬O5p1, so�DL5 O5O5p1 → O5p1 and�DL5 O5O5p1 →

O5p1 by completeness ofDL5. This also refutes(DT) since O5(O5p1 → p1) DL5-derivesO5O5p1 → O5p1 by the use of(C5), (Ext5), and(M5). The same setI refutestheoremhood of(DB), I = {p1} refutes(D4), and, e.g., an empty setI refutes(DE).

Now (DT is widely accepted in deontic logics modelled prohairetically. In semanticsemploying a deontic ‘ideality’ relationR, (DT) is valid if R is semi-reflexive, i.e., anyworld ideal from some standpoint is ideal with respect to itself. In Hansson’s analysis ofmonadic deontic logic, where a set of deontic formulas is modelled by a non-empty basis ofideal among possible worlds,(DT)—if admitted as well-formed—is immediate from theuniversality of ideality (cf. [9, pp. 381–382]). The same is true for Andersonian subsets ofsanction-free worlds within a set of worlds linked by an alethic accessibility relation, where(DT) is derivable as soon as alethic necessities are made to hold at the actual world (cf.[2, p. 187, Theorem OM45]). Føllesdal and Hilpinen [7] have called(DT) a “plausible-looking candidate for logical truth”. And should not, as Prior [18, pp. 255–256] hasclaimed, the acceptance of(DT) be trivial, reading it as ‘It ought to be the case that whatought to be the case is the case’?

The non-derivability of (DT) in any systemDLi brings to the fore the distinction be-tween what Barcan Marcus [15] has called the descriptive and evaluative uses of ‘ought’.Our definitions ofO i are descriptive, not evaluative: something is obligatory only if it is(at least) derivable from what has been (factually) commanded. ThusO3O3p1 means thatthere is an imperative that can be satisfied only ifO3p1 is true. But having no analyticalcontent, a violation of this imperative is not excluded, which means thatO3p1 is false, i.e.,that there is, as a matter of fact, no imperative that demands—among other things—p1. Touse Marcus’s example: that parking on highways ought to be forbidden does not entail, inthe semantics employed here, that parking on highways is forbidden.

For similar reasons, factual statements about the set of imperative demands cannot bemeaningfully employed for conditional commands. Consider the sentenceO3(¬O3A →

A). Does it entailO3A? Let I = {¬O3p1 → p1}, so I, v |= O3(¬O3p1 → p1) is true,whateverv is. However,�DL3 (¬O3p1 → p1) → O3p1, as can be seen fromI ′ = ∅,

v′(p1) = 1. SoI, v � O3p1. Possibly one may want to admit commands like ‘If it is not(already) obligatory to doα, then doα (by virtue of this command)!’, and the addresseeshould be able to conclude that (now) she has an obligation to doα. However, such con-ditional expressions are not expressible in the language and semantics presented so far.Exploring how imperative semantics can beextended to cover such and other conditionalexpressions, and how to sail around the maelstrom of self-referentiality lurking behind theabove example, must be left to further study.



8. Conclusion

When the question is what ought to be, according to some existing norms or impera-tives, it seems natural to employ a semantics that models the meaning of deontic operatorswith respect to these imperatives, rather than some notion of absolute or relative ‘better-ness’. Existing modal systems relate rather simply to ought-operators of different strength,defined with respect to a set of sentences thatdescribe what the imperatives demand. Tomix such operators is, however, generally not an option if the resulting semantics isto berepresented by a strongly complete axiomatic system, and the same holds for the use of re-strictions on the number and kind of imperatives to be considered. To cope with conflictingimperatives, we may define a van-Fraassen-type operator that prohibits the agglomerationof contradicting demands, anda ‘sceptical’ ought-operatoraccording to which somethingis obligatory only if there is no conflicting demand; both are best represented in systemsthat also contain operators which relate more directly to imperatives. In the light of thefact that natural languages do not prohibit the use of deontic modalities within imperativeexpressions, it is a welcome result that imperative semantics is possible, and sound andcompletely axiomatizable for logical languages that permit iterated and nested deontic op-erators. Here the roads of ‘ought’ defined evaluatively and ‘ought’ defined with respect togiven norms part, as demonstrated by the fact that statements like ‘it ought to be that whatought to be is the case’ remain well formed, but are true accidentally only.

References

[1] C.E. Alchourrón, The expressive conception of norms, in: R. Hilpinen (Ed.), New Studies in Deontic Logic,Reidel, Dordrecht, 1981, pp. 95–123.

[2] A.R. Anderson, The formal analysis of normative concepts, in: N. Rescher (Ed.), The Logic of Decision andAction, University Press, Pittsburgh, 1967, pp. 147–213.

[3] D.O. Brink, Moral conflict and its structure, Philos. Rev. 103 (1994) 215–247.[4] M.A. Brown, Conditional and unconditional obligations for agents in time, in: M. Zakharyaschev, et al.

(Eds.), Advances in Modal Logic, CSLI, Stanford, 2001, pp. 121–153.[5] E. Conee, Against moral dilemmas, Philos. Rev. 91 (1982) 87–97.[6] A. Donagan, Consistency in rationalist moral systems, J. Philos. 81 (1984) 291–309.[7] D. Føllesdal, R. Hilpinen, Deonticlogic: an introduction, in [10], pp. 1–35.[8] J. Hansen, Sets, sentences, and some logics about imperatives, Fundamenta Informaticae 48 (2001) 205–226.[9] B. Hansson, An analysis of some deontic logics, Noûs 3 (1969) 373–398. Reprinted in [10], pp. 121–147.

[10] R. Hilpinen (Ed.), Deontic Logic: Introductoryand Systematic Readings, Reidel, Dordrecht, 1971.[11] J.F. Horty, Moral dilemmas and nonmonotonic logic, J. Philos. Logic 23 (1994) 35–65.[12] J.F. Horty, Nonmonotonic foundations for deontic logic, in: D. Nute (Ed.), Defeasible Deontic Logic,

Kluwer, Dordrecht, 1997, pp. 17–44.[13] D. Jacquette, Moral dilemmas, disjunctive obligations, and Kant’s principle that ‘ought’ implies ‘can’, Syn-

these 88 (1991) 43–55.[14] S. Kanger, New foundations for ethical theory, in [10], pp. 36–58.[15] Marcus, R. Barcan, Iterated deontic modalities, Mind 75 (1966) 580–582.[16] Marcus, R. Barcan, Moral dilemmas and consistency, J. Philos. 77 (1980) 121–136.[17] I. Niiniluoto, Hypothetical imperatives and conditional obligation, Synthese 66 (1986) 111–133.[18] A.N. Prior, Formal Logic, Clarendon Press, Oxford, 1955.[19] T.J. Smiley, The logical basis of ethics, Acta Philos. Fennica 16 (1963) 237–246.[20] E. Stenius, The principles of a logic of normative systems, Acta Philos. Fennica 16 (1963) 247–260.[21] B. van Fraassen, Values and the heart’s command, J. Philos. 70 (1973) 5–19.


Journal of Applied Logic 3 (2005) 484–511

www.elsevier.com/locate/jal

Conflicting imperativesand dyadic deontic logic

Jörg Hansen

Institut für Philosophie, Universität Leipzig, Beethovenstraße 15, D-04107 Leipzig, Germany

Available online 2 June 2005

Editors: A. Lomuscio and D. Nute

Abstract

Often a set of imperatives or norms seems satisfiable from the outset, but conflicts arise whenways to fulfill all are ruled out by unfortunate circumstances. Semantic methods to handle normativeconflicts were devised by B. van Fraassen and J.F. Horty, but these are not sensitive to circumstances.The present paper extends these resolution mechanisms to circumstantial inputs, defines dyadic de-ontic operators accordingly, and provides a sound and (weakly) complete axiomatic system for suchdeontic semantics. 2005 Elsevier B.V. All rights reserved.

Keywords:Logic of imperatives; Deontic logic; Conflict of norms

1. The question of normative conflicts

Do moral conflicts exist? The orthodox belief in the 1950’s was that such conflicts onlyexist at first glance—the seemingly conflicting obligations arising from the application ofmerely incomplete principles. Instead, what is actually obligatory must be determined bya careful moral deliberation that involves considering and weighing all relevant facts andreasons, and cannot produce conflicting outcomes. Among the first that came to reject thisview were E.J. Lemmon[27] and B. Williams[43]: Lemmon observed that in cases of truemoral dilemma, one does not know the very facts needed to determine which obligationmight outweigh the other. Williams arguedin reductiothat if, in case of conflicting oughts,

E-mail address:[email protected](J. Hansen).

1570-8683/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jal.2005.04.005



there is just one thing one ‘actually’ ought to do, then feelings of regret about having notacted as one should have are out of place and one should not mind getting into similarsituations again. To avoid the derivation of the ought of a contradiction from two oughtsof equal weight but with contradictory contents, Williams argued that deontic logic shouldgive up the agglomeration principle

(C) OA ∧ OB → O(A ∧ B).

Lemmon had no such qualms: he advocated dropping the Kantian Principle ‘ought impliescan’

(KP) OA → ⋄A,

thus allowing for obligations to bring about the impossible, and concluded:

“I should like to see a proper discussion of the arguments that go to resolve moraldilemmas, because I do not believe that this is an area of total irrationality, though I donot believe that a traditional logical approach (the logic of imperatives, deontic logic,and whatnot) will do either.”

Regarding commands and legal norms, G.H. von Wright[45, Chapter 7], like H. Kelsen[23, p. 211]at the time, excluded the coexistence of conflicting norms from the samesource: The giving of two conflicting norms is the expression of an irrational will; it is aperformative self-contradiction and as such a pure fact that fails to create a norm. E. Stenius[40] and later C.E. Alchourrón and E. Bulygin[1] rejected this view: A system of normsthat is impossible to obey might be unreasonable and its norm-giver blameworthy, but itsexistence does not constitute a logical contradiction—conflicts are ubiquitous in systems ofpositive law and logic cannot deny this fact. In his later theory, von Wright[49] concedesthat existing normative systems may or may not be contradiction-free, and reformulatesdeontic principles as meta-norms for consistent norm-giving. Kelsen[24] later came toview logic as inapplicable to law.

2. Van Fraassen’s proposal and Horty’s variation

2.1. Van Fraassen’s operatorOF

Not taking sides,pro or contrathe existence of genuine normative conflicts, but arguingthat the view in favor seems at least tenable, B. van Fraassen[11] took up the burden offinding plausible logical semantics that could accommodate conflicting obligations. Theintended semantics should accept the possible truth of two deontic sentencesOA, O¬A

without committing the norm-subject to the absurd by makingO(A ∧ ¬A) true, for vanFraassen wanted to keep the Kantian Principle. Given the existence of certain imperativesin force, i.e., imperatives that are left as valid, relevant, not overridden, etc. by some un-specified deliberation process, van Fraassen’s idea was to make these imperatives part ofthe logical model, and to describe something as obligatory if it serves to satisfy some, notnecessarily all, imperatives. Formally, letI be the set of imperatives in force,B be the setof possible states of affairs, andi+ ⊆ B be the possible states of affairs where the impera-tive i ∈ I is considered fulfilled. Let‖A‖ ⊆ B be the set of possible states of affairs where



the indicative sentenceA is considered true. Finally, letscore(v) be the set of all impera-tives that are fulfilled in the state of affairsv: score(v) = {i ∈ I | v ∈ i+}. Van Fraassenthen defines:

[Df-F] OF A is true iff ∃v ∈ ‖A‖: ∀v′ ∈ ‖¬A‖: score(v) � score(v′).

SoA is obligatory if and only if (iff) there is some score that can be achieved whenA istrue, which is not included in any score that could be achieved when¬A is true. In otherwords,A is obligatory iff there are imperatives that can only be (collectively) satisfiedwhenA is true, but not whenA is false.

By slightly changing the viewpoint, van Fraassen’s proposal might also be describedin the following way: letI be a set not of imperatives, but of indicative sentences in thelanguageLBL of some basic logicBL. The motivation is thatI contains one sentenceAfor each imperativei in force that is true in exactly those states of affairs in which theimperative is fulfilled, i.e.,‖A‖ = i+. BL is assumed to be compact and the turnstile inΓ ⊢BL A means a classical consequence relation that characterizesBL, Γ ⊆ LBL, A ∈

LBL. Let theremainder setΓ ⊥ A be the set of all maximal subsets that do not deriveA,i.e., of allΓ ′ ⊆ Γ such that (i)Γ ′ �BL A, and (ii) there is noΓ ′′ such thatΓ ′ ⊂ Γ ′′ ⊆ Γ

andΓ ′′ � A. Then Df-F is equivalent to Df-F* (⊤ means an arbitrary tautology):1

[Df-F∗] OF A is true iff ∃I ′ ∈ I⊥¬⊤: I ′ ⊢BL A

SoA is obligatory iff it is derivable from a maximally consistent subset ofI . So somethingis obligatory if it is required for doing ‘the most’: if it is necessitated by a strategy to fulfillso many imperatives that no one who satisfies these as well could satisfy more. While aparallel operation for belief change is known as ‘credulous reasoning’, to call it ‘orthodox’might seem more appropriate: the agent is not released from any of her obligations as longas they are fulfillable, even if this fulfillment is at the expense of violating other norms.

To see how van Fraassen’s semantics work, first letI = {A,B}, whereA andB aresupposedly contingent and independent. There are no conflicts,I is consistent andOF A,OF B andOF (A ∧ B) are all true sinceI derivesA, B andA ∧ B. Thus agglomeration ofcontents is permitted so long as the underlying imperatives do not conflict. For the case ofconflict, changeI into {A ∧ C,B ∧ ¬C}; OF A andOF B are true sinceA andB derivefrom the maximally consistent sets{A∧C} and{B ∧¬C}, butOF (C ∧¬C) is false sinceno consistent subset derivesC ∧ ¬C. The same is true forOF (A ∧ B) though{A ∧ B} isconsistent: the truth ofA ∧ B is not necessary for maximal norm satisfaction.

An axiomatic systemDLF that is (weakly) complete with regard to van Fraassen’s se-mantics is defined by the following axiom-schemes, in addition toBL-instances andmodusponens(cf. [17, Section 5], ⊥ means an arbitrary contradiction, and the index ‘F ’ here andbelow indicates that the deontic operators occurring in the axiom scheme are thus indexed):

(MF ) OF (A ∧ B) → (OF A ∧ OF B)

(PF ) ¬OF ⊥

(NF ) OF ⊤

(ExtF ) If ⊢BL A ↔ B then ⊢DL OF A ↔ OF B

1 Cf. Horty’s [20, Theorem 2].



To van Fraassen’s own puzzlement, the cases where agglomeration remains permissible donot seem axiomatizable: object language does not reveal whether particularA andB ofsomeOF A andOF B are derived from the demands of imperatives that do not conflict andsoOF (A ∧ B) should be supported.2

2.2. The skeptical operatorOS

The invalidation of the agglomeration principle by van Fraassen’s semantics did notmake them popular (cf. Donagan[8, p. 298]). Moreover, let aP F -operator expressingpermission be defined in the usual way asP F A =def ¬OF ¬A, and consider againI =

{A ∧ C,B ∧ ¬C}: thenOF (A ∧ C) is true, and so isOF ¬C. Applying (MF ) and (ExtF ),OF ¬(A∧C) must be true, henceP F (A∧C) is false. So not even the obligatory is alwayspermitted, which seems strange (cf. Jacquette[22]).

In reaction to the dismissal of the agglomeration principle, Donagan[8] and Brink[7]have claimed that even if there could be a normative demand forA and a conflicting de-mand forB, with ⊢BL A → ¬B, it need not follow that the norm-subject has an obligationto realizeA andan obligation to realizeB. Rather, there should just be a disjunctive oblig-ation to realizeA or B. Given competing normative standards of equal weight, the strategyof this reasoning is not to trust a single standard, but to consider obligatory only whatallstandards demand. LetI be as before. Varying van Fraassen’s truth definition, Horty[20]has formalized this ‘skeptical’ ought as follows:3

[Df-S] OSA is true iff ∀I ′ ∈ I⊥¬⊤: I ′ ⊢BL A

SoOSA is true iff A is derivable from all maximally consistent subsets ofI . ‘skeptical’is the term used in the epistemic-oriented literature, yet ‘legalist’ also seems fitting, sincea norm violation is never pronounced as obligatory even if it is inevitable. This does notlet the agent off the hook: by doing what is obligatory in this sense, a maximum of normswill, by necessity, get satisfied.

Let againI = {A ∧ C,B ∧ ¬C}. OSA andOSB are false andOS(A ∨ B) is true: justA ∨ B, but neitherA norB are derived by both of the two consistent subsets{A ∧ C} and{B ∧ ¬C}. P S(A ∧ C) is also true:A,C were assumed to be contingent and independent,so the maximally consistent subset{A ∧ C} ⊆ I does not derive¬(A ∧ C). So what isOF -obligatory is at leastP S -permitted.

A complete axiomatic systemDLS is defined by the axiom-schemes (MS ), (CS ), (PS ),(NS ), and (ExtS ), together withBL-instances andmodus ponens. Since the truth definitionsfor OF A andOSA merely depend on a setI andBL, mixed expressions such asOF A ∧

¬OSA are meaningful and may be accepted as well-formed. Then

(CSF) OSA ∧ OF B → OF (A ∧ B)

2 So agglomeration requires some consistency check of the underlying imperatives’ contents. Van der Torre andTan[41,42]proposed a two-phase deontic logic, where ‘consistent aggregation’ must take place before weaken-ing. For the present imperative semantics, I suggested a bimodal approach in[17] with an operatorO2 that ‘moredirectly talks about the imperatives’. For comparisons and a new proposal cf.[15].

3 More in parallel to van Fraassen’s original definition, one may equivalently define:

[Df-S∗] OSA is true iff ∀v ∈ ‖¬A‖: ∃v′ ∈ ‖A‖: score(v) ⊂ score(v′)



is valid, and the mixed systemDL{F,S}—containing the axiom schemes forDLF , DLS ,the axiom scheme (CSF), all instances ofBL-theorems andmodus ponens—is sound and(weakly) complete (cf.[17, Section 6]).

3. Predicaments and dyadic deontic logic

Arguing for the possibility of moral conflicts, R. Barcan Marcus[33] explained:

“Under the single principle of promise keeping, I might make two promises in all goodfaith and reason that they will not conflict, but then they do, as a result of circumstancesthat were unpredictable and beyond my control”.

Note that there is no conflict at the outset: any dilemma could have been averted by notpromising anything. Moreover, there might have been some point in time at which keepingboth promises was possible: having 500 $ with me and another 1000 $ in the office, Ipromise Raoul and Johnny 500 $ each on Saturday with every intention of paying them onMonday, only to find out that the office had been burglarized over the weekend. Donagan[8] argues that this is not a genuine conflict, because three resolving principles apply:(i) one must not make promises one cannot or must not keep, (ii) all promises are made withthe implicit condition that they are void if they cannot or must not be kept, (iii) one mustnot make promises when one does not believe that the other party has fully understood (ii).But as I well knew beforehand, neither Raoul nor Johnny are going to let me off the hook,regardless of what may happen at the office. According to (iii), I was wrong to make thepromises. So am I entitled to break them (both)?—We have here what G.H. von Wright[48] terms a ‘predicament’: a situation from which there is no permitted way out, but towhich there also is no permitted inlet. The normative order is consistent, it is only throughone’s own fault that one finds oneself in a predicament.4 Von Wright then asks:

“The man in a predicament will, of necessity, react in some way or other, either dosomething or remain passive. Even though every reaction of his will be a sin, is it notreasonable to think that there is yet something he ought to do rather than anything else?To deny this would be to admit that it makes, deontically, no difference what he does.But is this reasonable? (. . .) If all our choices are between forbidden things, our duty isto choose the least bad thing”.[48, p. 80]

4 That predicamentsonly arise from an agent’s own faults, and not through misfortune or the wrongdoings ofothers, is a view von Wright and Donagan ascribe to Thomas Aquinas, but this does not seem quite correct: inthe discussion of oaths (Summa TheologicaII.II Qu. 89 art. 7 ad 2), Thomas considers the objection that it wouldsometimes be contrary to virtue, or an obstacle to it, if one were to fulfill what one has sworn to do—so oathsneed not always be binding. In answering, Thomas distinguishes oaths that are unlawful from the outset, where aman sinned in swearing, and oaths that could be lawful at the outset but lead to an evil result through some newand unforeseen emergency: fulfillment of such oaths is unlawful.



Sub-ideal demands are usually represented by a dyadic deontic sentenceO(A/C), mean-ing that it ought to be thatA given C is true. By accepting all instances of (DD-⊤)O(A/⊤) → P(A/⊤) as a logical truth in[48], von Wright dismisses an inconsistent nor-mative system as ‘conceptual absurdity’: ifA is obligatory on tautological conditions (i.e.,unconditionally obligatory), then there cannot be a likewise unconditional obligation to thecontrary. Although von Wright originally used the stronger (DD)O(A/C) → P(A/C) forarbitraryC (axiom A1 of the ‘old system’ in[44], and axiom B1 of the ‘new system’ in[46]), he later turned against it, arguing that while morality makes no conflicting claims,it is not a logical impossibility that conflicting promises can give rise to predicaments.5

Dyadic operators seem essential for even making this distinction.6

Turning object language oughts into a special sort of conditionals does not necessarilyimply a change in the formalization of the background imperatives: consider the setI =

{(¬C ∨ A), (¬C ∨ ¬A)}, corresponding to background imperatives in the usual way.I isalso its single maximally consistent subset, which derives¬C, soOF ¬C andOS¬C areboth true. But a single standard is no longer available onceC becomes true: the imperativeshave not all been fulfilled (otherwise one would not be in conditionC), and any maximalset of imperatives that is consistent with the given circumstances cannot contain all. So theproposal is to callA obligatory in caseC iff A is necessary for doing ‘the most’ that canbe achieved, given the truth ofC. Formally:

[Df-DF] OF (A/C) iff ∃I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL A

So OF (A/C) is true iff there is some set, among the maximal subsets ofI consistentwith C, that together withC derivesA. This is obviously a conservative extension of thedefinition given for the unconditional case, so we may defineOF A =def O

F (A/⊤).If a cautious, disjunctive approach were appropriate for cases of conflict, then it would

be hard to see why predicaments should be treated differently: that conflicts must be ac-counted for at the outset, but analogues of Buridan’s ass cannot be brought about by fateor unpredictable human nature, would hardly be plausible. Distrusting any single standard,such an approach would accept, given the circumstancesC, only what is necessary by anystandard that could still be met—no crying over spilled milk. Formally:

[Df-DS] OS(A/C) iff ∀I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL A

SoOS(A/C) is true iff all the maximal subsets ofI consistent withC deriveA, given thetruth of C. This is again a conservative extension of the unconditional case, so one maydefineOSA =def O

S(A/⊤).After a comparison of the above definitions with similar approaches namely in the study

of nonmonotonic reasoning, I will give an axiomatic dyadic deontic systemDDL{F,S},which I prove to be sound and (only) weakly complete with respect to the above semantics.

5 Cf. [47], [48, pp. 36, 81, 89].6 Below, I extend the treatment of conflicts to the area of predicaments, and do not follow von Wright in ruling

out conflicts. However, this can easily be done by axiomatically adding (DD-⊤) to the system presented below.



4. Comparisons

Though the truth definitions introduced in the preceding section naturally extend theproposals of van Fraassen and Horty for dealing with normative conflicts to the dyadiccontext and the related problem of predicaments, and though their resolution mechanismsare not exactly new (cf. below), there has not been much discussion of these conceptsin the deontic logic literature. Notably, Horty’s own dyadic operator in[21] is definedwith respect to (simply) maximally non-conflicting sets of prima facie oughts, and it isdisregarded that their joint demands may now be inconsistent with the situation. But themore general literature on nonmonotonic reasoning offers a range of parallel concepts.

RegardingOS , the most obvious parallel is Kratzer and Lewis’s premise semantics in[25] and[30] which has a set of formulasH (the premises) to define counterfactuals inmuch the same way as the setI is used here in the definition of deontic conditionals.Considering Kratzer’s definition, and setting aside the world-relativity ofH , let SH,C =

{H ′ ⊆ H | H ′ �BL ¬C} be the set of all subsets ofH that are, according to some basiclogic BL, consistent withC. A counterfactual conditional�→ is defined in the followingway:

H |= C �→ A iff ∀H ′ ∈ SH,C : ∃H ′′ ∈ SH,C : H ′ ⊆ H ′′,H ′′ ∪ {C} ⊢BL A

In other words,C �→ A is true iff each set inSH,C has a superset inSH,C that impliesC → A.7 The truth definition is tailored for a basic logic that may fail compactness and soaccommodates setsH with ever-largerC-consistent subsets, but no maximal ones. Here,BL was assumed to be compact, and we obtain:

Observation 1 (Relation to premise semantics). For any setI ⊆ LBL:

I |= OS(A/C) iff I |= C �→ A

Proof. Left-to-right. Suppose thatC �→ A is false, so there is someI ′ ∈ SI,C : ∀I ′′ ∈

SI,C : if I ′ ⊆ I ′′ then I ′′ ∪ {C} �BL A. I ′ �BL ¬C, so by definition there is someI ′′ ∈

I⊥¬C such thatI ′ ⊆ I ′′. SoI ′′ ∪ {C} �BL A, so¬∀I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL A and soOS(A/C) is false.

Right-to-left.SupposeOS(A/C) is false, so∃I ′ ∈ I⊥¬C: I ′ ∪ {C} �BL A. ThenI ′ ∈

SI,C , and by definition ofI⊥¬C there is no otherI ′′ ∈ SI,C : I ′ ⊆ I ′′, so∀I ′′ ∈ SI,C : ifI ′ ⊆ I ′′ thenI ′′ ∪ {C} �BL A, andC �→ A is false.

Then, the definition ofOS parallels that of a consequence relation associated to a Poolesystem without constraints[35]: This has two setsΓ,∆ of formulas, the facts and thedefaults. A scenario is a set∆′ ∪ Γ such that∆′ ⊆ ∆ and∆′ ∪ Γ �BL ⊥. A ‘maximal

7 Lewis’s [30] variation requires this property only of non-emptyH ′ ∈ SH,C . This corresponds to replacing

I ′ ∪ {C} ⊢BL A in the truth definition forOS with ∃B1, . . . ,Bn ∈ I ′: ⊢BL (B1 ∧ · · · ∧ Bn ∧ C) → A to producea regular instead of a normal operator. Lewis notes (p. 233) that for deontic conditionals, the premises of thepremise semantics might be understood to be “something that ought to hold”, so he is to be credited for theimperative semantics employed here.



scenario’ is one where∆′ ∈ E(Γ ), E(Γ ) being the set of all∆′ ⊆ ∆ such that∆′∪{Γ } �BL

⊥, and for all∆′′ ⊆ ∆, if ∆′ ⊂ ∆′′ then∆′′ ∪ {Γ } ⊢PL ⊥. Cn beingBL-consequence, apredictionA from the factsΓ and the defaults∆ is then defined as:

Γ |∼skept(∆) A =def A ∈⋂

∆′∈E(Γ )

Cn(∆′ ∪ Γ )

SoA is predicted fromΓ and∆ if all maximal scenarios deriveA. Likewise, a ‘credulousprediction’ can be defined as

Γ |∼cred(∆) A =def A ∈⋃

∆′∈E(Γ )

Cn(∆′ ∪ Γ )

(cf. Brass[5] for the analogy and notation). The following is then immediate:

Observation 2 (Relation to Poole systems). For any setI ⊆ LBL:

I |= OS(A/C) iff {C} |∼skept(I ) A

I |= OF (A/C) iff {C} |∼cred(I ) A

Regarding theOF -operator, it is perhaps not quite as obvious that its correspondingP F -operator is closely related to the ‘X-logics’ of Siegel and Forget[10,38]: The consequencerelation|∼X of these logics holds between a set of formulasΓ and a formulaA moduloasetX of formulas, where the definition is

Γ |∼X A iff Cn(

Γ ∪ {A})

∩ X = Cn(Γ ) ∩ X

As Makinson[31] pointed out,X can be understood as a set of ‘bad’ propositions that oneis to avoid. SoΓ |∼X A is true iff A can be realized together withΓ without increasingthe set of ‘bad’ proposition above those that were already true givenΓ . Here we have asetI of ‘desired’ propositions, so a statement seems ‘bad’ if it asserts that some desiredproposition be false, e.g.,¬A is true for someA ∈ I , or that at least oneA1, . . . ,An ∈ I isfalse, i.e.,¬(A1 ∧ · · · ∧¬An) is true. LetI� = {¬

∧

{A1, . . . ,An} | {A1, . . . ,An} ⊆ I,1�n � card(I )} be the ‘bad set’ corresponding toI . We then obtain:

Observation 3 (Relation toX-logics). For any setI ⊆ LPL, X = I�:

I |= P F (A/C) iff {C} |∼X A

Proof. Right-to-left. Suppose{C} |≁X A, so Cn(C ∧ A) ∩ I� �= Cn(C) ∩ I�, so bymonotony ofCn there is aB� ∈ Cn(C ∧ A) ∩ I� such thatB� /∈ Cn(C) ∩ I�. Bydefinition, B� = ¬(B1 ∧ · · · ∧ Bn) for someB1, . . . ,Bn ∈ I . The first fact provides{C ∧A} ⊢BL ¬(B1 ∧· · ·∧Bn) and by contraposition{B1, . . . ,Bn} ⊢BL C → ¬A. From thesecond fact we obtain{C} �BL ¬(B1 ∧ · · · ∧ Bn), so by contraposition{B1, . . . ,Bn} �BL

¬C, so {b1, . . . ,Bn} ⊆ I ′ for some maxi-consistentI ′ ∈ I⊥¬C, and so there is someI ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL ¬A, soI |= OF (¬A/C) and by definitionI � P F (A/C).

Left-to-right. SupposeI � P F (A/C), so there is aI ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL ¬A.By compactness ofBL there are{B1, . . . ,Bn} ⊆ I ′ with {B1, . . . ,Bn} ⊢BL C → ¬A,



and {B1, . . . ,Bn} �BL ¬C since they are inI ′. By definition there is aB� ∈ I� suchthat B� = ¬(B1 ∧ · · · ∧ Bn). Hence{C ∧ A} ⊢BL B�, so B� ∈ Cn(C ∧ A) ∩ I�, and{C} �BL B�, soB� /∈ Cn(C) ∩ I�. SoCn(C ∧ A) ∩ I� �= Cn(C) ∩ I�, and{C} |≁X A.

A unified treatment of both, skeptical and credulous consequence can be found inBochman’s ‘epistemic states’-semantics in[2–4]. Epistemic states, equivalent to the cumu-lative models in[26], are triplesE = 〈S,≺, ℓ〉, whereS is a set of objects (belief states),≺ some asymmetric ‘preference’ relation onS, andℓ a labeling function that assigns eachstates ∈ S a deductively closed theory. minS′ = {s ∈ S′ | ∀t ∈ S′, t �= s: t ⊀ s} is the setof minimal states inS′ ⊆ S. 〈A〉 = {s ∈ S | ¬A /∈ ℓ(s)} is the set of belief states consistentwith A. For eachA, 〈A〉 must be≺-smooth, i.e., for anys ∈ 〈A〉, eithers ∈ min〈A〉 or thereis somet ∈ min〈A〉 with t ≺ s. With BL as basic logic, Bochman’s definitions for skepticaland credulous consequence relations|∼ and|≈ are:

A |∼E B iff ∀s ∈ min〈A〉: ℓ(s) ⊢BL A → B

A |≈E B iff 〈A〉 = ∅ or ∃s ∈ min〈A〉: ℓ(s) ⊢BL A → B

Observation 4 (Relation to Bochman’s epistemic states). Let I ⊆ LPL, and let the cor-responding ‘epistemic state’EI = 〈S,≺, ℓ〉 be such that(i) S = P(I ), (ii) ℓ(s) = Cn(s),and (iii) s ≺ t iff t � s. Then

I |= OS(A/C) iff C |∼EIA

I |= OF (A/C) iff �PL ¬C andC |≈EIA

Proof. I prove first (a)I ′ ∈ min〈A〉 iff I ′ ∈ I⊥¬A, (b)EI is an epistemic state, (c)〈A〉 = ∅

iff ⊢BL ¬A: For (a), by definition〈A〉 = {I ′ ⊆ I | I ′ �BL ¬A}, so〈A〉 is the set of subsetsof I consistent withA. I ′ ∈ min〈A〉 means that for anyI ′′ ∈ min〈A〉, I ′ �= I ′′: I ′′ ⊀ I ′.By definition for anyI ′′ ∈ 〈A〉, I ′ �= I ′′: I ′ � I ′′. This means there is noI ′′ ∈ I consistentwith A such thatI ′ ⊆ I ′′, which meansI ′ ∈ I⊥¬A. For (b), if I ′ ⊆ I is in 〈A〉, i.e., it isconsistent withA, andI ′ /∈ I⊥¬A then by definition ofI⊥¬A there is someI ′′ ∈ I⊥¬A

such thatI ′ ⊂ I ′′, so there is someI ′′ ∈ min〈A〉 with I ′′ ≺ I ′. So EI is smooth, henceit is an epistemic state. For (c),〈A〉 = ∅ iff {I ′ ∈ P(I ) | I ′ �BL ¬A} = ∅ iff ∅ ⊢BL ¬A

holds by monotony ofBL. Putting together, we get:I |= OS(A/C) iff ∀I ′ ∈ I⊥¬C: I ′ ∪

{C} ⊢BL A iff ∀I ′ ∈ min〈C〉: I ′ ⊢BL C → A iff C |∼EIA. Likewise: I |= OF (A/C) iff

∃I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢BL A iff ∃I ′ ∈ min〈C〉: I ′ ⊢BL C → A iff 〈C〉 �= ∅ and [〈C〉 = ∅

or ∃I ′ ∈ min〈C〉: I ′ ⊢BL C → A] iff �BL ¬C andC |≈EIA.

A final parallel brings us back to deontic logic, namely the multiplex preference seman-tics of Goble in[12–14], where a multitude of preference relations enables definitions like‘all-best’ (universally preferred) and ‘some-best’ (existentially preferred), which are thenused in definitions of deontic operators. That, in the finite case, such semantics correspondsclosely to the present account will be explicated in Section6. Regarding meta-theory, fora somewhat more general semantic setting the skeptical consequence relation was axiom-atized by Kraus, Lehmann and Magidor[26], and the credulous consequence relation by



Bochman[2]. However, a completeness proof for a system that includes both8 seems to bemissing so far and this is what I shall now turn to.

5. The dyadic deontic logic DDL{F,S}

Let the basic logic be propositional logicPL: The alphabet has proposition lettersProp= {p1,p2, . . .}, operators ‘¬’, ‘ ∧’, ‘ ∨’, ‘ →’, ‘ ↔’ and parentheses ‘(’, ‘)’. The lan-guageLPL is defined as usual.

∧

,∨

in front of a set of sentences means their conjunctionand disjunction, and, e.g.,

∧ni=1 Ai further abbreviates

∧

{Ai, . . . ,An}. Semantically, val-uation functionsv : Prop→ {1,0} define the truth of sentencesA ∈ LPL as usual (writtenv |= A), B is the set of all such valuations, and‖A‖ is the extension{v ∈ B | v |= A} of A.PL is a sound and complete axiomatic system, and⊢PL A means thatA is provable inPL.

The alphabet of thelanguageLDDL{F,S} additionally has the operators ‘OF ’, ‘ OS ’, andthe auxiliary ‘/’. DDL{F,S} is then the smallest set such that

(a) for allA,C ∈ LPL, OF (A/C) andOS(A/C) ∈ L{F,S}

DDL ,

(b) if A,B ∈ L{F,S}

DDL , so are¬A, (A ∧ B), (A ∨ B), (A → B), (A ↔ B).

Outer parentheses will be mostly omitted. We defineP ∗(A/C) =def ¬O∗(¬A/C), where* is F or S. For simplification we do not permit mixed expressions and nested deonticoperators likep1 ∧ OS(p2/p1), P S(OF (p2/p2)/p1).

For DDL{F,S}-semantics, the truth ofDDL{F,S}-sentences is defined with respect to asetI ⊆ LPL (Boolean operators being as usual):

I |= OF (A/C) iff ∃I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A

I |= OS(A/C) iff ∀I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A

If I |= A, A is calledDDL{F,S}-satisfiable, andDDL{F,S}-valid if I |= A for all I ⊆ LPL(we write |=DDL{F,S} A).

Consider the followingaxiom-schemes(* is the uniform indexF or S):

(CExt∗) If ⊢PL C → (A ↔ B) then ⊢DDL{F,S} O∗(A/C) ↔ O∗(B/C)

(ExtC∗) If ⊢PL C ↔ D then ⊢DDL{F,S} O∗(A/C) ↔ O∗(A/D)

(DM∗) O∗(A ∧ B/C) → (O∗(A/C) ∧ O∗(B/C))

(DCS) OS(A/C) ∧ OS(B/C) → OS(A ∧ B/C)

(DCSF ) OS(A/C) ∧ OF (B/C) → OF (A ∧ B/C)

(DNS) OS(⊤/C)∣

∣ (DN-R∗) If �PL ¬C then ⊢DDL{F,S} O∗(⊤/C)

(DPF ) P F (⊤/C)∣

∣ (DP-R∗) If �PL ¬C then ⊢DDL{F,S} P ∗(⊤/C)

(Cond∗) O∗(A/C ∧ D) → O∗(D → A/C)

(CCMon∗) O∗(A ∧ D/C) → O∗(A/C ∧ D)

(RMonF ) P F (D/C) → (OF (A/C) → OF (A/C ∧ D))

8 One might add a third (monadic) deontic modalityO2 that ‘more directly talks about the imperatives’ toaxiomatize consistent agglomeration, but I must leave the details to future study (cf.[17, Section 6]for theresulting monadic systemDL{2,F,S}).



(RMonFSS) P F (D/C) → (OS(A/C) → OS(A/C ∧ D))

(RMonSSF) P S(D/C) → (OS(A/C) → OF (A/C ∧ D))

Thesystem DDL{F,S} is then the set such that (i) allLDDL{F,S} -instances ofPL-tautologiesare in DDL{F,S}, (ii) all LPL-instances of the above axiom schemes are inDDL{F,S},and (iii) DDL{F,S} is closed undermodus ponens. If A ∈ DDL{F,S} we write⊢DDL{F,S} A

and callA provable in DDL{F,S}. Γ ⊆ LDDL{F,S} is DDL{F,S}-inconsistentiff there areA1, . . . ,An in Γ , n � 1, with ⊢DDL{F,S} (A1 ∧ · · · ∧ An) → ⊥, otherwiseΓ is DDL{F,S}-consistent. A ∈ LDDL{F,S} is DDL{F,S}-derivablefrom Γ ⊆ LDDL{F,S} (writtenΓ ⊢DDL{F,S}

A) iff Γ ∪ {¬A} is DDL{F,S}-inconsistent.

Theorem 1. The following are DDL{F,S}-derivable(* is a uniform index or as indicated):

(RefS) OS(A/A)∣

∣ (Ref-R∗) If �PL ¬A then ⊢DDL{F,S} O∗(A/A)

(RW∗) If ⊢PL A → B then ⊢DDL{F,S} O∗(A/C) → O∗(B/C)

(Pres∗) O∗(⊥/C) → (O∗(A/D) → O∗(A ∧ ¬C/D))

(CMon∗) O∗(D/C) → (O∗(A/C) → O∗(A/C ∧ D))∣

∣ SSS, SFF, FSF(Cut∗) O∗(D/C) → (O∗(A/C ∧ D) → O∗(A/C))

∣

∣ SSS, SFF, FSF(Or∗) (O∗(A/C) ∧ O∗(A/D)) → O∗(A/C ∨ D)

∣

∣ SSS, SFF, FSF(DR∗) O∗(A/C ∨ D) → (O∗(A/C) ∨ O∗(A/D))

∣

∣ FFF, SFS,SSF(FH∗) P ∗(C/D) → (O∗(A/C ∨ D) → O∗(A/C))

∣

∣ FFF, FSS, SSF(FH+∗) P ∗(A → C/C ∨ D) → (O∗(A/C ∨ D) → O∗(A/C))

∣

∣ FFF, FSS, SSF(Trans∗) P ∗(A/A ∨ B) ∧ P ∗(B/B ∨ C) → P ∗(A/A ∨ C)

∣

∣ FFF, FSS, SFS(P -LoopF ) P F (A2/A1) ∧ · · · ∧ P F (An/An−1) ∧ P F (A1/An) → P F (An/A1)

(LoopS) OS(A2/A1) ∧ · · · ∧ OS(An/An−1) ∧ OS(A1/An) → OS(An/A1)

Proof. All easy and left to the reader.

Regarding axioms and theorems, (CExt∗) is a contextual extensionality rule for conse-quents, and (ExtC∗) an extensionality rule for antecedents. (DM∗) and (DC∗) are dyadicversions of their monadic analogues. TheOS -axioms are like those of Kraus, Lehmannand Magidor[26], but (CondS ) and (CCMonS ) equivalently replace (OrS ) and (CMonS ),and (DP-RS ) is added. TheOF -axioms are those of Bochman[2], where his (PresF ) isstrengthened to (DPF ). Instead of (CondF ), (CCMonF ) and (RMonF ), Goble[14] moreelegantly employs (TransF ) and (DRF ), which is equivalent given (DP-RF ). The ‘mixedschemes’ are again Bochman’s. Instead of (DCSF), (RMonFSS), and (RMonSSF), Goble has

(DKSF) OS(A → B/C) → (OF (A/C) → OF (B/C))

(TransFSS) and (TransSFS), which is again equivalent. The names are from the study of non-monotonic logics, namelyreflexivity, right weakening, preservation, (conjunctive) cautiousmonotony, conditionalizationanddisjunctive reasoning. (Ref∗) is Hansson’s[18, Theo-rem 2], (CCMon∗) Rescher’s[36, Theorem 4.4], (Or∗) is the right-to-left version of vonWright’s (B3) in [46], and (DR∗) Hansson’s Theorem 13. Føllesdal and Hilpinen[9] intro-duced the strong version (FH∗) of (RMon∗) (their Theorem 77). (FH+∗) is even stronger:its displayed versions could replace (CCMon∗) and (RMon∗∗), ∗∗=F,FSS,SSF. (Trans∗)is transitivity of weak preference given byA � B =def P

∗(A/A ∨ B) (Lewis [28, p. 54]).



Spohn[39] introduced (P -LoopF ) to define the relevant equivalence classes in his com-pleteness proof of Hansson’sDSDL3, and itsO-form was rediscovered by Kraus, Lehmannand Magidor[26] who put it to the same use. Note that (CCMon∗) is (P -Cond∗), and (Cut∗)is (P -RMon∗), where the deontic operators are swapped in theP -versions.

Theorem 2. DDL{F,S} is sound.

Proof. The validity of (DM∗), (DCS ), (DCSF), (CExt∗), and (ExtC∗) is immediate. (DNS ),(DPF ) are valid since any subset ofLPL derives⊤, and any maximally consistent subsetis consistent. If�PL ¬C then at least∅ is in I⊥¬C, henceI⊥¬C �= ∅ and then both(DN-RF ) and (DP-RS ) hold likewise.

(CondF ) AssumeOF (A/C ∧ D), so there is anI ′ ∈ I⊥¬(C ∧ D) such thatI ′ ∪ {C ∧

D} ⊢PL A and I ′ ∪ {C} ⊢PL D → A. SinceI ′ �PL ¬(C ∧ D), alsoI ′ �PL ¬C,so by maximality there is anI ′′ ∈ I⊥¬C such thatI ′ ⊆ I ′′, so there is anI ′′ ∈

I⊥¬C: I ′′ ∪ {C} ⊢PL D → A, soOF (D → A/C).(CondS) AssumeOS(A/C ∧ D). So for all I ′ ∈ I⊥¬(C ∧ D) : I ′ ∪ {C ∧ D} ⊢PL A.

If there is anI ′′ ∈ I⊥¬C: I ′′ ∪ {C} �PL D → A then I ′′ ∪ {C} �PL ¬D andI ′′ �PL ¬(C ∧ D). By maximality ∃I ′ ∈ I⊥¬(C ∧ D): I ′′ ⊆ I ′. SinceI ′ �PL

¬(C ∧ D), I ′ �PL ¬C, so there is anI ′′′ ∈ I⊥¬C: I ′ ⊆ I ′′′. ThenI ′′ ⊆ I ′′′ andby maximality ofI ′′ ∈ I⊥¬C, I ′′ = I ′′′ and henceI ′′ = I ′. SoI ′′ is in I⊥¬(C ∧

D) and I ′′ ∪ {C ∧ D} �PL A, but this violates the assumption. So for allI ′′ ∈

I⊥¬C: I ′′ ∪ {C} ⊢PL D → A, andOS(D → A/C).(CCMonF ) AssumeOF (A ∧ D/C), so ∃I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A ∧ D. ThenI ′ ∪

{C} �PL ¬D, for otherwiseI ′ ⊢PL ¬C which is excluded by the definition ofI⊥¬C. So I ′ �PL ¬(C ∧ D), by maximality∃I ′′ ∈ I⊥¬(C ∧ D): I ′ ⊆ I ′′ andI ′′ ∪ {C} ⊢PL A, I ′′ ∪ {C ∧ D} ⊢PL A. HenceOF (A/C ∧ D).

(CCMonS) AssumeOS(A ∧ D/C), so for allI ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A ∧ D, and soI ′ ∪ {C} �PL ¬D, for otherwiseI ′ ⊢PL ¬C contrary to the definition ofI⊥¬C,and so for allI ′ ∈ I⊥¬C: I ′ �PL ¬(C ∧ D). SupposeI ′′ ∈ I⊥¬(C ∧ D), soI ′′ �PL ¬(C ∧D) andI ′′ �PL ¬C. By maximality∃I ′ ∈ I⊥¬C such thatI ′′ ⊆ I ′.In turnI ′ �PL ¬(C ∧D) as just proved, so∃I ′′′ ∈ I⊥¬(C ∧D) such thatI ′ ⊆ I ′′′.But thenI ′′ = I ′′′ by maximality ofI ′′ ∈ I⊥¬(C ∧ D), soI ′′ = I ′ ∈ I⊥¬C andI ′′ ∪ {C} ⊢PL A as assumed. SoI ′′ ∪ {C ∧ D} ⊢PL A for anyI ′′ ∈ I⊥¬(C ∧ D).SoOS(A/C ∧ D) is true.

(RMonF ) AssumeOF (A/C), so ∃I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A. If P F (D/C) then∀I ′ ∈ I⊥¬C: I ′ ∪ {C} �PL ¬D. SoI ′ �PL ¬(C ∧ D). So by maximality∃I ′′ ∈

I⊥¬(C ∧D): I ′ ⊆ I ′′, soI ′′ ∪ {C} ⊢PL A, by monotonyI ′′ ∪ {C ∧D} ⊢PL A, soOF (A/C ∧ D) is true.

(RMonFSS) AssumeOS(A/C), so ∀I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A, and P F (D/C), so∀I ′ ∈ I⊥¬C: I ′ ∪{C} �PL ¬D, soI ′ �PL ¬(C∧D). SupposeI ′′ ∈ I⊥¬(C∧D),so alsoI ′′ �PL ¬C and by maximality∃I ′ ∈ I⊥¬C : I ′′ ⊆ I ′. We haveI ′ ∪

{C} ⊢PL A, so∃B1, . . . ,Bn ∈ I ′: {B1, . . . ,Bn} ∪ {C} ⊢PL A by PL-compactness.If I ′′ ∪ {C} �PL A then{B1, . . . ,Bn} � I ′′, by maximalityI ′′ ∪ {B1, . . . ,Bn} ⊢PL

¬(C ∧ D), butI ′′ ∪ {B1, . . . ,Bn} ⊆ I ′, soI ′ ∪ {C} ⊢PL ¬(C ∧ D) contrary to the



assumption. SoI ′′ ∪ {C} ⊢PL A, I ′′ ∪ {C ∧ D} ⊢PL A for anyI ′′ ∈ I⊥¬(C ∧ D).SoOS(A/C ∧ D) is true.

(RMonSSF ) AssumeOS(A/C), so ∀I ′ ∈ I⊥¬C: I ′ ∪ {C} ⊢PL A, and P S(D/C), so∃I ′ ∈ I⊥¬C: I ′ ∪ {C} �PL ¬D. So I ′ �PL ¬(C ∧ D). So by maximality∃I ′′ ∈ I⊥¬(C ∧ D): I ′ ⊆ I ′′, so I ′′ ∪ {C} ⊢PL A, so I ′′ ∪ {C ∧ D} ⊢PL A, soOF (A/C ∧ D) is true.

Theorem 3. DDL{F,S}-semantics are not compact.

Proof. In [17], I provided a counterexample to the compactness of semantics that onlyemploy the monadic deontic operatorOF . SinceOF A can be defined asOF (A/⊤), thisalso refutes the compactness ofDDL{F,S} and of the subsystem containing only the dyadicoperatorOF . The following counterexample is expressed in terms of the dyadic opera-tors OS only, which also refutes the compactness of the subsystem containing only thisoperator: let

Γ ={

OS(p2/⊤)}

∪{

P S(¬p2/p1)}

∪

∞⋃

i=3

{

OS(pi/p1)}

∪{

P S(¬p2/¬p1)}

∪

∞⋃

i=3

{

OS(pi/¬p1)}

∪{

P S(¬p2/p1 ↔ p2)}

∪

∞⋃

i=3

{

OS(pi/p1 ↔ p2)}

∪{

P S(¬p2/p1 ↔ ¬p2)}

∪

∞⋃

i=3

{

OS(pi/p1 ↔ ¬p2)}

Γ is finitely DDL{F,S}-satisfiable: letn be the greatest index of any proposition letter oc-curring in some finiteΓf ⊆ Γ . Then

If ={

pn+1 ∧ (p1 → ¬p2),¬pn+1 ∧ (¬p1 → ¬p2),p2,p3, . . . , pn

}

satisfiesΓf . For easy verification, I list the relevant sets of maximal subsets:

If ⊥¬⊤ =

{

{pn+1 ∧ (p1 → ¬p2),p2,p3, . . . , pn)},

{¬pn+1 ∧ (¬p1 → ¬p2),p2,p3, . . . , pn)}

}

If ⊥¬p1If − ⊥¬(p1 ↔ p2)

=

{

{pn+1 ∧ (p1 → ¬p2),p3, . . . , pn)},

{¬pn+1 ∧ (¬p1 → ¬p2),p2,p3, . . . , pn)}

}

If ⊥p1If ⊥¬(p1 ↔ ¬p2)

=

{

{ pn+1 ∧ (p1 → ¬p2),p2,p3, . . . , pn)},

{ ¬pn+1 ∧ (¬p1 → ¬p2),p3, . . . , pn)}

}

However, Γ is not DDL{F,S}-satisfiable: supposeI ⊆ LPL satisfiesΓ , and let A ∈

{p1,¬p1,p1 ↔ p2,p1 ↔ ¬p2}. Observe that



(i) There areI1, I2 ∈ I⊥¬⊤ such thatI1 ⊢PL p1 ∧ pi , I2 ⊢PL ¬p1 ∧ pi , i � 2.

Proof. From OS(p2/⊤),P S(¬p2/¬p1) ∈ Γ and the validity of (RMonFSS), fol-lows OF (p1/⊤), i.e., there is anI1 ∈ I⊥¬⊤: I1 ⊢PL p1. Likewise fromOS(p2/⊤)

and P S(¬p2/p1) ∈ Γ , it follows that there is anI2 ∈ I⊥¬⊤: I2 ⊢PL ¬p1. Tosatisfy OS(p2/⊤) it is necessary that for allI ′ ∈ I⊥¬⊤: I ′ ⊢PL p2, and fromOS(pi/p1),O

S(pi/¬p1) ∈ Γ , and the validity of (OrS ), it is obtained that for allI ′ ∈ I⊥¬⊤: I ′ ⊢PL pi , i � 3.

(ii) For eachA, there is anIA ∈ I⊥¬A : IA ∪ {A} ⊢PL ¬p2.

Proof. Let A ∈ {p1,p1 ↔ p2}. Then by observation (i)I1 ∈ I⊥¬A. SinceI1 ⊢PL

p2, to satisfyP S(¬p2/A) ∈ Γ there is anIA ∈ I⊥¬A such thatIA ∪ I1 ⊢PL ¬A.So IA ∪ {A} ⊢PL ¬(p1 ∧ p2 ∧ · · · ∧ pn) for somen. If n � 3, thenIA ∪ {A} ⊢PL

¬(p1 ∧ p2 ∧ · · · ∧ pn−1), sinceIA ∪ {A} ⊢PL pn is necessary forOS(pn/A) ∈ Γ .So IA ∪ {A} ⊢PL ¬(p1 ∧ p2), so IA ∪ {A} ⊢PL ¬p2. Likewise, the proof forA ∈

{¬p1,p1 ↔ ¬p2} is obtained fromI2 ∈ I⊥¬A.

(iii) If A ∈ {p1,p1 ↔ ¬p2} thenIA ∪ {p1,¬p2,p3,p4, . . .} �PL ⊥.If A ∈ {¬p1,p1 ↔ p2} thenIA ∪ {¬p1,¬p2,p3,p4, . . .} �PL ⊥.

Proof. SupposeA ∈ {p1,p1 ↔ ¬p2} andIA ∪ {p1,¬p2,p3,p4, . . .} ⊢PL ⊥. ThenIA ∪ {A,¬p2,p3,p4, . . .} ⊢PL ⊥. So IA ∪ {A} ⊢PL ¬(¬p2 ∧ p3 ∧ p4 ∧ · · · ∧ pn)

for somen. But alsoIA ∪ {A} ⊢PL ¬p2 ∧ p3 ∧ p4 ∧ · · · ∧ pn by observation (ii)and from the fact thatI satisfiesOS(pi/A) ∈ Γ , 3 � i � n. So IA ⊢PL ¬A, butthis contradictsIA ∈ I⊥¬A. The proof forA ∈ {¬p1,p1 ↔ p2} and the setIA ∪

{¬p1,¬p2,p3,p4, . . .} is done likewise.

It follows that Ip1 ∪ I(p1↔¬p2) �PL ⊥ and I¬p1 ∪ I(p1↔p2) �PL ⊥. This is most easilyseen by appealing toPL-semantics: somev ∈ B satisfies{p1,¬p2,p3,p4, . . .} and by (iii)all elements ofIp1 as well as all ofI(p1↔¬p2), so their union is satisfiable and thereforeconsistent (likewise for{¬p1,¬p2,p3,p4, . . .} andI¬p1 ∪ I(p1↔p2)). From (ii) it followsthat

Ip1 ∪ I(p1↔¬p2) ⊢PL (p1 → ¬p2) ∧(

(p1 ↔ ¬p2) → ¬p2)

I¬p1 ∪ I(p1↔p2) ⊢PL (¬p1 → ¬p2) ∧(

(p1 ↔ p2) → ¬p2)

But the conclusions are tautologically equivalent to¬p2, so there are consistent subsets ofI that derive¬p2, andI � OS(p2/⊤), althoughOS(p2/⊤) ∈ Γ .

Theorem 4. DDL{F,S} is weakly complete.

Proof. The proof follows the completeness proof of Spohn[39] for B. Hansson’s[18]preference-based dyadic deontic logicDSDL3. Since parts of this proof will be reused inthe next section for logics that might not include unrestricted (DN∗) or (DP∗), I will avoidtheir use up to the last step of this proof.



A. Preliminaries.We must prove that if|=DDL{F,S} A then⊢DDL{F,S} A for anyA ∈ LPL.We assume�DDL{F,S} A so ¬A is DDL{F,S}-consistent. We build a disjunctive normalform of ¬A and obtain a disjunction of conjunctions, where each conjunct isO∗(B/C)

or ¬O∗(B/C). One disjunct must beDDL{F,S}-consistent. Letδ be that disjunct. Let theδ-restricted languageL δ

PL be thePL-sentences that contain only proposition letters occur-ring in δ. Let r(L δ

PL) be 22n

mutually non-equivalent representatives ofLδPL, wheren is

the number of proposition letters inδ. By writing PL-sentences (including⊤ and⊥), wenow mean their unique representatives inr(L δ

PL). We construct a set∆ as follows:

(a) Any conjunct ofδ is in ∆.(b) For allB,C ∈ r(L δ

PL):– eitherP F (B/C) or OF (¬B/C) ∈ ∆, and– eitherP S(B/C) or OS(¬B/C) ∈ ∆.

(c) ∆ is DDL{F,S}-consistent.

It then suffices to find a setI ⊆ LPL that makes true allB ∈ ∆.B. Identifying the deontic bases.We identify syntactically what Hansson called thede-

ontic basisin an extension‖C‖ (Spohn[39] writesC). Monadic deontic logic has just onebasis, dyadic deontic logic usually has one basis for anyC, and here there may be severalbases, which expresses some conflict or predicament in caseC.

Definition 1. For anyC �= ⊥, C ∈ r(L δPL), let

– OSC =

∧{A ∈ r(L δ

PL) | OS(A/C) ∈ ∆},– OF

C = min{A ∈ r(L δPL) | OF (A/C) ∈ ∆}

where minΓ = {A ∈ Γ | ∀B ∈ Γ , if ⊢PL B → A then⊢PL B ↔ A}, Γ ⊆ LPL.

From (DCS ), (RW∗), andDDL{F,S}-consistency of∆ we obtain, for anyC �= ⊥:

(B1) OS(A/C) ∈ ∆ iff ⊢PL OSC → A,

(B2) OF (A/C) ∈ ∆ iff ∃O ∈ OFC : ⊢PL O→ A.

C. Identifying the relevant class of domains.We identify the most general circumstancesCA whereA is P F -permitted. To the same effect, Spohn employs equivalence classes[A]≈

defined using (P -LoopF ): A ≈ B iff B is in some{B1, . . . ,Bn} ⊆ r(L δPL) with P F (B1/A),

P F (B2/B1), . . . ,P F (Bn/Bn−1), P F (A/Bn) ∈ ∆. The set of all such classes is then{[C]≈ |

C ∈ C}.

Definition 2. For allA ∈ r(L δPL), let

CA = max{C ∈ r(L δ

PL) | P F (A/C) ∈ ∆}

C =⋃

A∈r(L δPL)

CA



where maxΓ = {A ∈ Γ | ∀B ∈ Γ : if ⊢PL A → B, then⊢PL B ↔ A}.

(C1) If P F (A/D) ∈ ∆, then there is aC ∈ CA such that⊢PL D → C.

Proof. Immediate from definition ofCA and finitude ofr(L δPL).

(C2) For allC ∈ C: CC = {C}.

Proof. By definitionC ∈ CA for someA ∈ r(L δPL), so by definitionP F (A/C) ∈ ∆,

andP F (C/C) due to (CExtF ). SupposeC′ ∈ CC : then by definitionP F (C/C′) ∈ ∆.With (FHF ) we getP F (A/C ∨ C′),P F (C/C ∨ C′) ∈ ∆, soC = (C ∨ C′) = C′ followsfrom maximality ofC,C′.

(C3) For allC ∈ C, if P F (C/D) ∈ ∆, then⊢PL D → C.

Proof. By definitionC ∈ CA for someA ∈ r(L δPL), so by definitionP F (A/C) ∈ ∆.

If P F (C/D) ∈ ∆, then we getP F (A/C ∨ D) ∈ ∆ with (FHF ). SoC = (C ∨ D) bymaximality, hence⊢PL D → C.

(C4) For allA �= ⊥: CA = {CA} for someCA ∈ CA and⊢PL A → CA.

Proof. If A �= ⊥ thenP F (A/A) ∈ ∆ due to (DP-RF ) and (CExtF ), so there is someC ∈ CA such that⊢PL A → C by (C1). AssumeC′ ∈ CA: By definition P F (A/C),P F (A/C′) ∈ ∆, so we getP F (C/C′) ∈ ∆ with (RWF ), andP F (A/C ∨ C′) ∈ ∆ with(FHF ), soC = (C ∨ C′) = C′ by maximality. SoC is the desiredCA.

(C5) For allA �= ⊥: ⊢PL OSA ↔ (A ∧OS

CA).

Proof. ⊢PL A → CA due to (C4), and by (B2)OS(OSA/A) ∈ ∆, so with (CondS ) we

obtainOS(A →OSA/CA) ∈ ∆ and thus the right-to-left direction⊢PL (A ∧OS

CA) →

OSA. For the opposite,⊢PL OS

A → A follows from (CExtS ), by definitions and (C4)P F (A/CA), OS(OS

CA/CA) ∈ ∆, so we getOS(OS

CA/A) ∈ ∆ with (RMonFSS). So

⊢PL OSA → (A ∧OS

CA).

(C6) For allA �= ⊥,OA ∈ OFA: ∃OCA

∈ OFCA

: ⊢PLOA ↔ (A ∧OCA).

Proof. Let OA ∈ OFA , so OF (OA/A) ∈ ∆. ⊢PL A → CA and (CondF ) derive

OF (A →OA/CA) ∈ ∆, so∃OCA∈ OF

CAwith ⊢PL (A ∧OCA

) →OA. If P F (¬OCA/

A) ∈ ∆, then fromOF (OCA/CA) ∈ ∆ and (RMonF ) we getOF (¬A/CA) ∈ ∆, but

by definitionP F (A/CA) ∈ ∆. SoOF (OCA/A) ∈ ∆, andOF (A ∧OCA

/A) ∈ ∆ by(CExt). Since⊢PL (A∧OCA

) →OA we obtain⊢PL OA ↔ (A∧OCA) by minimality

of OA.



(C7) For allC �= ⊥, C ∈ C: If {C →OC} ∪ {D} � ⊥, thenOF (C →OC/D) ∈ ∆.

Proof. {C →OC}∪ {D} � ⊥. If OF (¬C/D) ∈ ∆, then the conclusion is trivial. Oth-erwiseP F (C/D) ∈ ∆, so⊢PL D → C by (C3). For r.a.a. supposeP F (¬OC/D) ∈ ∆.With OF (OC/C) ∈ ∆ we obtainOF (OC ∧ ¬D/C) ∈ ∆ by (FH+F ), and⊢PLOC →

¬D by minimality ofOC . But then⊢PL D → (C ∧¬OC), which refutes the assump-tion. HenceOF (OC/D) ∈ ∆ andOF (C →OC/D) ∈ ∆ by use of (CExtF ).

D. Identifying the multiple system of spheres. If this were ‘ordinary’ dyadic deontic logicwith agglomeration and so just one basisOC for any C, we would be almost done: likeSpohn[39] orders his equivalence classes[C]≈ by a relationbefore, C could be orderedinto 〈C1, . . . ,Cn〉 with C1 = ⊤, andCi+1 = Ci ∧¬OCi

until this equals⊥. 〈S1, . . . , Sn〉 withSi = (Ci ∧ ¬Ci+1), 1� i < n, is then the ‘system of spheres’. Here this method fails sinceno C ∈ C is guaranteed to have a single basis. But as it turns out,C has the structure of a‘multiple’ system of spheres that is similarly identified.

(D1) {⊤} ⊆ C.

Proof. P F (⊤/⊤) ∈ ∆ by (DP-RF ), and⊢PL C → ⊤ for any P F (⊤/C) ∈ ∆, so⊤ ∈ C⊤, ⊤ ∈ C.

(D2) For allC ∈ C,O ∈ OFC : If C ∧ ¬O �= ⊥, thenC ∧ ¬O ∈ C.

Proof. If C ∧ ¬O �= ⊥ thenP F (⊤/C ∧ ¬O) ∈ ∆ by (DP-RF ), CC∧¬O ∈ CF . Weprove CC∧¬O = C ∧ ¬O: ⊢PL (C ∧ ¬O) → CC∧¬O is immediate from (CExtF )and (C1). If�PL CC∧¬O → (C ∧ ¬O) then {C → O} �PL ¬CC∧¬O , so OF (C →

O/CC∧¬O) ∈ ∆ follows from (C7).P F (C ∧ ¬O/CC∧¬O) ∈ ∆ by definition, so∆ isDDL{F,S}-inconsistent, but we assumed otherwise.

(D3) For allC ∈ C: If ⊢PL C → D, C �= D, then∃O ∈ OFD: ⊢PL C → (D ∧ ¬O).

Proof. EitherP F (C/D) ∈ ∆, so⊢PL D → C, C = D (C3). OrOF (¬C/D) ∈ ∆, so⊢PL O→ ¬C for someO ∈ OF

D and⊢PL C → (D ∧ ¬O).

(D4) For allD ∈ r(L δPL),O ∈ OF

D ∪ {OSD}: If D �= ⊥, thenD �= (D ∧ ¬O).

Proof. If D = (D ∧¬O), then⊢PL D → ¬O. But also⊢PL O→ D due to (CExt∗),soO = ⊥ andO∗(⊥/D) ∈ ∆ by (B1-2). SoD = ⊥ by (DP-R∗).

(D5) Let D be such that (i)⊤ ∈ D, and (ii) if D ∈ D, O ∈ OFD and(D ∧ ¬O) �= ⊥, then

(D ∧ ¬O) ∈ D. ThenD = C\{⊥}.

Proof. D ⊆ C is immediate from (D1), (D2). As forC ⊆ D, for eachC ∈ C, C �= ⊥,there is someD ∈ D such that (a)⊢PL C → D, and (b) for noO ∈ OF

D :⊢PL C →



(D∧¬O). (a) is guaranteed by⊤ ∈ D, and (b) follows from (D3), (D4) and finitenessof r(L δ

PL). SoC = D by (D3).

E. Canonical construction and coincidence lemma.

Definition 3 (Canonical construction). For allC ∈ C ∪ {⊥}, D ∈ r(L δPL)

– F-Succ(D) = {D ∧ ¬O |O ∈ OFD},

– F-Chain(C) be the set of sequences〈D1, . . . ,Dn〉, 1 � n, whereD1 = ⊤, Di+1 ∈

F-Succ(Di), Di �= Di+1 for any 1� i < n, andDn = C,– S-Chain(C,D) be the set of sequences〈D1, . . . ,Dk,Dk+1, . . . ,Dn〉, 1� k < n, where

〈D1, . . . ,Dk〉 ∈ F-Chain(C), Di+1 = Di ∧ ¬OSDi

, Di �= Di+1 for anyk � i < n, andDn = D.

For anyC ∈ C\{⊥}, C′ ∈ F-Succ(C), let

– π :C∪{⊥} → [Prop\L δPL] be a function that associates a unique proposition letter not

occurring inδ with each element ofC ∪ {⊥},– φ(C,C′) = π(C′) ∧

∧

{¬π(C′′) | C′′ ∈ F-Succ(C),C′ �= C′′},– σ(C) =

∧

{¬π(C′) | C′ ∈ F-Succ(C)}.

For anyC ∈ C ∪ {⊥} \ {⊤}, 〈C1, . . . ,Cn〉 ∈ F-Chain(C), let

– iF [〈C1, . . . ,Cn〉] = ¬C ∧∧n−1

i=1 φ(Ci,Ci+1).

For anyC ∈ C, 〈D1, . . . ,Dk,Dk+1, . . . ,Dn〉 ∈ S-Chain(C,D), Dk = C, let

– iS[〈D1, . . . ,Dk,Dk+1, . . . ,Dn〉] = ¬D ∧

{

σ(C) ∧∧k−1

i=1 φ(Ci,Ci+1) if C �= ⊤,

σ (C) otherwise.

Let IF be the set of all suchiF [〈C1, . . . ,Cn〉], and likewiseIS be the set of all suchiS[〈D1, . . . ,Dk,Dk+1, . . . ,Dn〉]. Then finallyI = IF ∪ IS .

The definition provides the construction of the canonical setI to make all of∆ true.F-Succ(C) is the set of immediate ‘contrary-to-duty’ successorsC′ of C, i.e.,∃O ∈ OF

C withC′ = C ∧ ¬O. (D2) showed eachC ∈ C to be such a successor of (a successor of. . .) ⊤,andF-Chain(C) is the set of all such chains beginning with⊤ and ending withC. φ is usedto make any twoiF [ch(C′)], iF [ch(C′′)], C′ �= C′′ being successors of (successors of. . .)C, inconsistent with each other and with anyiS[ch(C,D)] via σ . SinceC is finite, so isthe number of proposition letters introduced byπ , IF , IS andI . Regarding the sequencesused to constructI , I usech(C) for 〈D1, . . . ,Dn〉 ∈ F-Chain(C) with C ∈ C ∪ {⊥} \ {⊤},ch(C,D) for 〈D1, . . . ,Dn〉 ∈ S-Chain(C,D) with C ∈ C, D ∈ r(L δ

PL), andch, ch′, etc. forany sequence for which either holds. We obtain:

(E1) For allch= 〈D1, . . . ,Dn〉, ⊢PL Di+1 → Di and�PL Di → Di+1, 1� i < n.



(E2) If {iF [ch(C)], iF [ch(C′)]} �PL ⊥, thench(C) is a segment ofch(C′) or vice versa.(E3) If {iS[ch(C,D)], iS [ch(C′,D′)]} �PL ⊥, thenC = C′ and ch(C,D) is a segment of

ch(C′,D′) or vice versa.(E4) If {iF [ch(C)], iS [ch(C′,D)]} �PL ⊥, then ch(C) = 〈C1, . . . ,Ci〉 is a segment of

ch(C′,D) = 〈D1, . . . ,Dk,Dk+1, . . . ,Dn〉, whereDk = C′ and 1� i � k < n.(E5) NoiF [ch(C)] or iS[ch(C,D)] ∈ I is a contradiction.

Proof. (E1) is immediate from (D4). (E2–4) are immediate from the definitions ofφ andσ . For (E5), first note that eachi ∈ I consists of ar(L δ

PL)-conjunct and a[LPL\LδPL]-

conjunct. Since no proposition letter occurring in one occurs in the other, ifi is a con-tradiction, then so must be one of its conjuncts. Regarding ther(L δ

PL)-conjunct, for anyiF [ch(C)] it is ¬C which must be consistent sinceC = ⊤ is excluded. For anyiS[ch(C,D)]

the r(L δPL)-conjunct is¬D, andch(C,D) = 〈D1, . . . ,Dn〉 with D1 = ⊤, Dn = D and

n �= 1, so by (E1)D = ⊤ is excluded. For the[LPL\LδPL]-conjunct of anyiF [ch(C)],

ch(C) = 〈C1, . . . ,Cn〉 ∈ F-Chain(C), it is∧n−1

i=1 φ(Ci,Ci+1), a conjunction of conjunctionsof non-negated and negated proposition letters, which cannot be a contradiction:

– No conjunctφ(Ci,Ci+1), 1� i < n, is a contradiction: For anyC′ �= C′′ ∈ C, π(C′) �=

π(C′′), and noπ(C′) occurs negated and non-negated inφ(Ci,Ci+1).– If π(C′) occurs non-negated inφ(Ci,Ci+1) and negated inφ(Cj ,Cj+1), i < j ,

then C′ = Ci+1 and C′ ∈ F-Succ(Cj ). So there is ach(C′) ∈ F-Chain(C′), ch(C′) =

〈C1, . . . ,Ci,C′, . . . ,Cj ,C′〉, which violates (E1).– If π(C′) occurs negated inφ(Ci,Ci+1) and non-negated inφ(Cj ,Cj+1), i < j , thenC′ ∈ F-Succ(Ci) andC′ = Cj+1. ⊢PL Cj+1 → Ci+1, so ⊢PL C′ → Ci+1. So there areO1,O2 ∈ OF

Ciwith C′ = Ci ∧ ¬O1, Ci+1 = Ci ∧ ¬O2, and⊢PL (Ci ∧ ¬O1) → (Ci ∧

¬O2). Then⊢PLO2 → (Ci →O1), and with (CExtF ) ⊢PL O2 →O1. By minimalityO2 =O1 andC′ = Ci+1, butφ(Ci,Ci+1) left π(Ci+1) non-negated.

For the[LPL\LδPL]-conjunct ofiS[ch(C,D)], the case thatπ(C′) occurs non-negated in

φ(Ci,Ci+1) and negated inσ(C) is done like the second case above.

For anyB ∈ r(L δPL), I ′ ∈ I⊥¬B, I ′ �= ∅:

(E6) there is some designatedix ∈ I ′, ix = iF [ch] or ix = iS[ch], such that for alliF [ch′],iS[ch′] in I ′, ch is a segment ofch′,

(E7) ther(L δPL)-conjunct ofix PL-derives ther(L δ

PL)-conjunct of anyi ∈ I ′,(E8) ix is iF [ch] or iS[ch] with ch= 〈D1, . . . ,Dn〉 such that⊢PL B → Dn−1.

Proof. (E6) is immediate from (E2-4) and finiteness ofI ′. For (E7), ther(L δPL)-conjunct

of i is ¬Dn, whereDn is the last member of somech= 〈D1, . . . ,Dn〉 such thati = iF [ch]

or i = iS[ch]. By (E6) ther(L δPL)-conjunct of ix is ¬Dk for some 1� k � n, and by

(E1) ⊢PL ¬Dk → ¬Dn. For (E8), note thatDn−1 exists asch= 〈⊤〉 is excluded by theconstruction. IfDn−1 = ⊤, then⊢PL B → Dn−1 is trivial. Otherwise there must be somech′ = 〈D1, . . . ,Dn−1〉 such thati∗(ch′) ∈ I , * beingF or S. By (E6), i∗(ch′) cannot be in



I ′, though ther(L δPL)-conjunct¬Dn−1 derives ther(L δ

PL)-conjunct¬Dn of ix by (E1)and hence that of any otheri′ ∈ I ′ due to (E7), while its[LPL\L

δPL]-conjunct is derived

by that ofix. So it must be that{¬Dn−1} ∪ {B} ⊢PL ⊥, and⊢PL B → Dn−1.

Lemma 1 (Coincidence lemma). For all A,B ∈ r(L δPL):

I |= OF (A/B) iff OF (A/B) ∈ ∆

I |= OS(A/B) iff OF (A/B) ∈ ∆

Proof. Coincidence forOF .

Right-to-left. AssumeOF (A/B) ∈ ∆, so someOB ∈ OFB derivesA. By (C6)⊢PL ((CB →

OCB) ∧ B) ↔ OB for someCB ∈ C, OCB

∈ OFCB

. By (D2), CB ∧ ¬OCB∈ C ∪

{⊥}, so iF [ch] ∈ I for somech∈ F-Chain(CB ∧ ¬OCB). If {iF [ch]} PL-derives

¬B, only its r(L δPL)-conjunctCB →OCB

can be relevant, since the[LPL\LδPL]-

conjunct is consistent (E5) and has no proposition letter in common with¬B.If {CB → OCB

,B} ⊢PL ⊥, thenOB = ⊥ which contradictsP F (⊤/B) ∈ ∆ by(DPF ) andDDL{F,S}-consistency of∆. So {iF [ch]} �PL ¬B, so for someI ′ ∈

I⊥¬B : I ′ ∪ {B} ⊢PL CB →OCB, henceI ′ ∪ {B} ⊢PL A.

Left-to-right. AssumeP F (A/B) ∈ ∆ and for r.a.a. suppose∃I ′ ∈ I⊥¬B: I ′ ∪ {B} ⊢PL

¬A. SupposeI ′ �= ∅, so let ix be the designated member ofI ′ and ¬D itsr(L δ

PL)-conjunct. Then{¬D} ∪ {B} ⊢PL ¬A as ther(L δPL)-conjuncts of any

i ∈ I are PL-derived by¬D (E7), and the[LPL\LδPL]-conjuncts are not rel-

evant for a derivation of¬A ∈ r(L δPL). ix is iF [ch] or iS[ch] for somech =

〈D1, . . . ,Dn〉 with ¬D = ¬Dn = (Dn−1 → O), O ∈ OFDn−1

∪ {OSDn−1

}. So

{O} ⊢PL B → ¬A, soOF (B → ¬A/Dn−1) ∈ ∆ or OS(B → ¬A/Dn−1) ∈ ∆ by(B1), (B2). FromP F (A/B) ∈ ∆ we getP F (¬(B → ¬A)/B) ∈ ∆ with (CExtF ).By (E8) ⊢PL B → Dn−1, so with (FH+F ) or (FH+SSF ) we obtainOF ((B →

¬A) ∧ ¬B/Dn−1) ∈ ∆ or OS((B → ¬A) ∧ ¬B/Dn−1) ∈ ∆ respectively. Butthen⊢ O → ¬B follows from minimality ofO. So {B} ⊢PL Dn−1 ∧ ¬O, i.e.,{B} ⊢PL D, and since{ix} ⊢PL ¬D we get ix /∈ I ′. So by (E6)I ′ = ∅. Then{B} ⊢PL ¬A. With P F (A/B) ∈ ∆ and (CExtF ) we getP F (⊥/B), so by (DN-RF )B = ⊥ andI⊥¬B = ∅, completing the r.a.a.

Coincidence forOS .

Right-to-left. AssumeOS(A/B) ∈ ∆ and for r.a.a. suppose that there is someI ′ ∈

I⊥¬B: I ′ ∪{B} �PL A. AssumeI ′ �= ∅, so letix be the designated member ofI ′,and¬D its r(L δ

PL)-conjunct.ix is iF [ch] or iS[ch] for somech= 〈D1, . . . ,Dn〉

with ¬D = ¬Dn = (Dn−1 → O). Either O ∈ OFDn−1

, then ⊢PL O → OSDn−1

follows from (DCSF), or trivially if O = OSDn−1

. By (E8) ⊢PL B → Dn−1, so

by (CondS ) ⊢PL OSDn−1

→ (B → OSB), and alsoI ′ ∪ {B} ⊢PL O. Chaining the

results, we getI ′ ∪ {B} ⊢PL OSB , and I ′ ∪ {B} ⊢PL A by definition of OS

B ,contrary to what was assumed. SoI ′ = ∅. For anyB ∈ r(L δ

PL), B �= ⊥, we



have iS[ch(CB , (CB ∧ ¬OSCB

))] ∈ IS by (C4) and definition ofIS . If I ′ = ∅,

then {iS[ch(CB , (CB ∧ ¬OSCB

))]} ∪ {B} ⊢PL ⊥, and {CB → OSCB

} ∪ {B} ⊢PL ⊥

since only ther(L δPL)-conjunct is relevant. By (C4) we have⊢PL B → CB , so

{B ∧ OSCB

} ⊢PL ⊥, and by (C5)OSB = ⊥. From (DP-RS ) we get B = ⊥, so

I⊥¬B = ∅, which completes the r.a.a.Left-to-right. AssumeOS(A/B) /∈ ∆. B �= ⊥ due to (DNS ) and (CExtS ), so iS[ch(CB ,

(CB ∧ ¬OSCB

))] ∈ I ′ for some I ′ ∈ I⊥¬B (otherwise againB = ⊥). If

iF [ch(C′)] ∈ I ′, thench(C′) is a segment ofch(CB , (CB ∧ ¬OSCB

)) by (E4), so

⊢PL ¬C′ → ¬CB and as⊢PL B → CB also⊢PL ¬C′ → ¬B. So iF [ch(C′)] /∈ I ′

and I ′ ∩ IF = ∅. If iS[ch(C′,D′)] ∈ I ′, then ch(CB , (CB ∧ ¬OSCB

)) is a seg-

ment of ch(C′,D′) by (E3), so itsr(L δPL)-conjunct ¬D′ is derived by that

of iS[ch(CB , (CB ∧ ¬OSCB

))]. The [LPL\LδPL]-conjuncts are not relevant, so if

I ′ ∪ B ⊢PL A, then{CB →OSCB

} ∪ {B} ⊢PL A. Since⊢PL B → CB , by (C4) then

{B ∧OSCB

} ⊢PL A, and by (C5)⊢PL OSB → A, soOS(A/B) ∈ ∆ by (B1), contrary

to the assumption. SoI ′ ∪ B �PL A, so not for allI ′ ∈ I⊥¬B: I ′ ∪ {B} ⊢PL A,soI |= OS(A/B) is false.

6. A link to multiplex preference semantics

In the preceding section, the completeness theorem was proved by identifying a multiplesystem of spheres. This multiple system of spheres can just as well be used to construct amultitude of preference relations, which—as originated with Goble[13,14]—can then inturn be used to define the deontic operators: letP be a non-empty set of preference relationsP ⊆ B × B such that eachP is transitive, connected, and satisfies the ‘limit assumption’:

(LA) If ‖A‖ �= ∅ thenbestP (‖A‖) �= ∅

where bestP (‖A‖) = {v ∈ ‖A‖ | ∀v′ ∈ ‖A‖: vPv′}. For Hansson-type operators, letL

DDL{F+,S} be like LDDL{F,S} except thatOF+replacesOF , and let the truth definitions

for the deontic operators read:

P |= OF+

(A/C) iff ∃P ∈ P: bestP (‖C‖) ⊆ ‖A‖

P |= OS(A/C) iff ∀P ∈ P: bestP (‖C‖) ⊆ ‖A‖

Likewise, for Lewis-type operators, letLDDL{F,S−} be likeLDDL{F,S} except thatOS−

re-placesOS , and the truth definitions now read:

P |= OF (A/C) iff ∃P ∈ P: ∃v ∈ ‖C ∧ A‖: ∀v′ ∈ ‖C ∧ ¬A‖: notv′Pv

P |= OS−

(A/C) iff ∀P ∈ P: ∃v ∈ ‖C ∧ A‖: ∀v′ ∈ ‖C ∧ ¬A‖: notv′Pv

Theaxiomatic system DDL{F+,S} is like DDL{F,S} except that (DNF ) replaces (DN-RF )and

(DP-RF ) replaces (DPF ). Similarly, DDL{F,S−} is like DDL{F,S} except that (DN-RS )



replaces (DNS ) and (DPS ) replaces (DP-RS ). So all systems only differ on the ‘mind-boggling’ [29] question whether everything or nothing is obligatory in impossible cir-cumstances.DDL{F+,S} and DDL{F,S−} are sound (cf.[14], also Arrow’s axiom: ifbestP (‖C ∨D‖)∩‖C‖ �= ∅, thenbestP (‖C‖) = bestP (‖C ∨D‖)∩‖C‖, is helpful). I haveno counterexample to compactness, so the semantics might just be compact. Weak com-pleteness is easily obtained from the previous constructions, but seems not to have beenstated before, so I shall give the proof in full.

Theorem 5 (Completeness ofDDL{F+,S} and DDL{F,S−}). The systems DDL{F+,S} and

DDL{F,S−} are weakly complete with respect to the above multiplex preference semantics.

Proof. In provingDDL{F,S}-completeness, up till the coincidence lemma no use was madeuse of unrestricted (DPF ) and (DNS ) missing inDDL{F+,S} andDDL{F,S−} respectively.So we can reuse and continue that construction with all pertaining lemmas in the canon-ical construction forDDL{F+,S} andDDL{F,S−}, with the implicit understanding that forDDL{F+,S} the index meant isF+ rather thanF , and forDDL{F,S−} the index meant isS−

instead ofS.Let F-Chain(⊥), S-Chain(C,⊥), be defined as before,C ∈ C. We only considerch=

〈D1, . . . ,Dn〉 that are in such a set. LetOchDi

= Di ∧ ¬Di+1, for any 1� i < n. Note that

Di+1 = Di ∧ ¬O for someO ∈ OFDi

∪ {OSDi

}, and⊢O→ Di by (CExt∗), so⊢PL OchDi

↔

O. For any 1� i < j � n:

(S1) ‖Dj‖ ⊆ ‖Di‖

(S2) ‖Dj‖ ∩ ‖OchDi

‖ = ∅ (and‖OchDj

‖ ∩ ‖OchDi

‖ = ∅ with CExt∗, j �= n)(S3) n �= ∞

(S4) ‖OchD1

‖ ∪ · · · ∪ ‖OchDn−1

‖ = B

Proof. (S1) and (S2) are immediate from (E1) and the definition ofch. n is finite (S3)sincer(L δ

PL) is finite and repetitions inch are excluded. For (S4),

B\∥

∥OchD1

∥

∥ ∪ · · · ∪∥

∥OchDn

∥

∥ =∥

∥Dn−1 ∧ ¬OchDn−1

∥

∥ =∥

∥Dn−1 ∧ (Dn−1 → Dn)∥

∥

andDn = ⊥ by definition.

For anych= 〈D1, . . . ,Dn〉, v, v′ ∈ B, let Pch be such that

vPchv′ iff v ∈ ‖Och

Di‖, v′ ∈ ‖Och

Dj‖, i � j < n

By (S2) and (S4), eachv must belong to exactly one sphere. The index of eachCi istransitive and connected, soPch is as well.LA holds due to (S3) and (S4).

Let ch = 〈D1, . . . ,Dn〉 be as described above. LetDi be its “smallestA-permittingsphere”, i.e., aDi with ‖Och

Di‖ ∩ ‖A‖ �= ∅ and∀j , 1� j < i < n: ‖Och

Dj‖ ∩ ‖A‖ = ∅ (we

write DA for Di ). We then obtain, for anych andA �= ⊥:

(S5) There is aDA ∈ ch,



(S6) ‖A‖ ⊆ ‖DA‖, and(S7) bestPch(‖A‖) = ‖Och

DA‖ ∩ ‖A‖.

Proof. (S5) is immediate from (S1), (S4) andD1 = ⊤. For (S6) letDA = Di ∈ ch, 1� i <

n: If ‖A‖ � ‖DA‖, thenA∧¬DA �= ⊥, so by (S5) there is aDA∧¬DA= Dj ∈ ch. If DA =

DA∧¬DA, then‖Och

DA‖ = ‖Och

DA∧¬DA‖, by construction⊢PL Och

DA→ DA, so‖Och

DA∧¬DA‖ ∩

‖A∧¬DA‖ = ∅ contrary to the definition ofDA∧¬DA. If i < j , then‖DA∧¬DA

‖ ⊆ ‖DA‖,but then‖DA∧¬DA

‖ ∩ ‖A ∧ ¬DA‖ = ∅ and again‖OchDA∧¬DA

‖ ∩ ‖A ∧ ¬DA‖ = ∅. So

j < i, but then‖OchDA

‖ ∩ ‖A ∧ ¬DA‖ �= ∅ implies‖OchDA

‖ ∩ ‖A‖ �= ∅, soDA was not the

smallestA-permitting sphere. (S7) then follows from the definitions ofOchDA

andPch.

Finally, let P = {Pch | ch ∈ F-Chain(⊥) ∪⋃

C∈C S-Chain(C,⊥)}. Note thatP �= ∅: by(D1) ⊤ ∈ C, so even if⊤ ∧O⊤ = ⊥ for all O ∈ OF

⊤ ∪OS⊤, then〈⊤,⊥〉 ∈ F-Chain(⊥),

〈⊤,⊥〉 ∈ S-Chain(⊤,⊥), henceP〈⊤,⊥〉 = B × B ∈ P. SoP is as required.The next lemma holds for allA ∈ r(L δ

PL), A �= ⊥ andP ∈ P and saves us from havingto do separate proofs for the two systems:

(S8) bestP (‖A‖) ⊆ ‖B‖ iff ∃v ∈ ‖A ∧ B‖: ∀v′ ∈ ‖A ∧ ¬B‖: notv′Pv.

Proof. AssumebestP (‖A‖) ⊆ ‖B‖: A �= ⊥, so bestP (‖A‖) �= ∅ due to (LA). So ∃v ∈

bestP (‖A‖) s.t.v ∈ ‖A∧B‖. Supposev′Pv for somev′ ∈ ‖A∧¬B‖: sov′ ∈ ‖A‖ andv′ ∈

bestP (‖A‖) by transitivity ofP and definition ofbest. But thenbestP (‖A‖) ∩ ‖¬B‖ �= ∅,contradicting the assumption. Assume∃v ∈ ‖A ∧ B‖: ∀v′ ∈ ‖A ∧ ¬B‖: notv′Pv, andfor r.a.a. suppose thatbestP (‖A‖) ∩ ‖¬B‖ �= ∅: So∃v′ ∈ bestP (‖A‖) ∩ ‖¬B‖. Thenv′ ∈

‖A ∧ ¬B‖ by definition of best, and notv′Pv, as assumed. Butv ∈ ‖A‖, so v′Pv bydefinition ofbest, which completes the r.a.a.

Lemma 2 (Coincidence lemma). For all A,B ∈ r(L δPL):

P |= OF (A/B) iff OF (A/B) ∈ ∆

P |= OS(A/B) iff OF (A/B) ∈ ∆

Proof. Coincidence forOF .

CaseB = ⊥. If B = ⊥, then in the case ofDDL{F+,S}, OF+(⊤/⊥) ∈ ∆ holds due to(DNF+

), so OF+(A/⊥) ∈ ∆ due to (CExtF

+). Also, if B = ⊥, then for any

P ∈ P, bestP (‖B‖) = ∅. P �= ∅, so ∃P ∈ P: bestP (‖B‖) ⊆ ‖A‖ holds for anyA, and both sides of the iff-clause are true, and so is the iff-clause. In the case ofDDL{F,S−}, if B = ⊥, thenP F (⊤/⊥) ∈ ∆ due to (DPF ), so P F (¬A/⊥) ∈ ∆

due to (CExtF ), and so by definition of∆, OF (A/⊥) /∈ ∆. Also, if B = ⊥,then ‖A ∧ B‖ = ∅, so for anyP there is nov ∈ ‖A ∧ B‖ such that∀v′ ∈

‖A ∧ ¬B‖: notv′Pv. So both sides of the iff-clause are false, and the iff-clausetrue.



CaseB �= ⊥. Right-to-left. OF (A/B) ∈ ∆, so ∃OB ∈ OFB : ⊢PL OB → A, so by (C6)

∃OCB∈ OF

CB: OCB

∧ B = OB . SinceCB ∈ C, there is ach∈ F-Chain(⊥) suchthat ch = 〈D1, . . . ,Di,Di+1, . . . ,Dn〉, 1 � i < n, D1 = ⊤, Di = CB , Di+1 =

C ∧ ¬OCB, andDn = ⊥. By definitionOch

Di=OCB

. For any 1� j < i, ‖OchDj

‖ ∩

‖CB‖ = ∅ due to (S2), so also‖OchDj

‖ ∩ ‖B‖ = ∅, and‖OchDi

‖ ∩ ‖B‖ �= ∅, for

otherwiseOCB∧B = ⊥ =OB , by (B2)OF (⊥/B) ∈ ∆, and by (DP-RF ) B = ⊥,

contrary to what was assumed. SoDi = DB , andbestPch(‖B‖) = ‖OchDB

‖∩‖B‖ =

‖OCB∧ B‖ = ‖OB‖, and sobestPch(‖B‖) ⊆ ‖A‖ and by (S8)P |= OF (A/B).

Left-to-right.SupposeP |= OF (A/B), so∃P ∈ P: bestP (‖B‖) ⊆ ‖A‖ by (S8).By construction there is somech = 〈D1, . . . ,Dn〉 such thatP = Pch, ch ∈

F-Chain(⊥) or ch∈ S-Chain(C,⊥) for someC ∈ C. As B �= ⊥, by (S5) there issome ‘smallestB-permitting sphere’DB in ch, with bestP (‖B‖) = ‖Och

DB‖∩‖B‖

and‖OchDB

‖ ∩ ‖B‖ �= ∅. EitherDB ∈ C andOchDB

∈ OFDB

: then{DB →OchDB

} �PL

¬B, soOF (DB →OchDB

/B) ∈ ∆ by (C7),⊢PL B → DB by (S6), soOF (OchDB

∧

B/B) ∈ ∆ with (CExtF ) and OF (A/B) ∈ ∆ by (RWF ). Or OchDB

= OSDB

, so

OS(OchDB

/DB) ∈ ∆, andOS(B → A/DB) ∈ ∆ by (RWS ). AssumeOF (A/B) /∈

∆, then P F (¬A/B) ∈ ∆ by definition of ∆, so P F (¬(B → A)/B) ∈ ∆ by(CExtF ), and so with (FH+SSF ) we obtainOS((B → A) ∧ ¬B/DB) ∈ ∆, soby definition⊢PL OS

DA→ ¬B, but then‖Och

DA‖ ∩ ‖B‖ = ∅, contrary to what was

assumed. SoOF (A/B) ∈ ∆.

Coincidence forOS .

CaseB = ⊥. If B = ⊥, then in the case ofDDL{F+,S}, OS(⊤/⊥) ∈ ∆ holds due to(DNS ), soOS(A/⊥) ∈ ∆ due to (CExtS ). Also, if B = ⊥, then for anyP ∈ P,bestP (‖B‖) = ∅, so for allP ∈ P: bestP (‖B‖) ⊆ ‖A‖ holds for anyA, and bothsides of the iff-clause are true, as is the iff-clause. In the case ofDDL{F,S−}, ifB = ⊥, thenP S(⊤/⊥) ∈ ∆ due to (DPS ), soP S(¬A/⊥) ∈ ∆ due to (CExtS ),and by definition of∆, OS(A/⊥) /∈ ∆. Also, if B = ⊥, then‖A ∧ B‖ = ∅, andsinceP �= ∅ there is someP for which it is false that∃v ∈ ‖A ∧ B‖: ∀v′ ∈

‖A ∧ ¬B‖: notv′Pv, so it is not true for allP . So both sides of the iff-clauseare false, and the clause true.

CaseB �= ⊥. Right-to-left.AssumeOS(A/B) ∈ ∆, and for r.a.a.P � OS(A/B), so by(S8) ∃P ∈ P: bestP (‖B‖) ∩ ‖¬A‖ �= ∅. By construction there is somech =

〈D1, . . . ,Dn〉 such thatP = Pch, ch ∈ F-Chain(⊥) or ch ∈ S-Chain(C,⊥) forsomeC ∈ C. SinceB �= ⊥, by (S5) there is some ‘smallestB-permitting sphere’DB in ch, with bestP (‖B‖) = ‖Och

DB‖ ∩ ‖B‖ and ‖Och

DB‖ ∩ ‖B‖ �= ∅. Either

DB ∈ C andOchDB

∈ OFDB

, or OchDB

= OSDB

: In both cases, since⊢PL B → DB

by (S6), we haveOS(B → A/DB) ∈ ∆ from OS(A/B) ∈ ∆ and (CondS ), and{Och

DB} ⊢PL B → A either by (DCSF) and minimality ofOch

DB∈ OF

DB, or by defin-

ition of OSDB

. So‖OchDB

‖ ∩ ‖B‖ = bestP (‖B‖) ⊆ ‖A‖, which completes the r.a.a.

Left-to-right. SupposeP |= OS(A/B), OS(A/B) /∈ ∆, so P S(¬A/B) ∈ ∆.



B �= ⊥, so CB ∈ C by (C4), so letch = 〈D1, . . . ,Dn〉 be in S-Chain(CB ,⊥).Then DB = CB : SupposeDB = Di and CB = Dj for j < i. By constructionof ch, Och

Dj= OS

CB, andOS

CB∧ B �= ⊥ for otherwiseOS

B = ⊥ by (C6), which

with (DP-RS ) derivesB = ⊥, but this was excluded. So‖OSCB

‖ ∩ ‖B‖ �= ∅,so DB is not the ‘smallestB-permitting sphere’. SupposeDB = Di andCB =

Dj for j > i, then ‖CB‖ ∩ ‖OchDB

‖ = ∅ by (S2), as⊢PL B → CB by (C4F ),

‖B‖ ∩ ‖OchDB

‖ = ∅, and againDB is not the ‘smallestB-permitting sphere’.

So DB = CB and by construction ofch, OchDB

= OSCB

. SinceP S(¬A/B) ∈ ∆,

�PL OSB → A follows from (B1) and construction of∆. With (C5), (S7) we get

‖OSB‖ = ‖OS

CB∧ B‖ = ‖Och

DB‖ ∩ ‖B‖ = bestPch(‖B‖). SobestPch(‖B‖) � ‖¬A‖

and by (S8)P � OS(A/B), contradicting the assumption.

7. The puzzle is still incomplete

From a complete picture of dyadic deontic reasoning about conflicting imperatives, atleast two pieces are still missing. The first is that an imperative itself may be conditionalin a way irreducible to a material implication in its content: e.g., if I’m to throw rice asthe wedding party leaves the church, but Huey, Dewey and Louie have stolen the bag,blocking the doors won’t garner me any praise. It has been argued that such conditionalimperatives have two associated propositions, the antecedent and the consequent, and thatobligations are only ‘triggered’, if the antecedents hold, thus providing the opportunity fornorm satisfaction or violation. Secondly, even though weighing out the relevant factorsmay not always produce an unequivocal result, imperativescanbe ordered by rank of theissuing authority or normative weight: e.g., finding the victim of an accident on my wayto a crucial appointment, it seems clear what my obligations are and to not be the time forskeptical or credulous reasoning.

In an attempt to tackle these complexities, Horty[21] proposed the following definitionof the imperatives ‘binding’ in some circumstancesA:

Binding(I ,<)(A) =def{

i ∈ I | (1) i ∈ TriggeredI (A),

(2) there is noj ∈ TriggeredI (A) such that

(a) i < j, and

(b) {consequent(i),consequent(j)} ⊢BL ⊥}

whereTriggeredI (A) =def {i ∈ I | A ⊢BL antecedent(i)}, and< is some strict partial or-der onI , i < j meaning thati ranks higher thanj . The truth of the (skeptical or credulous)dyadic deontic formulaO∗(B/A) is then defined with respect to the set of consequents ofthe imperatives inBinding(I ,<)(A).

Yet Horty’s proposal is problematic for several reasons. First, the triggering condi-tion does not capture all senses in which antecedents may hold. Consider the situation(C ∨ D) and letI = 〈C ⇒!A,D ⇒!A, !(¬A ∧ B)〉, where for shortA ⇒!B meansthe imperative withantecedent(A) and consequent(B), !(¬A ∧ B) is the unconditional



imperative⊤ ⇒! (¬A ∧ B), and the sequence represents the ordering<. Though wedo not know which imperative overrides the weakest imperative!(¬A ∧ B), we knowfor sure that it is overridden in these circumstances and so should not be included inBinding(I ,<)(C ∨ D), but with Horty’s definition it is. I suggest that, for a better defi-nition of the setTriggeredI (C ∨ D), we need an operation like Makinson and van derTorre’s [32] ‘basic output’, which is expressly tailored to process such disjunctive inputs(triggering conditions) intelligibly.

Secondly, the inconsistency check seems both too rigid and not rigid enough. For thelatter, letI1 = 〈!(A ∧ ¬B), !(B ∧ C)〉 andI2 = 〈!((A ∧ ¬B) ∨ D), !¬D, !(B ∧ C)〉: inboth cases more important imperatives are in conflict with the weakest, but it is rejectedonly in the first. For the former, letI = 〈C ⇒!¬D,C ⇒! (B ∧ D)〉 and the situation be(C ∧ D). 〈C ⇒! (B ∧ D)〉 is not in Binding(I ,<)(C ∧ D), its consequent contradictingthat of a more important imperative. But this has become unfulfillable, which intuitivelyclears the way for obligatoriness ofB. For a solution, I propose to leave inconsistencychecks entirely to the (credulous or skeptical) reasoning strategy defined via sets consistentwith the circumstancesC: let each of these include a maximallyC-consistent subset ofthe most important triggered imperatives’ consequents, a maximal subset of the secondmost important triggered imperatives’ consequents that can beC-consistently added to theformer, etc. This is the incremental maximizing employed for belief revision by Brewka[6] and Nebel[34] (to work, < must be well-founded). Drawing on a parallel result byRott [37, Theorem 7], as long as conflicts between incomparable or equally importantimperatives are allowed, the logic for accordingly defined deontic operators should still beDDL{F,S}.

Acknowledgement

I thank David Makinson for inspiring remarks, and Lou Goble and Leon van der Torrefor comments on an earlier version presented at�EON 2004. The paper is part of a projectbegun in[16] to relate deontic logic to reasoning about imperatives.

References

[1] C.E. Alchourrón, E. Bulygin, Unvollständigkeit, Widersprüchlichkeit und Unbestimmtheit der Normenord-nungen, in: A.G. Conte, R. Hilpinen, G.H. von Wright (Eds.), Deontische Logik und Semantik, Athenaion,Wiesbaden, 1977, pp. 20–32.

[2] A. Bochman, Credulous nonmonotonic inference, in: T. Dean (Ed.), Proceedings of the 16th InternationalJoint Conference on Artificial Intelligence, IJCAI ’99, Stockholm, Sweden, 1999, Morgan Kaufmann, SanFrancisco, 1999, pp. 30–35.

[3] A. Bochman, A Logical Theory of Nonmonotonic Inference and Belief Change, Springer, Berlin, 2001.[4] A. Bochman, Brave nonmonotonic inference and its kinds, Ann. Math. Artificial Intelligence 39 (2003)

101–121.[5] S. Brass, On the semantics of supernormal defaults, in: R. Bajcsy (Ed.), Proceedings of the 13th International

Joint Conference on Artificial Intelligence, IJCAI ’93, Chambery, France, Morgan Kaufmann, New York,1993, pp. 578–583.



[6] G. Brewka, Belief revision in a framework for default reasoning, in: A. Fuhrmann, M. Morreau (Eds.), TheLogic of Theory Change, Workshop, Konstanz, 1989, in: Lecture Notes in Artificial Intelligence, vol. 465,Springer, Berlin, 1991, pp. 206–222.

[7] D.O. Brink, Moral conflict and its structure, Philos. Rev. 103 (1994) 215–247.[8] A. Donagan, Consistency in rationalist moral systems, J. Philos. 81 (1984) 291–309.[9] D. Føllesdal, R. Hilpinen, Deontic logic: An introduction, in[19], pp. 1–35.

[10] L. Forget, V. Risch, P. Siegel, Preferential logics are X-logics, J. Logic Comput. 11 (2001) 71–83.[11] B. van Fraassen, Values and the Heart’s command, J. Philos. 70 (1973) 5–19.[12] L. Goble, Multiplex semantics for deontic logics, Nordic J. Philos. Logic 5 (2000) 113–134.[13] L. Goble, Preference semantics for deontic logics: Part I—simple models, Logique Anal. 46 (2003) 383–418.[14] L. Goble, Preference semantics for deontic logics: Part II—multiplex models, Logique Anal. 47 (2004).[15] L. Goble, A logic for deontic dilemmas, J. Appl. Logic 3 (2005).[16] J. Hansen, Sets, sentences, and some logics about imperatives, Fund. Inform. 48 (2001) 205–226.[17] J. Hansen, Problems and results for logics about imperatives, J. Appl. Logic 2 (2004) 39–61.[18] B. Hansson, An analysis of some deontic logics, Noûs 3 (1969) 373–398, reprinted in[19], pp. 121–147.[19] R. Hilpinen, Deontic Logic: Introductory and Systematic Readings, Reidel, Dordrecht, 1971.[20] J.F. Horty, Nonmonotonic foundations for deontic logic, in: D. Nute (Ed.), Defeasible Deontic Logic,

Kluwer, Dordrecht, 1997, pp. 17–44.[21] J.F. Horty, Reasoning with moral conflicts, Noûs 37 (2003) 557–605.[22] D. Jacquette, Moral dilemmas, disjunctive obligations, and Kant’s principle that ‘Ought’ implies ‘Can’,

Synthese 88 (1991) 43–55.[23] H. Kelsen, Reine Rechtslehre, second ed., Deuticke, Wien, 1960.[24] H. Kelsen, Allgemeine Theorie der Normen, Manz, Wien, 1979.[25] A. Kratzer, Conditional necessity and possibility, in: R. Bäuerle (Ed.), Semantics from Different Points of

View, Springer, Berlin, 1979, pp. 116–147.[26] S. Kraus, D. Lehmann, M. Magidor, Nonmonotonic reasoning, preferential models and cumulative logics,

Artificial Intelligence 44 (1990) 167–207.[27] E.J. Lemmon, Moral dilemmas, Philos. Rev. 71 (1962) 139–158.[28] D. Lewis, Counterfactuals, Basil Blackwell, Oxford, 1973.[29] D. Lewis, Semantic analyses for dyadic deontic logic, in: S. Stenlund (Ed.), Logical Theory and Semantic

Analysis, Reidel, Dordrecht, 1974, pp. 1–14.[30] D. Lewis, Ordering semantics and premise semantics for counterfactuals, J. Philos. Logic 10 (1981) 217–

234.[31] D. Makinson, Bridges between classical and nonmonotonic logic, Logic J. IGPL 11 (2003) 69–96.[32] D. Makinson, L. van der Torre, Input/output logics, J. Philos. Logic 29 (2000) 383–408.[33] R.B. Marcus, Moral dilemmas and consistency, J. Philos. 77 (1980) 121–136.[34] B. Nebel, Syntax-based approaches to belief revision, in: P. Gärdenfors (Ed.), Belief Revision, Cambridge

University Press, Cambridge, 1992, pp. 52–88.[35] D. Poole, A logical framework for default reasoning, Artificial Intelligence 36 (1988) 27–47.[36] N. Rescher, An axiom system for deontic logic, Philos. Stud. 9 (1958) 24–30.[37] H. Rott, Belief contraction in the context of the general theory of rational choice, J. Symbolic Logic 58

(1993) 1426–1450.[38] P. Siegel, L. Forget, A representation theorem for preferential logics, in: L.C. Aiello, J. Doyle, S.C. Shapiro

(Eds.), Principles of Knowledge Representation and Reasoning (Proceedings of the 5th International Con-ference, KR 96), Morgan Kaufmann, San Francisco, CA, 1996, pp. 453–460.

[39] W. Spohn, An analysis of Hansson’s dyadic deontic logic, J. Philos. Logic 4 (1975) 237–252.[40] E. Stenius, The principles of a logic of normative systems, Acta Philos. Fennica 16 (1963) 247–260.[41] L. van der Torre, Reasoning About Obligations, Thesis Publishers, Amsterdam, 1997.[42] L. van der Torre, Y.-H. Tan, Two-phase deontic logic, Logique Anal. 43 (2000) 411–456.[43] B. Williams, Ethical consistency, Proc. Aristotelian Soc. (supp.) 39 (1965) 103–124.[44] G.H. von Wright, A note on deontic logic and derived obligation, Mind 65 (1956) 507–509.[45] G.H. von Wright, Norm and Action, Routledge & Kegan Paul, London, 1963.[46] G.H. von Wright, A new system of deontic logic, Danish Yearbook Philos. 1 (1964) 173–182, reprinted in

[19], pp. 105–115.



[47] G.H. von Wright, A correction to a new system of deontic logic, Danish Yearbook Philos. 2 (1965) 103–107,reprinted in[19], pp. 115–120.

[48] G.H. von Wright, An Essay in Deontic Logic and the General Theory of Action, North-Holland, Amsterdam,1968.

[49] G.H. von Wright, Deontic logic—as I see it, in: P. McNamara, H. Prakken (Eds.), Norms, Logics and Infor-mation Systems, IOS, Amsterdam, 1999, pp. 15–25.


Deontic logics for prioritized imperatives

JORG HANSENInstitut fur Philosophie, Universitat Leipzig, Beethovenstraße 15, D-04107, Leipzig, Germany

e-mail: [email protected]

Abstract. When a conflict of duties arises, a resolution is often sought by use of an ordering of

priority or importance. This paper examines how such a conflict resolution works, compares

mechanisms that have been proposed in the literature, and gives preference to one developed by

Brewka and Nebel. I distinguish between two cases – that some conflicts may remain unresolved,

and that a priority ordering can be determined that resolves all – and provide semantics and

axiomatic systems for accordingly defined dyadic deontic operators.

Keywords: deontic logic, logic of imperatives, priorities

1. Introduction

W. D. Ross (1930) argued that whenever there appears to be a conflict of

duties, through careful study of all aspects of the situation one will arrive

at the conclusion – or rather: the considered opinion – that one of these

duties is ‘‘more pressing’’ than others, and this duty is then one’s duty sans

phrase, whereas the others were prima facie only. Ross gives the following

example:

EXAMPLE (The road accident). ‘‘If I have promised to meet a friend at

a particular time for some trivial purpose, I should certainly think myself jus-

tified in breaking my engagement if by doing so I could prevent a serious

accident or bring relief to the victims of one.’’

There are two conflicting obligations: to keep the promise, and to prevent the

accident or help its victims. The second takes priority: it is in these circum-

stances ‘‘more of a duty’’ than keeping the appointment.

While in the example the determination of the priority ordering seems to

rely on a comparison of the outcomes of satisfying or violating the conflicting

duties under considerations of utility and possible harm, in the case of legal

Artificial Intelligence and Law (2006) 14: 1–34 � Springer 2006

DOI 10.1007/s10506-005-5081-x


obligations or individual imperatives the ordering can often be directly

obtained from the norm’s position in a normative hierarchy or the rank of

the issuing authority. These factors may also relate to each other, e.g. when

the decision of a commander in the field overrules that of her superior due to

some unforeseen danger or opportunity. I will leave aside the question of

how a particular ordering is determined, and also not address Ross’s

notoriously problematic distinction between a prima facie duty and a duty

sans phrase. What interests me here is rather how a conflict resolution

based on (established) priorities works, i.e. what the resolution mechanism

looks like, or should look like, when an ordering of priority or importance

of possibly conflicting norms is assumed. Section 2 introduces the formal

framework and explains how it is used to define deontic operators. After

pointing out counterintuitive results of a conflict resolution based on a

method by Horty (2003), I show in Section 3 that a method developed for

the resolution of inconsistencies in prioritized theory bases by Brewka

(1989, 1991) and Nebel (1991, 1992) fares better (Section 3). A broader

comparison includes ordering based mechanisms by Alchourron and

Makinson (1981), Gardenfors (1984), Alchourron (1986), and variants

(Section 4). Section 5 explores what a priority ordering must be like to

resolve all possible conflicts, and provides a sound and weakly complete

axiomatic system (which readers might find familiar) for a corresponding

dyadic deontic operator. All formal proofs are delegated to the Appendix.

Section 6 concludes.

2. Imperative semantics and deontic logic

When a conflict between norms is resolved by an appeal to some priority

ordering, I assume that what is thus conceived as ordered are the norms

themselves, though their ordering may reflect a ranking of their sources, or an

axiological order of the states realized when fulfilling the norms. So for a

logical analysis, some formal representation of norms is required. I only

consider unconditional imperatives,1 like ‘‘Invite Jones to dinner!’’, and I is a

set of such imperatives. To each imperative corresponds a descriptive sentence

like ‘‘You invite Jones to dinner,’’ which – grammatically similar, but in the

indicative, not the imperative mood – describes what must be the case if and

only if (iff) the imperative is satisfied. Any such descriptive sentence is as-

sumed to have a formalization in the language of a basic logic, which I let be

propositional logic PL.2 A function f : I ! LPL assigns every imperative in I

theLPL-formalization of its corresponding descriptive sentence, and the tuple

hI; f i is called a basic imperative structure. I write !A for an i in I with f (i)=A,

and use the superscripted i f;C f instead of fðiÞ; f ðCÞ for better readability. Inanalogue to the usual concept of remainders, let IfA be the maximal sets of

JORG HANSEN2


imperatives such that the sets of corresponding descriptive sentences do not

derive A (I also call these ‘‘A-remainders’’ of I), i.e. IfA contains all C � I

such that (i) Cf0PL A and (ii) there is no D � I : C � D and D

f0PL A.

In the ‘imperativist tradition’ of deontic logic, authors used such

semantics to interpret deontic formulas, rather than employing the usual

possible worlds semantics.3 Let deontic formulas be those of a language

LDL, based on an alphabet like the one for LPL, except that it additionally

contains the operator symbol ‘O’, whereby OA formalizes the (true or false)

statement that what A describes is obligatory. LDL is then the smallest set

such that

(a) for all A 2 LPL;OA 2 LDL,

(b) if A;B 2 LDL, so are :A; ðA ^ BÞ; ðA _ BÞ; ðA ! BÞ; ðA $ BÞ.

Interpretations of Boolean operators being as usual, the truth definition

ðtd-1Þ hI; f i � OA iff I f ‘PL A:

defines a normal modal logic, i.e. the set of LDL-sentences defined as true for

all tuples hI; f i equals the axiomatically defined set that contains all LDL-

instances into tautologies, furthermore all LPL-instances into

ðExtÞ If ‘PL A $ B; then OA $ OB is in the set:

ðMÞ OðA ^ BÞ ! ðOA ^OBÞ

ðCÞ ðOA ^OBÞ ! OðA ^ BÞ

ðNÞ O>

and is closed under modus ponens. Furthermore, the above truth definition

defines standard deontic logic SDL, which adds the ‘‘deontic’’ scheme (D):

ðDÞ OA ! PA

iff hI; fi is required to be such that I f is consistent (as usual, PA abbre-

viates :O:A). Requiring I f to be consistent excludes conflicts between

imperatives and is thus a severe and in this case unwanted restriction, for to

show how conflicts are resolved they must first be semantically modeled.

But if e.g. two imperatives !p1 and !( p2 ^ :p1) can both be in I, then not

only does (D) fail, but also (td-1) is not very useful, making OA true for

any A 2 LPL. Instead, the following definition for a ‘‘disjunctive’’ ought

operator was put forward:4

ðtd-2Þ hI; f i � OA iff 8C 2 If ? : Cf ‘PL A:

So OA is true when all maximally consistent subsets of what the imperatives

demand derive A. It is apparent that the definition tolerates conflicts and e.g.

if !p1 and !(p2 ^ :p1) are in I, then O(p1� p2), but not O^ is true. Moreover,

DEONTIC LOGICS FOR PRIORITIZED IMPERATIVES 3


this solution is easily adapted to the dyadic case and the related problem of

dilemmas. Dyadic deontic logic uses a language LDDL that employs the

additional auxiliary sign ‘‘/’’ and is likeLDL, except that clause (a) now reads

(a) for all A;C 2 LPL;OðA=CÞ 2 LDDL,

where O(A/C) is read as ‘‘A is obligatory in the circumstances characterized

by C’’. The truth definitions for dyadic deontic formulas should allow some

influence of the circumstances, so e.g. if John must either not impregnate

Suzy Mae or marry her, and she is in fact pregnant by him, then marrying her

seems to be what he must do. But simply putting

ðtd-3Þ hI; f i�OðA=CÞ iff I f [ fCg ‘PL A:

will not suffice if subjects can get (themselves) into dilemmas, i.e. situations

where the norms are collectively satisfiable at the outset, but due to misfor-

tune or failure they cannot all be satisfied anymore. To handle such situa-

tions, and to e.g. prevent the derivation of O(^/p1) when I contains

!ð:p1 _ p2Þ and !ð:p2 ^ p3Þ , the truth definition for a ‘‘disjunctive’’ dyadic

ought operator can be given as:

ðtd-4Þ hI; f i � OðA=CÞ iff 8C 2 If:C : Cf [ fCg ‘PL A:

So A is obligatory in the situation described by C if A is what the imperatives

in any :C-remainder demand, given C. With usual truth conditions for

Boolean operators, this semantics has a sound and (weakly) complete axiom

system PD defined as containing all LDDL-instances into tautologies, all

LPL-instances into

ðCExtÞ If ‘PL C ! ðA $ BÞ then ‘PD OðA=CÞ $ OðB=CÞ

ðExtCÞ If ‘PL C $ D then ‘PD OðA=CÞ $ OðA=DÞ

ðDMÞ OðA ^ B=CÞ ! ðOðA=CÞ ^OðB=CÞÞ

ðDCÞ OðA=CÞ ^OðB=CÞ ! OðA ^ B=CÞ

ðDNÞ Oð>=CÞ

ðDD-RÞ If 0PL:C then ‘PD OðA=CÞ ! PðA=CÞ

ðCondÞ OðA=C ^DÞ ! OðD ! A=CÞ

ðCCMonÞ OðA ^D=CÞ ! OðA=C ^DÞ

and closed under modus ponens.5 PD resembles the system P defined by

Kraus et al. (1990) with the (restricted) dyadic ‘‘deontic’’ scheme (DD-R)

added, hence the name.

For the present purposes, I define a prioritized imperative structure to be a

tuple hI; f; <i that is like a basic imperative structure, except that it addi-

tionally includes an ordering relation < on I, where the formal properties of

this relation are for the moment left open. Unfortunately, authors disagree

JORG HANSEN4


on the direction in which ‘i1 < i2’ is to be read, if it means that i1 takes

priority over i2 (reading < like a preference relation), or that i1 is less

important than i2 (reading < like a utility function). I assume the former,

and adapt differing definitions to this convention, so e.g. a tuple

i1 < i2 < i3. . . is read like a list that starts with what is most important. For

any ordering < on some set C; I define min<C ¼ fi 2 C j 8i0 2 C : if i0 6¼ i,

then i0hig, so min<I is the set of the highest ranking, most important, etc.

imperatives, and max<C ¼ fi 2 C j 8i0 2 C : if i0 6¼ i, then ihi0g, so max<I are

the imperatives that come last, are least important, rank lowest, etc.

3. Reasoning with prioritized imperatives

As explained by Ross, in the case of a normative conflict one proceeds by

examining the situation for clues to an ordering of the obligations involved,

e.g. by considering the rank of the issuing authority, notions of urgency or a

gross difference in the utilities of the outcomes. The example of the road

accident illustrates that the disjunctive ought operator defined in the previous

section, which pays no attention to priorities, produces inadequate results:

EXAMPLE (The road accident: disjunctive reasoning). Let A be

helping the accident victims, B keeping the promise, and T a conjunction of

actual necessities, including the agent’s present physical and psychical capa-

bilities (I write for :T ). An imperative interpretation produces I={!A, !B}

and ‘PL T ! ðA ! :BÞ as helping causes me to miss the meeting. so

(td-1) makes O true and the impossible obligatory, so it is not very useful.

so (td-4) makes O(A�B/T) true but O(A/T) false, as

fB;Tg0PLA. So there is only a disjunctive obligation to help or proceed to the

meeting. But intuitively, helping takes priority over anything else.

The situation looks like a conflict: there exist requirements which cannot

all be satisfied. But the conflict is avoided by (so far intuitively) giving

priority to the norm of greater weight. Note that if symmetrical or incom-

parable obligations are not ruled out, then a demand that takes priority can

not only dissolve a dilemma, but also create a conflict for an otherwise

conflict-free situation:

EXAMPLE (The road accident II). It is Tuesday afternoon, and like on

all Tuesdays, Mirjam must fetch her grandmother from the day care center

before it closes at 6:30. Today, Mirjam was also asked by her boss to bring the

office mail to the post office after hours, which also closes at 6:30, but lies in the

opposite direction. However, when she told of her other duty, she was allowed

to leave early. Driving at 5:30 in the direction of the post office, Mirjam



becomes involved in a traffic accident. The law requires her to stay at the

accident site until the police have recorded it, which won’t happen before 6:00.

Then she can only get to one place, the post office or the day care center, on

time. The law takes priority over her other duties, but a ranking of these is not

obvious; in particular it is difficult to say which violation could have worse

consequences, and Mirjam will have a hard time making up her mind.

To formalize the reasoning about priorities when faced with conflicting

demands, Horty (2003) proposed that the priority ordering is used to first

determine a set of ‘‘binding imperatives’’ in the set of all imperatives:

DEFINITION 1 (Binding imperatives). Let hI; f; <i be a prioritized

imperative structure. Then

So an imperative is ‘‘binding’’ if there is no higher ranking imperative with a

materially inconsistent demand (cf. Horty 2003, p. 560). If it is also higher-

ranking than any such imperative, Horty calls it ‘‘overriding’’. Binding,

instead of I, is then used to define a disjunctive dyadic ought operator:

ðtd-5Þ hI; f;<i � OðA=CÞ iff 8C 2 Bindingf:C : Cf [ fCg ‘PL A

I examine how this definition6 copes with the examples and variants:

EXAMPLE (The road accident: Horty’s solution). A is helping the

accident victims, and B keeping the promise. So I={!A, !B} models the logical

situation, where ‘PL T ! ðA ! :BÞ, as the situation excludes both helping at

the accident site and meeting my friend. The ordering is !A < !B, so !A over-

rides !B and Binding = {!A}. (td-5) makes OðA=TÞ;Oð:B=TÞ and Pð:B=TÞtrue, so helping is obligatory, keeping the appointment forbidden, and not

keeping the appointment permitted, which is as it should be.

EXAMPLE (The road accident II: Horty’s solution). Let A be

Mirjam’s staying at the accident site, B taking her boss’s mail to the post

office, and C fetching her grandmother from the day care center. I={!A, !B, !C}

is the logical model of the normative situation. Mirjam left early enough

to get to both the post office and the day care center in time, so

0PLT ! ðB ! :CÞ, but ‘PL T ! ðA ! :ðB ^ CÞÞ as waiting excludes her

accomplishing both. The legal obligation takes priority, so !A < !B and

!A < !C, while the ranking between !B and !C unclear. Binding = I, as the

truth of A only excludes satisfying both !B and !C, but getting to one place

remains possible. Binding f

T

¼ ff!A; !Bg; f!A; !Cg; f!B; !Cgg, (td-5) making

JORG HANSEN6


OðA=TÞ false as fT;B;Cg0PL A, and only OððA ^ ðB _ CÞÞ _ ðB ^ CÞ=TÞtrue, so it seems Mirjam can choose to wait or go on driving to her destinations,

which is counterintuitive.

EXAMPLE (The road accident III, with Horty’s solution).Things are as in variant II, but suppose some grave danger arises fromMirjam’s

not being at the day care center before it closes, as the disturbed old lady will

wander off on her own and may fall or get lost. Hence, fetching her is much

more urgent than posting the letters, of which the important ones were very

likely faxed beforehand. Fetching her grandma may be even more important

than waiting for the police, but Mirjam is not sure about that and can do both

anyway. So beside !A < !B we have !C < !B, with the ranking between !A and

!C unclear. Intuitively, Mirjam must stay until the police are finished and then

fetch her grandma. But as it is two higher-ranking imperatives that exclude, if

satisfied, the satisfaction of the lower-ranking one, still Binding = {!A, !B,

!C}=I, making OðA=TÞ false and even PðB ^ :C=TÞ true even though !C ranks

higher than !B.

EXAMPLE (The road accident IV, with Horty’s solution).Mirjamdid not dare askher boss for permission to leave early, sneakingout at 5:45

instead, but thatwas too late to get to both places in time, i.e.‘PL T ! ðC ! :BÞ.Again, Mirjam gets involved in a traffic accident and is required to wait for the

police. Fetching her grandmother takes priority over posting the mail, so I={!A,

!B, !C}, !C < !B and !A < !B. Imagine the accident left her car a wreck,

making it impossible to get to the day care center in time, but when the police

finish around 6:15 she can still get to the post office. Let S be this situation

(including T), so ‘PL S ! :C. Binding = {!A, !C}, as !B is not reinstated

when satisfying !C is excluded, so Binding f:S ¼ ff!Agg, making Pð:B=SÞtrue. But it is hard to see why Mirjam should not have to post the letters.

EXAMPLE (The road accident V, with Horty’s solution). As invariant IV, Mirjam left too late to make it to both the post office and the day

care center on time, so ‘PL T ! ðC ! :BÞ. Again, fetching grandma takes

priority over posting the letters, i.e. !C< !B. Suppose it is not the damage, but

the time required by the police that makes it impossible to get to the day care

center on time (it is too far from the accident site, while the post office is just a

block away), so ‘PL T ! ðA ! :CÞ. Making up her mind, Mirjam decides

that the legal obligation to wait for the police probably takes priority over her

familial duty, i.e. !A < !C < !B. Both !B and !C are overridden by higher

ranking imperatives and so are not in Binding f:S ¼ ff!Agg, making

Pð:B=SÞ true. But again it is hard to see why Mirjam is relieved from posting

the letters.



Thus Horty’s set Binding solves simple cases, but is not adequate for

complex hierarchies wheremore than two imperativesmay be in conflict, and it

makes life too easy when conflicting higher ranking imperatives become un-

fulfillable or are themselves overridden.7 To overcome these difficulties when

formalizing that disregard for a lower ranking imperative can (only) be excused

by obedience to higher-ranking ones (and not vice versa), I suggest that neither

remainder sets of I, nor of a fixed subset Binding, but an ‘‘incremental’’ max-

imizing strategy should be used. For a situation C, the relevant sets are con-

structed by first adding a maximal set of the most important imperatives such

that their demands do not derive :C, then adding a maximal subset of the

second most important imperatives that can be added without the corre-

sponding demands now deriving :C, etc. Introduced by Rescher (1964, p. 50),

such incremental maximizing was more rigorously defined and employed for

the purpose of theory revision by Brewka (1989, 1991) andNebel (1991, 1992).

Both employ a strict partial order, i.e. < is irreflexive and transitive. Nebel

additionally assumes < to be the asymmetric part of a complete preorder £,

i.e. obtained from a reflexive, transitive and connected ordering £ via defining

i<j iff i £ j and j 6� i. Both agree that < must be well-founded, i.e. infinite

descending chains are excluded.8 For any <, Brewka defines a full prioritiza-

tion� to be any (strict) well-order on the given set that preserves<, i.e. for all i,

j: if i< j then i � j. Clearly:

THEOREM 1 (Existence of full prioritizations). For every well-

founded strict partial order < on a set C there is a full prioritization �, i.e. a

strict well-order that is order-preserving with respect to <.

Brewka then defines subsets of the set as ‘preferred subtheories’. Calling

them preferred remainders (they are not theories here), his definition trans-

lates thus:9

DEFINITION 2 (Brewka’s preferred remainders). Let hI; f; <i be aprioritized imperative structure, where < is a well-founded strict partial order

on I. Then C � I belongs to the preferred remainder set I + A iff (i) Cf0PLA,

and (ii) C is obtained from a full prioritization � by defining

SA½�#i� ¼

S

j�i SA½�#j� [ fig if ½

S

j�i SA½�#j��

f [ fif g0PL A, andS

j�i SA½�#j� otherwise,

(

for any i 2 I, and letting SA� ¼

S

i2I SA½�#i� and C ¼ SA

�.

(i) bans the empty set from I + A for tautological A, and (ii) recursively

defines S½�#i� to include all elements of some such set for a prior element j,

adding i if possible without the corresponding set deriving A. C is the union

JORG HANSEN8


of all such sets. I drop superscripts if the meaning is clear. The following is

almost immediate:

THEOREM 2 (Preferred remainders are remainders). Let hI; f; <ibe a prioritized imperative structure, where < is a well-founded strict partial

order on I. Then I + A � IfA.

As noted, Nebel’s (1992) approach defines < as the asymmetric part of a

complete, well-founded preorder £. For each i 2 I, the priority class

[i ]={j 2 I | i £ j and j £ i} contains all j 2 I of the same £-priority as i. A

preference-ordering�N between all subsets D;C of I is then defined by letting

D �N C iff 9i 2 I : 8j < i : C \ ½ j� ¼ D \ ½ j� and C \ ½ i� � D \ ½i�

i.e. by preferring D over C iff both agree for all priority classes up to some [i],

of which D contains all elements of C plus more. Then choosing a maximally

�N-preferred set among all C � I with Cf0PLA equals choosing from I + A:

THEOREM 3 (Nebel’s prioritized removals). Let hI; f; <i be a

prioritized imperative structure, where < is the asymmetric part of a complete,

well-founded preorder £ on I. Then I + A equals

fC � I j Cf0PLA and 8D � I : ifD �N C then D

f ‘PL Ag:

There is an alternative, non-constructive definition of Brewka’s pre-

ferred remainders, attributed to Ryan (1992) by Rintanen (1994) and also

appearing in Sakama and Inoue (1996): Let hI; f; <i be a prioritized imper-

ative structure, where < is a well-founded strict partial order on I, and define

p ðIfAÞ to be the set

fC 2 IfA j 9 �: 8D 2 IfA n fCg : 9i 2 C n D : 8j 2 D n C : i � jg:

So some A-remainder C is in p ðIfAÞ iff for some full prioritization �;C

contains for any other A-remainder D some exclusive element that �-ranks

higher than any element exclusively in D. The following holds.

THEOREM 4 (Preferred remainders, after Ryan and Sakama &

Inoue). Let hI; f; <i be a prioritized imperative structure, < a well-founded

strict partial order on I and I + A and pðIfAÞ be as defined. Then

I + A ¼ pðIfAÞ.

Proposing use of Brewka’s and Nebel’s concept of preferred remainders,

based on some strict partial, well-founded ordering <, as the resolution

mechanism for conflicts between imperatives or dilemmas that arise in certain

situations, a disjunctive ought operator can be defined parallel to (td-4) as

follows:



ðtd-6Þ I � OðA=CÞ iff 8C 2 I + :C : Cf [ fCg ‘PL A:

This truth definition fares better in dealing with the above examples:

EXAMPLE (The road accident: the Brewka/Nebel solution). A is

helping the accident victim, B keeping the promise, I={!A, !B} and

‘PL T ! ðA ! :BÞ. The ordering is !A < !B, being its only full prioritization.

The construction of S< includes !A but rejects !B, so I +

T

={{!A}} and (td-6)

makes true OðA=TÞ;Oð:B=TÞ and Pð:B=TÞ, which is as it should be.

EXAMPLE (The road accident II: the Brewka/Nebel solution).I ¼ f!A; !B; !Cg: ‘PL T ! ðA ! :ðB ^ CÞÞ, as waiting allows Mirjam to get to

one place, the post office or the day care center in time, but not both. The

ordering is !A < !B and !A < !C, and for !B, !C unclear. Its two full priori-

tizations are !A � !B � !C or !A � !C � !B, producing I +

T

={{!A, !B}, {!A,

!C}}. Then (td-6) makes true O(A � (B � C)/T), so waiting and then going to

one place, the post office or the day care center, is obligatory as it should be.

EXAMPLE (The road accident III: the Brewka/Nebel solution).Still ‘PL T ! ðA ! :ðB ^ CÞÞ, i.e. waiting excludes getting to both places.

Fetching her grandma now takes priority over going to the post office, so

!C < !B and !A < !B, this time the ranking between !A, !C being unclear. The

two full prioritizations are !A � !C � !B and !C � !A � !B, so I +

T

={{!A,

!C}}, and (td-6) makes O(A � C/T ) true. So Mirjam must stay at the site until

the police are finished with her and then go to fetch her grandmother, as it

should be.

EXAMPLE (The road accident IV: the Brewka/Nebel solution).Getting to both the post office and the day care center on time was never

possible, so ‘PL T ! ðC ! :BÞ. The car being wrecked, the assumed situation

S excludes getting to the day care center on time, so ‘PL S ! :C. Again!C < !B and !A < !B, with the relation between !A and !C unclear, so

!A � !C � !B and !C � !A � !B are the full prioritizations, producing

I + :S ¼ ff!A; !Bgg. Hence O(A � B/S), i.e. Mirjam must wait and then hurry

to the post office, which is as it should be.

EXAMPLE (The road accident V: the Brewka/Nebel solution).Again, Mirjam cannot get to both the post office and the day care center on

time, so ‘PL T ! ðC ! :BÞ. Waiting for the police excludes getting to the day

care center on time, i.e. ‘PL T ! ðA ! :CÞ. Mirjam decides that the law

JORG HANSEN10


overrides her familial duty, so !A < !C < !B, which, being its only full

prioritization, yields I +

T

={{!A, !B}}. So Mirjam must wait and post the

letters, as it should be.

Let a semantics be called a prioritized imperative semantics iff it defines the

truth of LDDL-sentences using (td-6), with respect to arbitrary prioritized

imperative structures hI; f; <i. Then it may be surprising – though Rott

(1993, Theorem 7) already proved a similar result – that the logical properties

of such a semantics are not different from that defining the deontic operator

using (td-4), i.e. with respect to basic imperative structures and simple

remainders, since the system PD remains sound and (only) weakly complete:

THEOREM 5 (Soundness, completeness of PD). PD is sound and

(only) weakly complete with respect to prioritized imperative semantics.

4. Alternative resolution mechanisms

4.1. LEAST EXPOSURE AND ITS VARIANTS

Alchourron and Makinson (1981) seem to have been the first to logically

examine the idea of resolving contradictions in a body of norms, or con-

tradictions that arise from such a body together with some set of true

empirical facts, by imposing an order upon that body. The object of their

study is a set of regulations that is partially ordered by a relation £, which

does not necessarily stand for an ordering by priority or importance. Rather,

i £ j means that j is as much exposed, or more exposed, to the risk of

legislative derogation as i. If a conflict occurs between two parts of the code,

or between the code and some empirical facts, the aim is to find a (possibly

maximal) non-conflicting subset that is most secure from the changes which

the law-giver will presumably enact upon learning of this situation. Their

definition translates to the present framework as follows:

DEFINITION 3 (Alchourron and Makinson’s strict exposure).Let hI; f; <i be a prioritized imperative structure, where < is a strict partial

order on I (like the asymmetric part of a partial order £). Then for all

C;D � I :

C �AM D iff D 6¼ [ and 8i 2 C : 9j 2 D : i < j:

So a subset is strictly less exposed than some other if for any member

of the first there is a member of the second which is strictly more exposed.

To see how this approach compares to Brewka and Nebel’s, consider three

cases:



Case 1. Let hI; f; <i be !p1< !ðp2 ^ :p3Þ<!p3, the imperative in the

‘middle’ conflicting with the lowest. Then we have

– If ?¼ ff!p1; !ðp2 ^ :p3Þg; f!p1; !p3gg and

– I +?¼ ff!p1; !ðp2 ^ :p3Þgg.

The exposure criterion yields f!p1; !ðp2 ^ :p3Þg �AM f!p1; !p3g: for each left

member, a right member is strictly more exposed, namely !p3.

Case 2. Let hI; f; <i be !p1 < !ð:p1 ^ p2Þ < !p3, so the ‘‘middle’’ now

conflicts with the higher-ranking imperative. Then we have

– If ?¼ ff!p1; !p3g; f!ð:p1 ^ p2Þ; !p3gg and

– I +?¼ ff!p1; !p3gg.

But f!p1; !p3g 6�AM f!ð:p1 ^ p2Þ; !p3g: from the left set, !p3 is not less exposed

than !ð:p1 ^ p2Þ from the right. The authors recognize that a conflict between

higher-ranking norms excludes lower-ranking norms from a least exposed set

and propose to use relevant logic for determining conflicts as a cure (Al-

chourron and Makinson 1981, p. 139).

Case 3. Let hI; f; <i be !p1 < !ð:p1 ^ p2 ^ :p3Þ < !p3, the ‘‘middle’’ now in

conflict with both ends of the hierarchy (by whatever logic). Then we have

– If ?¼ ff!p1; !p3g; f!ð:p1 ^ p2 ^ :p3Þgg and

– I +?¼ ff!p1; !p3gg.

Yet f!ð:p1 ^ p2 ^ :p3Þg �AM f!p1; !p3g: from the right set, !p3 is more ex-

posed than any left member. Mediocrity rules! But even if !p3 is more exposed

to legislative change, if that change came about and removed !p3, the right set

would still contain a member that ranks higher than any in the left.

Prakken, pursuing an argumentative approach, wants to employ Al-

chourron and Makinson’s criterion at the heart of his ‘‘rebuttal’’ mechanism

used to determine justified arguments (derivations from facts and defaults.

But the criterion he presents (Prakken 1997, p. 192) translates differently:

DEFINITION 4 (Prakken’s criterion for hierarchical rebuttal).Let hI; f; <i be a prioritized imperative structure, where < is a strict partial

order on I (like the asymmetric part of a partial order £). Then for all C;D � I:

C �P D iff 9j 2 D : 8i 2 C : i<j:

So D can be improved by exchanging some member with any member of

C. Giving the rationale here and in Prakken and Sartor (1997, p. 36)10, the

change in the order of the quantifiers seems intentional – yet it makes a

difference: let I be f!p1; !p2; !ðp3 ^ :p1Þ; !ðp4 ^ :p2Þg and !p1 < !ðp3 ^ :p1Þ and

JORG HANSEN12


!p2 < !ðp4 ^ :p2Þ. E.g. p1, p2 may be primary targets and p3 ^ :p1; p4 ^ :p2respective secondary ones, where reaching the secondary target includes

failing to reach the (better) primary one. Reaching both primary targets

seems best, and in fact f!p1; !p2g �AM f!ðp3 ^ :p1Þ; !ðp4 ^ :p2Þg: for every

member in the left set there is a lower-ranking one in the right. Also

I +?¼ ff!p1; p2gg, since all four full prioritizations yield this preferred

remainder. But f!p1; !p2g 6�P f!ðp3 ^ :p1Þ; !ðp4 ^ :p2Þg as no member in the

right set ranks lower than all in the left. So Prakken’s criterion appears even

less suited to our task than Alchourron & Makinson’s.11

Sartor (1991) used Alchourron and Makinson’s criterion for a

‘‘prevailing’’ relation between subsets modulo a rejected sentence A,12 as

follows:

DEFINITION 5 (Sartor’s ‘‘prevailing’’ relation). Let hI; f; <i be aprioritized imperative structure, where < is a strict partial order on I, and

�AM be as defined above. Then for all C;D � I :

C �AS D iff ½C [ D� f ‘PL A; and 8D0 � D such that ½C [ D

0� f ‘PL A :

9C0 � C : ½C0 [ D0� f ‘PL AandC0 �AM D

0:

Finally pref ðIfAÞ ¼ min�ASðIfAÞ.

To see how his definition works, consider first the ‘‘mediocrity rules’’

example: hI; f;<i is !p1<!ð:p1 ^ p2 ^ :p3Þ<!p3. Then f!p1; !p3g �?S f!ð:p1

^ p2 ^ :p3Þg, as the only subset of the right set conflicting with the left set is

the right set itself, and for this set some conflicting subset of the left set,

namely {!p1}, is strictly less exposed than the right set. So now the result is as

it intuitively should be. Sartor’s relation also handles the example against

Prakken’s criterion well: here I ¼ f!p1; !p2; !ðp3 ^ :p1Þ; !ðp4 ^ :p2Þg, with

!p1<!ðp3 ^ :p1Þ and !p2<!ðp4 ^ :p2Þ. Then f!p1; !p2g � ?S f!ðp3 ^ :p1Þ;

!ðp4 ^ :p2Þg, as the three subsets of the right set conflicting with the left

set are f!ðp3 ^ :p1Þg; f!ðp4 ^ :p2Þg and the right set itself, with which

the following respective subsets of the left set are both in conflict and strictly

less exposed: {!p1}, {!p2}, and the left set itself. So reaching the primary

targets is best, as it should be. In fact, it can be proved that pre-

ferred remainders are always prevailing remainders, but the converse does

not hold:

EXAMPLE (Counterexample to prefðIfAÞ � I + AÞ. Let hI; f;<ibe such that I consists of

i1 : !ðp1 ^ ððp2 _ p3Þ ! qÞÞ i3 : !p2

i2 : !ð:p1 ^ ððp2 ^ p3Þ ! :qÞÞ i4 : !ðp3 ^ ðp1 ! :p2ÞÞ



and the ordering i1 < i2 < [i3, i4]. Intuitively, to satisfy i1 takes priority, and

then the only choice is the one between the equally ranking i3 and i4, and indeed,

I +?¼ ffi1; i3g; fi1; i4gg. But the remainder {i2, i3, i4}, missing the most

important imperative i1, is also in prefðI f ?Þ, as

ðaÞ fi1; i3g 6� ?S fi2; i3; i4g; ðbÞ fi1; i4g 6�?

S fi2; i3; i4g:

For (a), consider {i4}, which is a subset of the right hand set: it conflicts with the

left hand set, so for the relation to hold, a strictly less exposed subset of the left

set must also conflict with {i4}. The only such subset is the left set itself, but since

for its member i3 there is no strictly more exposed member in {i4}, it is not

strictly less exposed. The refutation of (b) works similarly using {i3}.

So Sartor’s definition also produces counterintuitive results where Brewka

and Nebel’s approach does not.13

4.2. UTILITY-REFLECTING PRIORITIES

Regarding the neighboring realm of epistemic logic, and the related problem

of revising belief sets in the face of conflicting information, such information

often finds the reasoner less willing to give up some beliefs than others. In an

attempt to allocate this ordering of ‘‘epistemic importance’’ a role in deter-

mining which of the contradictory beliefs should be given up, Gardenfors

(1984) proposed the following: Let K be a belief set (set of descriptive sen-

tences) that is the logical closure of some finite basis, and £ a relation (of

epistemic importance) that is a complete preorder on this set, which addi-

tionally ranks logically equivalent beliefs equally. For any remainder

C 2 K ? A there is then a ‘‘spanning sentence’’ SC in C that derives any

element in C. Then for any C;D 2 K ? A:

C �G D iff SC<SD

So a remainder is preferred to some other iff its spanning sentence is epi-

stemically at least as important as that of the other. It is essential for the

construction that < is a complete ordering on K, which due to logical clo-

sure includes the spanning sentence that is the ‘‘sum’’ of a remainder. But, the

logical philosophers not being kings, a set of imperative-contents is rarely

logically closed, which precludes a direct parallel. Yet, choosing subsets that

‘‘in sum’’ are the most important has an analogue if the ordering of the

imperatives reflects not so much their importance or rank of the source, but a

measure of ‘‘goodness’’ or utility of the outcome when satisfying the

imperative. For this, let the (well-founded, strict partial) order <u corre-

spond to a function u : X ! R, with I f � X � LPL, that assigns a real

JORG HANSEN14


number to (at least) what the imperatives demand in a manner conversely

respecting <u, i.e. if i <u j then u(i f) > u( j f). Then let

I uA ¼ fC � IjCf 6‘PL A and 8D � I : ifX

uðD fÞ �X

uðCfÞ thenD f ‘PL Ag:

So an A-consistent subset is preferred to another if the good brought about by

satisfying all of its demands sums up to a higher value than by doing so for the

other set. I uA includes the maximally preferred among such sets. Obviously

I uA � IfA if u assigns just positive values. TheO-operator is then defined by

ðtd-7Þ hI; f;<ui � OðA=CÞ iff 8C 2 I u:C : Cf [ fCg ‘PL A:

If a set of imperatives that requires A to be true in the situation C constitutes

a ‘‘reason’’ to call A obligatory in this situation, then O(A/C) is true if the

reasons for obligatoriness A have more collective weight (sum up to a higher

value) than any such reasons for :A.14 Well-foundedness of <, with the

condition that u respects <, excludes infinitely increasing utilities and thus

corresponds to the limit assumption in preference semantics, which avoids

counterintuitive results but may be criticized as superficial (cf. Fehige 1994).

Still, infinitely lower and lower ranking imperatives, whose satisfaction

nevertheless produces some good, are not excluded, and thus not the scenario

of McNamara (1995) where a bad act like killing your mum will eventually

become permitted by (only then) piling up good deeds of small value. To

protect imperatives from getting overruled by inferiors in this manner, one

could assign to an imperative’s satisfaction a utility that is absolutely higher

than the sum of the utilities assigned to the satisfaction of any number of

lower-ranking imperatives (think of the sequence 1, 0.5, 0.25, 0.125, ...). It is

immediate that if <u is thus protected, i.e. for all i 2 I, u(if) ‡P

u({ j 2 I | i

<u j}f), and u assigns just positive values, then I u A � I + A.

4.3. SAFE CONTRACTION AND A MODIFICATION

What characterizes Brewka and Nebel’s approaches is that they try to

maximize the number of higher-ranking imperatives in a set that avoids a

conflict. This intuition has a counterpart formulated by Alchourron (1986):15

‘‘It is logical to believe that the reasonable way of overcoming a

conflict of obligations is to leave aside the less important norms

contributing to its creation.’’

Alchourron’s proposal is then to remove from the set of norms all those that

are least-ranking in a minimal conflicting subset, thus removing the conflict

as well. For the formal description of this ‘‘safe contraction’’, let IgA be the

minimal sets of imperatives such that the corresponding sentences derive

A (the ‘‘A-kernels’’ of I), i.e. the set of all C � I such that (i) Cf ‘PL A and



(ii) for all D � I, if D � C then Df0PLA. < being a strict partial ordering on

I, the following defines the set of all the least important imperatives of

A-kernels of I:

rðIgAÞ ¼def fmax<C j C 2 IgAg:

Due to PL-compactness, any C 2 I gA is finite and so max<C 6¼ if

C 6¼ . Then the set I/A of elements of I that are ‘‘safe’’ with respect to A is

defined by

I=A ¼ def InrðI gAÞ:

If C characterizes the situation, the dyadic deontic operator can then be

defined as in (td-3), but using I=:C instead of I:16

ðtd-8Þ hI; f;<i � OðA=CÞ iff ½I=:C� f [ fCg ‘PL A:

To see how this works, consider again the example of the road accident:

EXAMPLE (The road accident: safe contraction). I={!A, !B},

with !A < !B, as helping the victims of the accident is more important than

proceeding to the appointment. ‘PL T ! ðA ! :BÞ, because staying excludes

meeting my friend. hence

= {!A}. Hence I must stay and help, as it should be.

EXAMPLE (The road accident V: safe contraction). I={!A, !B, !C}.

Mirjam left too late for both the post office and the day care center, so

‘PL T ! ðC ! :BÞ. Waiting makes her too late for the day care center, so

‘PL T ! ðA ! :CÞ. The legal requirement to stay and her duty to fetch her

grandma are both more important than posting the letters, and Mirjam decides

that the law also overrides her familiar duty, so !A<!C<!B. Intuitively

Mirjam must wait and then hurry to the post office. But

¼f!A; !Cg; f!B; !Cgg; max< f!A:!Cg ¼ f!Cg; max<f!B; !Cg ¼ f!Bg: Soand = {!A}. Hence a solution by safe contraction only

requires Mirjam to wait for the police, though she could still post the letters when

the police are finished.

The last example illustrates that safe contraction removes elements too

liberally; even when it has already removed some element of a kernel, further

elements get removed as well even though under the definition of a kernel

removing one suffices: !C was removed due to its conflict with !A, so there

was no need to also remove !B to avoid the conflict with !C. This makes life

JORG HANSEN16


too easy for the norm subjects, and some moderation appears necessary.

Following the idea that the removal mechanism should somehow be adjusted

to the set’s shrinking it brings about, a moderated version of safe contraction

can be defined as follows:

DEFINITION 6 (Moderated safe contraction). Let hI; f;<i be a

prioritized imperative structure, where < is a well-founded strict partial order

on I. For each A 2 LPL, let M½<;a� � I; 0 � a � card(I), be

MA½<;a� ¼def

[b<a

MA½<;b� [min<rð½In

[b<a

MA½<;b��gAÞ:

Finally MA< ¼def

ScardðIÞa¼1 MA

½<;a� and .

So if X are the <-minimal elements in rðIgAÞ, then the moderated mech-

anism first puts X in the set M< of elements to be removed, then the

<-minimal elements in rð½In X �gAÞ, etc. Thus elements get removed in

each step until there is noA-kernel left in Iminus the last version ofM<, which

alsomeans that the cardinality of I suffices for the indices (I omit superscripts if

themeaning is clear). To see how this works, consider again the above example:

EXAMPLE (The road accident V: moderated safe contraction).!C is the minimal element in Removing !C is unavoidable:

the only other element !A in the -kernel in which !C is maximal ranks

before !C, so if !A was maximal in some -kernel, !C would not be in

So !C is in M½<;1� ¼ f!Cg, equalling M< since no -kernel is left

in Inf!Cg. Hence , so both obligations – to wait and post the

letters – remain.

If < is not a well-order, the moderated method still removes too much:

if it removes anything from a kernel, it removes all <-maximal members,

but by definition of a kernel, one is enough. Instead, one might again consider

the full prioritizations that preserve <, rather than <. The relation between

moderated safe contraction and Brewka’s method is then the following:

THEOREM 6 (Moderated safe contractions and preferred

remainders). Let hI; f;<i be a prioritized imperative structure, where < is

a well-founded strict partial order on I. Then is some full

prioritization of <}.

5. Uniquely prioritized imperatives

As demonstrated, the method proposed by Brewka and Nebel seems adequate

for a resolution of normative conflicts by use of priority orderings of the



underlying norms. Whether all conflicts and dilemmas are thus avoidable is a

matter of dispute. W. D. Ross may be understood as claiming that conflicts

are merely apparent and that by weighing all relevant facts and reasons, it can

be decided which of the conflicting prima facie duties are really our duties (cf.

Ross 1930, Searle 1980, p. 242). G. H. von Wright stated that an axiological

order can ‘‘provide a safeguard against any genuine predicament’’ (cf. von

Wright 1968, p. 68, 80). And Hare’s description of ‘‘critical moral thinking,’’

that lets principles override other, less important principles, suggests that this

process can overcome all moral conflicts (Hare 1981, p. 43, 50). On the other

hand, Barcan Marcus (1980) and Horty (2003, p. 564) have argued that if it is

the presence of certain facts that determines the ordering, i.e. this is not an

arbitrary hacking through the Gordian knot, then situations might be

incomparable (if all such facts are missing), or be completely symmetrical (e.g.

identical obligations towards identical twins), so conflicts remain possible.

Rather than take sides in this controversy, I will examine what is required if

the method of Brewka and Nebel is to resolve all conflicts and dilemmas. For

all possible17 situationsC; I + :Cmust then be a singleton: otherwise there are

C;D 2 I + :C such that Cf [ Df ‘PL :C, and there is a dilemma. So I define:

DEFINITION 7 (Uniquely prioritized imperatives). Let hI; f;<ibe a prioritized imperative structure, Then hI; f;<i is called uniquely priori-

tized iff for all C 2 LPL such that, 0PL:C, cardðI + :CÞ ¼ 1.

The question of a resolution of all conflicts can now be rephrased to ask

what an imperative structure hI; f;<i must be like to be uniquely prioritized.

The most obvious way to avoid all conflicts for any situation C is to let the

strict partial order < be total, i.e. for any two i, j 2 I, either i < j or j < i.

Then there is just one full prioritization � that preserves <, namely < itself.

This result was also noted by Nebel (1992, Proposition 11), who con-

structs < as the strict part of a complete preorder £ ) then requiring < to

be total makes each equivalence class [i] a singleton. Gardenfors (1984,

p. 146), who also constructs < as the strict part of some complete preorder

£, points out that it is enough if the only choice left is between equivalents,

i.e. here either i < j or j < i for any i, j 2 I with 0PLif $ j f. But this still

requires too much. It suffices that the demands of elements in each [i] are

chained, so for all j1; j2 2 ½i� :‘PL jf1 ! j

f2 or ‘PL j

f2 ! j

f1 – then it does not

matter in which order j1 and j2 appear in a full prioritization � of <.18 And

one can be even more lax, as demonstrated by the following cases, which also

are of a sort in which no ambiguity ever arises:

Case 1: Let hI; f;<i be ½!p1; !p2�<!ðp1 ^ p2Þ, so the demands of the two

higher-ranking imperatives are ‘‘doubled’’ by a lower one. If both, !p1, !p2 are

in S� 2 I + :C, then adding !ðp1 ^ p2Þ adds nothing. Otherwise

½S½�#!ðp1^p2Þ� [ f!ðp1 ^ p2Þg�f ‘PL :C, so !:ðp1 ^ p2Þ cannot be in S�.

JORG HANSEN18


Case 2: Let hI; f;<i be ½!p1; !p2�<!:ðp1 ^ p2Þ, so the demands of the two

higher-ranking imperatives run contrary to the lower one’s. Either both

!p1,!p2 are in S� 2 I + :C, so !:ðp1 ^ p2Þ cannot be consistently added, or

½S½�#!:ðp1^p2Þ��f [ fCg ‘PL :ðp1 ^ p2Þ, so adding !:ðp1 ^ p2Þ adds nothing.

Case 3: Let hI; f;<i be !ðp1 ^ p2Þ<!ðp1 ^ :p2Þ<½!p1; !p2�. If p1 ^ p2is consistent with C, then I + :C is ff!ðp1 ^ p2Þ; !p1; !p2gg. Otherwise,

fCg ‘PL :ðp1 ^ p2Þ. If p1 ^ :p2 is consistent with C, then I + :C is

ff!ðp1 ^:p2Þ; !p1gg. Otherwise, also fCg ‘PL;:ðp1 ^:p2Þ. Hence fCg ‘PL :p1.Then any set in I + :C contains at most !p2, depending on whetherC is consistent

with p2.

Unable to make out a necessary and sufficient requirement for

cardðI + :CÞ ¼ 1, without reference to particular C, I can only rephrase its

definition as follows:

THEOREM 7 (Property of uniquely prioritized imperatives).Any hI; f; <i,where < is the strict part of some complete preorder £, is uniquely

prioritized iff for all consistent C 2 LPL;C 2 I + :C; i 2 I and j1, j2 2 [i],

fi0 2 Cji0<igf [ fCg ‘PL jf1 ! j

f2 or fi0 2 Cji0<igf [ fCg ‘PL j

f2 ! j

f1 :

Let a semantics be called a uniquely prioritized imperative semantics iff it

defines the truth of LDDL-sentences using (td-6), but only considers uniquely

prioritized imperative structures. It validates the additional axiom scheme:

ðRMonÞ PðD=CÞ ! ðOðA=CÞ ! OðA=C ^DÞÞ:

Consider the system PD that was sound and (weakly) complete with respect

to prioritized imperative semantics. The system that results when (RMon) is

added is Hansson’s (1969) DSDL3 as axiomatized by Spohn (1975):

THEOREM 8 (PD+RMon equals DSDL3). Let PD+(RMon) be

like PD, except that (RMon) is added as art axiom scheme. Then

PD+(RMon) = DSDL3, which is the smallest set that contains all LDDL-

instances into tautologies as well as all LPL-instances of the following schemes:

ðA0Þ OðA=AÞ

ðA1Þ If 0PL A then ‘DSDL3 :Oð? =AÞ

ðA2Þ OðB ^ C=AÞ $ ðOðB=AÞ ^OðC=AÞÞ

ðA3Þ If ‘PL A $ A0 and ‘PL B $ B0 then ‘DSDL3 OðB=AÞ $ OðB0=A0Þ

ðA4Þ PðB=AÞ ! ðOðC=A ^ BÞ $ OðB ! C=AÞÞ

and is closed under modus ponens.



Hansson’s DSDL3, which is also the core of Aqivist’s (1986) system G,

usually characterizes a preference-based dyadic deontic semantics, i.e. an

interpretation of deontic formulas using a ‘‘betterness relation’’ between

valuations or possible worlds, and not using explicitly given norms as it is

here. However, this extreme interpretational change did not result in a

changed logical behavior of the deontic operators, i.e. DSDL3 can be

reconstructed quite naturally using an imperative semantics with an axio-

logical order in the spirit of von Wright (1968):

THEOREM 9 (Soundness, completeness of DSDL3). DSDL3 is

sound and (only) weakly complete for uniquely prioritized imperative

semantics.

The construction used to prove the completeness theorem also exhibits

the following relation between priorities and contrary-to-duty norms: sup-

pose there is a finite, or finitely based, set of deontic truths D � LDDL, and

hI; f; <i is a uniquely prioritized imperative structure that makes true all of

D. The construction used to prove DSDL3-completeness provides a un-

iquely prioritized imperative structure hI0; f0;<0i that also makes true all of

D, but the demands of these imperatives are now chained and so the pri-

ority relation can remain empty or let all imperatives rank equally (cf.

Theorem 7). E.g. if hI; f;<i is !p1 < !p2, then hI0; f0;<0i is

½!ðp1 ^ p2Þ; !ð:ðp1 ^ p2Þ ! p1Þ; !ðð:ðp1 ^ p2Þ ^ :p1Þ ! p2Þ�. These can be

viewed as contrary-to-duty norms, where the primary obligation is: to make

p1 ^ p2 true, if that is not possible, to make p1 true, and if that is also not

possible, to make p2 true. So instead of using ranks and priorities to avoid

conflicts, contrary-to-duty formalizations can be employed to produce the

same effect. While there may be pragmatic reasons to attach higher priority

to the commands of the king than the wishes of his jester, from the

standpoint of logic, exception clauses like ‘‘if the king did not say other-

wise’’ suffice.

6. Conclusion

Describing how a resolution of normative conflicts using priorities works is

surprisingly difficult. A method based on a proposal by Horty could not solve

complex cases where more than two norms conflict or overriding norms are

no longer satisfiable or are themselves overridden. A method developed for

theory revision by Brewka and Nebel, which creates maximally non-con-

flicting sets by starting with a maximal set of what is most important and

JORG HANSEN20


incrementally adding maximally to it, is able to resolve these difficulties.

Alternative mechanisms discussed in normative theory, namely Alchourron

and Makinson’s exposure criterion and variants by Prakken and Sartor, have

counterintuitive results in cases that Brewka and Nebel’s method adequately

solves. The same holds for Alchourron’s ‘‘safe contractions’’, but the intui-

tion underlying his construction, that in a conflict the least important norms

should be set aside, is captured in a moderated version that is equivalent to

Brewka’s. I explain how a semantics that models explicitly given imperatives

can be used to define deontic operators. When the ‘‘preferred remainders’’

from Brewka and Nebel’s method are thus used for the definition of a dis-

junctive (skeptical) dyadic deontic operator, then such a semantics is char-

acterized by the axiom system PD, which resembles Kraus, Lehmann and

Magidor’s system P with the dyadic D-axiom added. Whether all conflicts

can be resolved using priorities is left to philosophical dispute, but conditions

are discussed that guarantee priority orderings, which do just that. For a

semantics that defines its dyadic deontic operators with respect to such

‘‘uniquely prioritized’’ imperatives, Hansson’s axiom system DSDL3 is

proved to be sound and complete. The proof’s construction also exhibits the

fact that priorities are dispensable and that contrary-to-duty constructions

can take their place.

Most of the approaches discussed here include conditional entities,

which pose different problems like the following (rephrased from Rintanen

1994):

(1) a says: if you drink anything, then don’t drive.

(2) b says: if you go to the party, then you do the driving.

(3) c says: if you go the party, then have a drink with me.

(4) You go the party.

Suppose that b does not mind if you have one drink with c, and c does not

care that you may be driving, and let the three imperatives be ranked in des-

cending order. One may be tempted to reason as follows: consider first the

imperative in line (1), but it has not yet been ‘triggered’ as you have not yet

drunk anything, so it is set aside. Regarding (2), its condition is true, so you

must do the driving. Still, only (3) is triggered, so you should have a drink with

c. But satisfying (2) and (3) both triggers and violates the highest-ranking

imperative. Is it not more prudent to violate one of the lower-ranking imper-

atives instead of the higher-ranking one? For a solution, we need an adequate

definition of triggering (that can handle e.g. disjunctive inputs, like Makinson

and van der Torre’s ‘‘basic output’’ (Makinson and van der Torre 2000, 2001),

and to find a maximizing strategy that is consistent with the above intuition. It

is clear that the present discussion has not provided the tools to properly

address such problems, so these must be left to further study.



Acknowledgments

I thank David Makinson for valuable help with an earlier version, and Lou Goble,

John F. Horty and Henry Prakken for helpful discussions and clarifying corre-

spondence. An earlier version of the paper was presented to the working group Law

and Logic of the XXII World Congress of Philosophy of Law and Social Philosophy

(IVR 2005) 24–29: May 2005, Granada.

Notes

1 Though some discussed approaches cover conditional imperatives, or entities that can be

interpreted as such, these cause problems that are best considered separately.2 PL is based on a language LPL, defined from a set of proposition letters Prop ={p1,

p2,...}, Boolean connectives ‘‘ :’’, ‘‘ ^’’, ‘‘ _’’, ‘‘ fi ’’, ‘‘M’’ and brackets ‘‘(’’,‘‘)’’ as usual,

The truth of a LPL-sentence A is defined recursively using a valuation function v :

Prop ! f1; 0g (I write v � A), starting with v � p iff v(p)=1 and continuing as usual. If

A 2 LPL is true for all valuations it is called a tautology. PL is the set of all tautologies,

and this set is used to define provability, consistency and derivability (I write C ‘PL A) as

usual. > is an arbitrary tautology, and ? is :>.3 E.g. (td-1) most closely resembles definitions of Kanger (1957) and Alchourron and

Bulygin (1981). For authors belonging to this tradition cf. Hansen (2001), Section 1 and

Hansen (2004), fn. 1, in addition to which Ziemba (1971) must be mentioned.4 Cf. Horty (1997). The ‘‘disjunctive’’ ought is more commonly referred to as ‘‘skeptical’’

non-monotonic inference. Horty (2003) attributes the proposal to Brink (1994), yet the idea

to use such a definition for (dyadic) deontic logic already appears in Lewis (1981). For

alternatives in the deontic-logical treatment of normative conflicts cf. Goble (2005).5 Cf. Kraus et al. (1990), where, however, the proofs are done in a more general setting. Also

cf. my (Hansen 2005) for constructive proofs in the manner of Spohn (1975) as well as more

comparisons and truth definitions and axioms for an alternative ‘‘credulous’’ O-operator.6 Horty’s definition only employs circumstances to derive consequents from a set of condi-

tional imperatives, but this has no effect on the solution of the examples.7 Horty is preparing a refined version of Binding that solves all of the examples (private

correspondence).8 Brewka (1991) and for Nebel cf. Rott (1993, fn. 9). For the rationale, let the ordered I

be hi0; . . . ; i0:125; i0:25; i0:5; i1i, with i1 ¼ !p; i0:5 ¼ !:p; i0:25 ¼ !p; i0:125 ¼ !:p, etc., and i0=!q.

We cannot tell whether p or :p is obligatory, but this is not a case of conflict either, all

imperatives !p being overridden by ones demanding :p, and vice versa.9 I use notation from both, Brewka (1989, 1991), Brewka and Eiter (1999), and Nebel

(1991, 1992).10 Also cf. Sartor (2005, p. 734): ‘‘preference must be given to the argument such that its

weakest defeasible subreasons are better than the weakest defeasible subreasons in the other’’.11 Prakken could argue that he only compares minimal conflict pairs (subargu-

ments), which f!p1; !p2g; f!ðp3 ^ :p1Þ; !ðp4 ^ :p2Þg is not, while f!p1g �P f!ðp3 ^ :p1Þg and

f!p2g �P f!ðp4 ^ :p2Þg hold. But let I ¼ f!p1; !p2; !ðp3 ^ ðp4 ! :ðp1 ^ p2ÞÞÞ; !ðp4 ^ ðp3 ! :ðp1 ^ p2ÞÞÞg, with !p1 < !ðp3 ^ ðp4 ! :ðp1 ^ p2ÞÞÞ and !p2 < !ðp4 ^ ðp3 ! :ðp1 ^ p2ÞÞÞ, so

with the primary targets one ‘‘bonus’’ secondary target is reachable. Still

f!p1; !p2g 6�P f!ðp3 ^ ðp4 ! :ðp1 ^ p2ÞÞÞ; !ðp4 ^ ðp3 ! :ðp1 ^ p2ÞÞÞg, and this is a minimal

conflict pair. Yet intuitively, the argument for p1 � p2 should win over any for :ðp1 ^ p2Þ.

JORG HANSEN22


12 Sartor (1991) simply rejects contradictions, yet the adjustment to any A is immediate.13 The indicative version of the example also shows that Prakken cannot avoid counterin-

tuitive results by replacing, in his definition of a rebuttal in (1997), his own relation �P

by the relation �AM of Alchouron and Makinson for the comparison of minimal conflict

pairs: then all arguments for q are rebutted by arguments for :q as demonstrated, but

intuitively any consistent argument including i1 cannot be defeated and so the argument

for q should win.14 For resolving legal arguments by summing up weights of reasons cf. Hage (1991, 1996).15 Also cf. Iwin (1972, p. 486): ‘‘When we are in a situation compelled to satisfy two

obligations requiring contradictory actions, then the most natural way out of this diffi-

culty consists in comparing the two obligations and not satisfying the less important

one.’’16 I=:C is the notation in Alchourron and Makinson (1985), whereas the notation in

Alchourron (1986) would be I/C. Alchourron’s own truth definition for deontic operators

employs a deontic logic as basic logic and conditional imperatives that are not treated

here.17 If C is a contradiction, then by definition I + :C ¼ [.18 I owe this insight to Leon van der Torre (private correspondence).

Appendix: Proofs

THEOREM 1 (Existence of full prioritizations). For every well-

founded strict partial order < on a set C there is a full prioritization �, i.e. a

strict well-order that is order-preserving with respect to <.

Proof. Let < be a well-founded strict partial order on the set C. Let each

x 2 C be assigned an ordinal ax in the following way: ax=0 for the elements

in min<C, and for any other x 2 C; ax ¼ supfayjy<xg þ 1, where sup denotes

the supremum of a set of ordinals. Transfinite induction available for well-

founded partial ordered sets tells us that ax is well-defined for any x 2 C. Let

the equivalence class ½x� ¼ fy 2 Cjay ¼ axg. Finally � is an arbitrary strict

well-order on elements of the same equivalence class, and x � y if ax is

smaller than ay. Clearly � is a strict well-order on C. To prove x � y if x<y,

for any x; y 2 C, suppose x<y. Then ax is in {az|z<y}, so ay=sup

{az|z<y}+1 is at least ax+1. Hence x � y.

THEOREM 2 (Preferred remainders are remainders). Let hI; f; <ibe a prioritized imperative structure, where < is a well-founded strict partial

order on I. Then I + A � IfA.

Proof. Suppose S� 2 I + A. Since ½S��f0A there is some C 2 IfA such

that S� � C. Suppose C 6� S�. Let i be some element such that i 2 C, but

i 62 S�. Then by the construction of S�; ½S

j�i S½�#j��f [ fifg ‘PL A, and sinceS

j�i S½�#j� � S�, also ½S��f [ fi fg ‘PL A. But then C

f ‘PL A, which is ex-

cluded by C 2 IfA. So C ¼ S� and S� 2 IfA.



THEOREM 3 (Nebel’s prioritized removals). Let hI; f;<i be a

prioritized imperative structure, where < is the asymmetric part of a complete,

well-founded preorder £ on I. Then I + A equals

fC � I j Cf0PLA and 8D � I : ifD �N C thenDf ‘PL Ag:

Proof. Left-to-right: Suppose C 2 I + A. Suppose for some

D � I : D �N C. Then there is a priority class [i] such that for all

j < i;C \ ½j� ¼ D \ ½j� and C \ ½i� � D \ ½i�. Let � be the full prioritization

used to construct C ¼ S� and i be the �-first element of [i] with i 62 C, but

i 2 D. By construction of S�; ½S

j�i S½�#j��f [ fi fg ‘PL A, and alsoS

j�i S½�#j� ¼ fj 2 S� j j � ig. Since � is order-preserving, for all j 2 S�, if

j � i, then j< i or j 2 [i]. Then j 2 D, by definition of �N or by choice of i.

Hence Df ‘PL A.

Right-to-left: Suppose C � I; Cf0PLA and 8D � I. If D �N C,

then Df ‘PL A. Let� be a well ordering on I respecting < which puts i � j for

any i 2 C and j 2 ½i�nC, i.e. an ordering that positions the elements of C before

their £-equals, and D ¼ S�. We prove 8i 2 I : D \ ½i� ¼ C \ ½i� by induction

on <:

For the induction basis, let i be some £-least element. By definition of �the elements of C \ ½i� are positioned �-before any other elements of D,

and by assumption Cf \ ½i�0PLA, so C \ ½i� � D due to the construction of

S� ¼ D. Suppose there is some j 2 D \ ½i� such that j 62 C \ ½i�. Then

C \ ½i� � D \ ½i�, and since trivially 8j < i : C \ ½j� ¼ D \ ½j� as i was £-least andso no such j exists, D �N C holds. Then it must be that Df ‘PL A, but by

definition of S� ¼ D this is excluded. So D \ ½i� � C \ ½i� and, hence,

D \ ½i� ¼ C \ ½i�.For the induction step, let i be some arbitrary element of I. By definition of

�, the elements of C \ ½i� are positioned �-before any other elements of

D \ ½i�. The induction hypothesis guarantees that for all j 2 I with

j<i : D \ ½j� ¼ C \ ½j�, so fj 2 D j j � ig ¼ fj 2 C j j � ig. As fj 2 C j j � igf

[½C \ ½i��f0PLA (otherwise C ‘PL AÞ, for any j 2 C \ ½i� : ½S

k�j S½�#k��f

[fj fg0PLA, so j 2 D and hence C \ ½i� � D. Suppose there is some j 2 D \ ½i�with j 62 C \ ½i�. Then C \ ½i� � D \ ½i�, and since also 8j<i : C \ ½i� ¼ D \ ½i�by the induction hypothesis, D �N C holds. Then it must then be that

Df ‘PL A, but by definition of S� ¼ D this is excluded. So D \ ½i� � C \ ½i� and

hence D \ ½i� ¼ C \ ½i�.Since 8i 2 I : D \ ½i� ¼ C \ ½i� we have C ¼ D ¼ S� and so C 2 I + A.

THEOREM 4 (Preferred remainders, after Ryan and Sakama &

Inoue). Let hI; f; <i be a prioritized imperative structure, < a well-founded

strict partial order on I and I + A and pðIfAÞ be as defined. Then

I + A ¼ pðIfAÞ.

JORG HANSEN24


Proof. Left-to-right: Suppose S� 2 I + A. Assume for r.a.a. that there is a

C 6¼ S� in IfA such that for all i 2 S�nC there is a j 2 CnS� with j � i. Let i

be the �-least element in S�nC, which is not empty, because otherwise

S� � C, but this is excluded by maximality of S�, so existence of i is guar-

anteed by well-foundedness of �. ThenS

k�i S½<#k� � C: otherwise there is

some k 2 S� with k � i but k 62 C and so i would not be the �-least. Since

j 62 S� it must be that ½S½<#j��f [ fj fg ‘PL A. But j � i, so S½<#j� �

Sk�i S½<#k�,

and chaining results we get Cf ‘PL A, which contradicts C 2 IfA.

Right-to-left: Suppose C 2 pðIfAÞ. Let � be a full prioritization for which

the statement in the definition of pðIfAÞ is true. By definition, S� is in I + A

and so also in IfA. Assume for r.a.a. that S� 6¼ C. Then there is some

i 2 CnS� such that for all j 2 S�nC; i � j. SupposeS

j�i S½�#j� 6� fj 2 C j j � ig.Then there is a j 2 S� nC : j � i and so i 6� j by the antisymmetry of �, and

ihj since � preserves <, but we assumed otherwise. SoSj�i S½�#j� � fj 2 C j j � ig. Since i 2 CnS� it must be that i 62 S�, so by

definition ½S

j�i S½�#j��f [ fi fg ‘PL A. So C

f ‘PL A, which contradicts

C 2 IfA.

THEOREM 5 (Soundness, completeness of PD). PD is sound and

(only) weakly complete with respect to prioritized imperative semantics.

Proof. (Sketch) We must prove that PD is (a) sound with respect to

prioritized imperative semantics, (b) (weakly) complete with respect to pri-

oritized imperative semantics and (c) only weakly complete, i.e. not compact.

(a) Soundness. The validity of (CExt), (ExtC), (DM) and (DC) is

immediate. (DN) is unrestrictedly valid since 8C 2 I + :C : Cf ‘PL > is

trivial. Theorem 1 guarantees the existence of a generating �, so if C is

not a contradiction (then I + :C ¼ [ by definition) at least [ is in I + :C,so I + :C 6¼ [ and (DD-R) must hold. Leaving details to the reader, for

(Cond), suppose OðA=C ^DÞ, so 8C 2 I + :ðC ^DÞ : Cf [ fC ^Dg ‘PL A

and assume for r.a.a. that O(D fi A/C) is false, which yields

9D 2 I + :C : Df [ fCg0PLD ! A and also includes D

f0PL:ðC ^DÞ.

Likewise, for (CCMon) suppose OðA ^D=CÞ, which yields

8D 2 I + :C : Df [ fCg ‘PL A ^D and also includes D

f0PL:ðC ^DÞ. D is

obtained from some full prioritization � via Definition 2, i.e. D ¼ S:C� .

Appealing to the �-first element on which they might differ, prove that

S:C� ¼ S

:ðC^DÞ� .

(b) Weak completeness. We must prove that for any A 2 LDDL such

that :A 62 PD there is a prioritized imperative structure hI; f; <i that

models A. I proved PD to be weakly complete for basic imperative

semantics and (td-4) in Hansen (2005), so take the basic imperative struc-

ture hI; f i that models A and define <¼ [. Then I + :C ¼ If:C for any

C. Hence hI; f; <i also models A using (td-6).



(c) Non-compactness. Leaving details to the reader, I gave a counterex-

ample to compactness of basic imperative semantics in Hansen (2005, The-

orem 3), i.e. a non-satisfiable set C � LDDL of which all finite subsets are

satisfiable, which applies here as well. The tricky part is to show that the sets

Ip1 [ Iðp1$:p2Þ and I:p1 [ Iðp1$p2Þ are in I +?, which works using the complete

preorder induced by ordinal labels from the proof of Theorem 1 (for each

union choose a full prioritization � that puts its elements �-first in any

equivalence class and prove that S� in I +? constructed from � equals the

respective union).

THEOREM 6 (Moderated safe contractions and preferred

remainders). Let hI; f; <i be a prioritized imperative structure, where < is a

well-founded strict partial order on I. Then is some full

prioritization of <}.

Proof. I first give a construction next(i), containing the – compared to i –

‘‘next-larger’’ element in M�, and a helpful lemma:

LEMMA 1. Let � be a full prioritization of <;A 2 LPL and M� be

accordingly defined (cf. Definition 8). For any i 2 I, let

nextðiÞ ¼df min�fj 2 M� j i ¼ j or i � jg

contain i if i 2 M� or else the element in M� that comes �-next to i. Let

Mi ¼min�fM½�;a� j nextðiÞ � M½�;a�g if nextðiÞ 6¼ [,

M� otherwise

�

be the first set in the construction of M� that includes next(i) if that is non-

empty and otherwise the whole of M�. Then

Mi n nextðiÞ ¼ M� \ fj 2 I j j � ig:

Proof. For next(i), note that due to well-orderliness, the �-minimum of a

non-empty set is a singleton, and so is next(i), unless there is no j in M� with

i=j or i � j, in which case it is[. ForMi, note that sinceM½�;a� increases with

each step a, 0 £ a < card(I), the inclusion-minimum is well defined. To prove

the lemma, if neither i 2 M� nor a �-next member in M� exists and so

Mi ¼ M�, then the equivalence is trivial. Otherwise Mi ¼ M½�;a� for some a,

and we must prove that (i) if j 2 I is in M½�;a� n nextðiÞ then j � i, and (ii) that

if j 2 M� is not in M½�;a� n nextðiÞ then j =� i.

ad i): Let j 2 M½�;a�. We assumed next(i) to be non-empty, so let next(i)={i0}.

By definition, i0 is the element of M½�;a� that was added at step a and so if i0=j,

then j is not in M½�;a�A n nextðiÞ. Otherwise, by the construction of M½�;a�; j is in

some M½�;b� with b < a. Let b be the smallest of these. To be in the increase of

JORG HANSEN26


M½�;a�; i0 must be �-maximal in some A-kernel X in I n

Sc<aM½�;c�. Since the

process is incremental, we have I nS

c<a M½�;c� � I nS

c<b M½�;c�. So

X � I nS

c<bM½�;c�. Hence, if i0 � j, then j is not �-minimal among the �-

maximal members of A-kernels in I nS

c<b M½�;c�, but this contradicts that j is the

element by whichM½�;b� was increased at step b. So i0 =� j, we supposed i0 „ j, and

so by �-connectedness j � i0. Also, if i=j or i � j, then i0=j or i0 � j by the

definition of next(i) and j 2 M�. By the previous result and �-transitivity, we

obtain j � j, contradicting �-irreflexivity. So i =� j; i 6¼ j and hence by connect-

edness, j � i.

ad ii): Assume j 2 M� is not in M½�;a� n nextðiÞ. Again let next(i)={i0}. So

i0 is the element of M½�;a�, which is added at step a. If j=i¢, then i=j or i � j

and in both cases j =� i. Otherwise j must be in someM½�;b� with a<b. Let b be

the smallest of these. To be in the increase of M½�;b�; j must be �-maximal in

some A-kernel X in I nS

c<b M½�;c�. Since the process is incremental, we have

I nS

c<bM½�;c� � I nS

c<a M½�;c�. So X � I nS

c<aM½�;c�. Suppose j � i0, then

i0 is not �-minimal among the �-maximal members of A-kernels in

I nS

c<aM½�;c�. But this contradicts that i0 is the element by which M½�;a� was

increased at step a. So j =� i0, and we supposed j „ i0. Thus by �-connect-

edness, i0 � j and hence i � j by �-transitivity and definition of next(i).

For the theorem, it now suffices to prove that for each full prioritization

�;A 2 LPL and accordingly constructed sets S� (Definition 2) and M�

(Definition 8), I nM� ¼ S�; which is done by induction over �:

Induction basis: Let i0 be the �-least element in I. Suppose i0 is in

I nM� : I nM� contains no A-kernels, so fi0g0PLA and so i0 is in S�. Sup-

pose i0 is not in I nM�. Then i0 is �-maximal in some A-kernel of I. So for all

other j in this A-kernel, j � i, but also i � j because i0 is �-least. So by �-

antisymmetry this A-kernel equals {i0}, so fif0g ‘PL A and by definition

i0 62 S�. Hence i0 2 I nM� iff i0 2 S�.

Induction step: Right-to-left: Suppose i 62 I nM�, so i 2 M�. Let a be

the step that has i in its increase, i.e. M½�;a� ¼S

b<aM½�;b� [ fig.Since i 2 M�; nextðiÞ ¼ i, so M½�;a� is Mi as defined above. So

Mi n fig ¼S

b<aM½�;b� ¼ M� \ fj 2 I j j < ig by Lemma 1. Hence

½I nS

b<aM½�;b�� \ fj 2 I j j � ig ¼ ½I n ½M� \ fj 2 I j j � ig�� \ fj 2 I j j � ig ¼½I nM�� \ fj 2 I j j � ig which by the induction hypothesis equals

S� \ fj 2 I j j � ig, and bydefinition this equalsS

j�i S½�#j�. To be in the increase

of M½�;a�; i must be �-maximal in some A-kernel X in I nS

b<aM½�;b�, so

X � ½I nS

b<a M½�;b�� \ fj 2 I j j � ig [ fig. So by definition of anA-kernel andthe above equation, ½

Sj�i S½�#j��

f [ fifg ‘PL A. Hence i 62 S½�#i� and i 62 S�.

Left-to-right: Suppose i 62 S�. So by definition ½S

j�i S½�#j��f [ fifg ‘PL A

and ½S

j�i S½�#j��f0PLA. By the induction hypothesis

Sj�i S½�#j� ¼

S� \ fj 2 I j j � ig ¼ ½I nM�� \ fj 2 I j j � ig. By the above lemma, M� \fj 2 I j j � ig ¼ Mi n nextðiÞ and so ½I nM�� \ fj 2 I j j � ig ¼ ½I n ½M� \fj 2 I j j � ig�� \ fj 2 I j j � ig ¼ ½I n ½Mi n nextðiÞ�� \ fj 2 I j j � ig, which,



since nextðiÞ � I, equals ½½I nMi� [ nextðiÞ� \ fj 2 I j j � ig, which equals

½I nMi� \ fj 2 I j j � ig since nextðiÞ \ fj 2 I j j � ig ¼ [ by definition of nex-

t(i). So we obtain the results that ½½I nMi� \ fj 2 I j j � ig� f [ fifg ‘PL A and

½½I nMi� \ fj 2 I j j � ig� f0PLA. First observe nextðiÞ 6¼ [ – otherwise

Mi ¼ M� by definition, so ½I nM��f [fifg ‘PL A, but by definition

½I nM��f0PLA, so i 62 ½I nM�� and, hence, i is in M�, but then

nextðiÞ ¼ fig 6¼ [. So Mi ¼ M½�;a� for some a. Next assume i 62 M�. Then

i 2 I nMi. By the first result obtained above, i is �-maximal in some A-

kernel in I nMi. Suppose i is not �-minimal among the �-maximal mem-

bers of A-kernels in I nMi: then some A-kernel must lie completely

in ½I nMi� \ fj 2 I j j � ig, but the other result excluded this. So

min�rð½I nS

b<a M½�;b��gAÞ ¼ fig, so i is in the increase at step a, i.e. i 2 Mi,

so i 2 M�, which contradicts its assumed negation and so is true. So

i 62 I nM�.

THEOREM 7 (Property of uniquely prioritized imperatives).Any hI; f; <i, where < is the strict part of some complete preorder £, is un-

iquely prioritized iff for all consistent C 2 LPL;C 2 I + :C; i 2 I and

j1; j2 2 ½i�;

fi0 2 C j i0<igf [ fCg ‘PL jf1 ! j

f2 or fi0 2 Cj i0<igf [ fCg ‘PL j

f2 ! j

f1:

Proof. For the left-to-right direction (in contraposition), suppose there are

j1, j2 2 [i], a consistent D 2 LPL and C 2 I + :D with neither fi0 2 C j i0<igf

[fDg ‘PL jf1 ! j

f2 nor fi0 2 C j i0<igf [ fDg ‘PL j

f2 ! j

f1. Consider C ¼

D ^ :ðj f1 ^ jf2Þ. If fi0 2 C j i0<igf ‘PL :C, then fi0 2 C j i0<igf ‘PL :D_

ðj f1 ^ jf2Þ, which includes fi0 2 C j i0<igf [ fDg ‘PL j

f1 ! j

f2, contrary to what

we supposed. Let �1 be a full prioritization of < that for all i¢ <i is like C,

i.e. it contains fi0 2 C j i0<ig, and then puts j1 �1-first among the members of

[i]. Likewise let �2 be a full prioritization of < that contains fi0 2 C j i0<ig

and puts j2 �2-first among the members of [i]. fi0 2 C j i0<igf ‘PL :C is ex-

cluded, so regarding the constructions S�1;S�2

2 I + :C;S

k�1j1S½�1#k� ¼

fi0 2 C j i0<ig ¼S

k�2j2S½�2#k�. If ½

Sk�1j1

S½�1#k��f [ fj f1g ‘PL :C, then fi0 2 C j

i0<igf [ fj f1g ‘PL :D _ ðj f1 ^ jf2Þ or equivalently fi0 2 C j i0 < igf[

fDg ‘PL jf1 ! j

f2, but that was excluded. So ½

Sk�1j1

S½�1#k��f [ fj f1g0PL:C and

so j1 2 S�1, and likewise j2 2 S�2

is proved. But then S�16¼ S�2

– otherwise

½S�1� f ‘PL j

f1 ^ j

f2 and so ½S�1

� f ‘PL :C, which is excluded. So card

ðI + :CÞ 6¼ 1 for some non-contradictory C 2 LPL.

For the right-to-left direction, assume for r.a.a. that cardðI + :CÞ 6¼ 1 for

some consistent C 2 LPL, so there are two full prioritizations �1 and �2

JORG HANSEN28


such that S�16¼ S�2

for the according S�1;S�2

2 I + :C. Then there is a i 2 I

such that S�1\ ½i� 6¼ S�2

\ ½i�: let [i] be the <-first such equivalence class. So

there is a j1 2 [i] with either j1 2 S�1and j1 62 S�2

or vice versa. This being

equal, assume the former, so ½S

k�2j1S½�2#k��

f [ fj f1g ‘PL :C. So there is a

(jf1 ! :C)-kernel X in

Sk�2j1

S½�2#k�, and X \ ½i� 6¼ [, else X � S�1by choice

of [i] and since j1 2 S�1we obtain ½S�1

� f ‘PL :C, which is excluded. We get

fj 2 S�2j j < igf [ fCg ‘PL j

f2 ! j

f1 or fj 2 S�2

j j < igf [ fCg ‘PL jf1 ! j

f2

from the right side of the iff-clause for any j2 2 X \ ½i�. Suppose the first caseholds for a j2 2 X \ ½i�. Since j2 2 X �

Sk�2j1

S½�2#k� we obtain

½S

k�2j1S½�2#k��

f ‘PL C ! jf1, and since the ( j

f1 ! :C)-kernel X is in

Sk�2j1

S½�2#k�, this yields ½S

k�2j1S½�2#k��

f ‘PL :C, which is excluded. So for all

j2 2 X \ ½i�: fj 2 S�2j j<igf [ fCg ‘PL j

f1 ! j

f2. For all other k 2 X, i.e.

k 62 ½i�, we have k �2 i by X �S

k�2j1S½�2#k� and so k 2 S�1

since j1 2 [i] and

due to the choice of i. So since j1 is in S�, and also fj 2 S�2j j<ig � S� by the

choice of i, for any k 2 X : ½S��f [ fCg ‘PL kf. Hence, ½S��

f ‘PL :C, whichthe construction of the antecedent excludes. This completes the r.a.a.

THEOREM 8 (PD + RMon equals DSDL3). Let PD+(RMon) be

like PD, except that (RMon) is added as an axiom scheme. Then

PD+(RMon) = DSDL3.

Proof. Easy and left as exercise to the reader.

THEOREM 9 (Soundness, completeness of DSDL3). DSDL3 is

sound and (only) weakly complete for uniquely prioritized imperative seman-

tics.

Proof. We must prove that DSDL3 is (a) sound with respect to prioritized

imperative semantics, (b) (weakly) complete with respect to prioritized

imperative semantics, and (c) only weakly complete, i.e. not compact.

(a) Soundness. Only the validity of (RMon) needs to be proved, as The-

orem 5 proved validity of the PD-axioms with respect to all, not necessarily

uniquely, prioritized imperative structures. The proof is like the one for

(CCMon) in Theorem 5, i.e. we assume O(A/C ) and obtain that for all

D 2 I + :C : Df [ fCg ‘PL A and obtain the additionally included fact that

for all such D;D f [ fCg0PL:D by assuming P(D/C) and from card

ðI + :CÞ ¼ 1 in case C is consistent (otherwise DN and CExt make the

conclusion OðA=C ^DÞ trivially true).

(b) Weak completeness. The proof is a condensed version of its more

widely applicable variants in Spohn (1975) and my (2001, 2005), to which I

refer for further features of the construction. We must prove that for

any DSDL3-consistent. A 2 LDDL, i.e. :A 62 DSDL3, there is a uniquely



prioritized imperative structure hI; f; <i that models A. We build a disjunc-

tive normal form of A and obtain a disjunction of conjunctions, where each

conjunct is O(B/D) or :OðB=DÞ. One disjunct must then be DSDL3-con-

sistent. Let d be that disjunct. Let the d-restricted language Ld

PL be the PL-

sentences that contain only proposition letters occurring in d. Let rðLd

PLÞ be22

n

mutually non-equivalent representatives of Ld

PL, n being the number of

proposition letters in d. LPL-sentences now mean their representatives in

rðLd

PLÞ. Construct a set D � LDDL such that:

(a) Any conjunct of d is in D.

(b) For all B;D 2 rðLd

PLÞ: either P(B/D) or Oð:B=DÞ 2 D.

(c) D is DSDL3-consistent.

It then suffices to find a uniquely prioritized imperative structure that

makes true all of D. We identify what Hansson (1969) called the deontic basis

in an extension kCk (Spohn 1975 writes ~C) by letting OC ¼VfA 2

rðLd

PLÞ jOðA=CÞ 2 Dg be the ‘‘sum’’ of everything demanded in the situation

C. From this definition and (b), (c) it follows immediately that OðOC=CÞ 2 D.

Furthermore, observe

(O1) OðA=CÞ 2 D iff ‘PL OC ! A:

System of spheres: Let S ¼ hC1; . . . ;Cni be defined recursively by letting

(i) C1 ¼ >,

(ii) if Ci 2 S and Ci 6¼? then Ciþ1 ¼ Ci ^ :OCi(otherwise i=n).

Observe that for all Ci, Cj, 1 £ i < j £ n:

(O2) ‘PL Cj ! Ci

(O3) Cj ^ OCi¼?

Proof. Immediate (due to A0 and O1, ‘PL OCj! Cj so also OCj

^ OCi¼?).

(O4) Ci 2 rðLd

PLÞ; 1 � i � n

(O5) The sequence is finite, i.e. n 6¼ 1

(O6) OC1_ . . . _ OCn

¼ >

Proofs. (O4) is immediate from the definitions. Regarding (O5), rðLd

PLÞ is

finite, so if S is infinite, then Ci=Cj for some 1 £ i < j £ n. Then

Ci ^ OCi¼? due to (O3), but since ‘PL OCi

! Ci due to (A0) and (O1), then

OCi¼? and so due to (A1) Ci ¼?. But then the sequence ends with i. For

(O6), if OC1_ . . . _ OCn

6¼ > then > ^ :O> ^ . . . ^ :OCn1^ :OCn

6¼?. The

left side equals Cn ^ OCn, so Cn is not last in the sequence, contrary to what

was assumed.

Smallest A-permitting sphere: Let CA be the first Ci with

0PLðA ^ OCiÞ !?, i.e. for all j < i : ðA ^ OCj

Þ ¼?. Then furthermore:

JORG HANSEN30


(O7) For any rðLd

PLÞ-sentence A 6¼? there is some CA 2 S:

(O8) ‘PL A ! CA

(O9) IfA 6¼?; then PðA=CAÞ 2 D:

(O10) ‘PL OA $ ðOCA^ AÞ

Proofs. (O7) is immediate from (O6). Regarding (O8) let CA=Ci, so for all

j < i : OCj^ A ¼?, so ‘PL A ! ð> ^ :O> ^ . . . ^ :OCi1

Þ and thus ‘PL A

! Ci. For (O9), if CA exists, then by the construction of D either

PðA=CAÞ 2 D or Oð:A=CAÞ 2 D, but in the second case OCA^ A ¼? due to

(O1), which is excluded by the definition of CA. Regarding (O10): due to (O8)

and derivability of (Cond) we obtain OðA ! OA=CAÞ 2 D from

OðOA=AÞ 2 D, so ‘PL ðOCA^ AÞ ! OA, which is the right-to-left direction. If

A ¼?, then OA ¼? since ‘PL OA ! A due to (A0) and (O1) and then the left-

to-right direction is trivial. Otherwise PðA=CAÞ 2 D due to (O9) and

OðOCA=AÞ 2 D is obtained from OðOCA

=CAÞ, (O8) and derivability of

(RMon). Thus ‘PL OA ! OCAand due to (A0) and (O1) also ‘PL OA ! A.

Both include the left-to-right version.

Canonical construction of hI; f; <i: Let I ¼ fC ! OC jC 2 Sg; f be iden-

tity and < be the strict part of £=I� I (or [, al gusto). Verification: due to

the constructions of S and I it is immediate that for all i; j 2 I :‘PL i f ! j f or

‘PL j f ! i f, so the demands of the imperatives are chained and I is uniquely

prioritized (Theorem 7). All imperatives are equally ranked, so I + A ¼ IfA.

Coincidence: We must prove that for all B;D 2 rðLd

PLÞ , hI; f; <i models

O(B/D) iff OðB=DÞ 2 D. Right-to-left: Assume OðB=DÞ 2 D. If D =^, then

by definition I + :D ¼ If:D ¼ [, so (td-6) makes O(B/D) trivially true.

Otherwise, by (O7) there is some CD 2 S. By its definition, D ^ OCDis con-

sistent, so D ^ ðCD ! OCDÞ is also consistent and so CD ! OCD

must be in

(the) C 2 If:D. Since ‘PL D ! CD by (O8) and ‘PL ðOCD^DÞ ! OD by

(O10) we have ‘PL ððCD ! OCDÞ ^DÞ ! OD , so C [ fDg ‘PL OD and due to

(O1), C [ fDg ‘PL B and so (td-6) makes O(B/D) true. Left-to-right: Assume

that for C 2 I + :D ¼ If:D we have C [ fDg ‘PL B. If D=^, then

OðB=DÞ 2 D follows trivially from (A0), (A2) and (A3). Otherwise, by (O7)

there is a CD 2 S: D ^ OCDis by definition consistent, so D ^ ðCD ! OCD

Þ isconsistent and CD ! OCD

must be in (the) set C. CD=Ci for some i in the

construction of S. For all j < i;‘PL D ! Cj due to (O8) and (O2), and

‘PL D ! :OC jby definition of CD, so Cj ! OC j

62 C. For all

j > i;‘PL OCD! :Cj by (O3), so with (O8) we obtain fCD ! OCD

g[fDg ‘PL Cj ! OC j

for any Cj ! OC j2 C. So if C [ fDg ‘PL B, then

fCD ! OCDg [ fDg ‘PL B. Hence ‘PL ðOCD

^DÞ ! B by (O8), ‘PL OD ! B

by (O10), so OðB=DÞ 2 D by (O1).



(c) Non-Compactness. Compactness is disproved by any infinite set D of

LDDL-sentences, of which any finite subset Df is satisfiable, but not D.

Consider

D ¼ fOðp1=>Þg [ fPð:p1=:ðp1 ^ . . . ^ pnÞÞ j n > 1g

[ fPðA ^ :pn=:ðp1 ^ . . . ^ pnÞÞ j n > 1 and fAg0PL png

Let Df � D be a finite subset of D. Let n be the highest index such that pnoccurs in some formula in Df. Then hI; f; <i with I ¼ f!ðp1 ^ . . . ^ pnÞg,< being [, satisfies Df : I +?¼ fIg ¼ ff!ðp1 ^ . . . ^ pngg. So for all sets C in

I +? : Cf ‘PL p1, so Oðp1=>Þ is true. The only C 2 I + ðp1 ^ . . . ^ pnÞ, n > 1,

is the empty set and that plus :ðp1 ^ . . . ^ pnÞ is consistent with :p1and A ^ :pn if fAg0PL pn. So Pð:p1=:ðp1 ^ . . . ^ pnÞÞ and PðA ^ :pn=:ðp1 ^ . . . ^ pnÞÞ are true for any n > 1. Hence D is finitely satisfiable.

Suppose D is satisfiable, i.e. there is a uniquely prioritized impera-

tive structure hI; f; <i that makes all D true. The sets I +? and

I + ðp1 ^ . . . ^ pnÞ must be singletons (Definition 9). To make Oðp1=>Þtrue, there must be some full prioritization � such that ½S?

��f ‘PL p1,

where S?� 2 I +? is as described in Definition 2. Due to PL-compactness

there must be some i 2 I such that ½S?½�#i��

f ‘PL p1. Let i be the �-first such

element. Its existence is guaranteed by well-foundedness of � since 0PL p1. So

½S

j�i S?½�#j��

f0PLp1 and i 2 S?

�. Let n be the smallest index such that fi fg0PL pn.

i f must be consistent since i 2 S?�, so the finite length of if alone guarantees the

existence of such an n. We have ½S

j�i S?½�#j��

f0PL p1 ^ . . . ^ pn and so

Sj�i S

?½�#j� � S

ðp1^...^pnÞ� , where S

ðp1^...^pnÞ� 2 I + ððp1 ^ . . . ^ pnÞÞ is as described

in Definition 2. S?½�#i� 6� S

ðp1^...^pnÞ� , for otherwise Pð:p1=:ðp1 ^ . . . ^ pnÞÞ

could not be true, so it must be that ½S?½�#i��

f ‘PL p1 ^ . . . ^ pn. But then

½Sðp1^...^pnÞ� � f ‘PL i f ! ðp1 ^ . . . ^ pnÞ, which includes ½S

ðp1^...^pnÞ� � f ‘PL i f ! pn.

Hence Pði f ^ :pn=:ðp1 ^ . . . ^ pnÞÞ cannot be true. But PðA ^ :pn=:ðp1 ^ . . . ^ pnÞÞ must be true for all A such that fAg0PL pn and if meets the

condition. So D is not satisfiable.

References

Alchourron, C. E. (1986). Conditionality and the Representation of Legal Norms, in Martino,

A. A. and Socci Natali, F. (eds.), Automated Analysis of Legal Texts: Edited Versions of

selected papers from the Second International Conference on ‘‘Logic, Informatics, Law’,

Florence, Italy, September 1985. North Holland, Amsterdam, 173–186.

Alchourron, C. E. and Bulygin, E. The Expressive Conception of Norms, in Hilpinen, R.

(1981). New Studies in Deontic Logic. Reidel: Dordrecht, 95–124.

Alchourron, C. E. and Makinson, D. Hierarchies of Regulations and Their Logic, in Hilpinen,

R. (1981). New Studies in Deontic Logic. Reidel: Dordrecht 95–124, 125–148.

JORG HANSEN32


Alchourron, C. E. and Makinson, D. (1985). On the Logic of Theory Change: Safe Con-

traction. Studia Logica 44: 405–422.

Aqvist, L. (1986). Some Results on Dyadic Deontic Logic and the Logic of Preference.

Synthese 66: 95–110.

Brewka, G. (1989). Preferred Subtheories: An Extended Logical Framework for Default

Reasoninig, in Sridharan, N. S. (ed.), Proceedings of the Eleventh International Joint

Conference on Artificial Intelligence IJCA1–89, Detroit, Michigan, USA, August 20–25,

1989, San Mateo, Calif.: Kaufmann, 1043–1048.

Brewka, G. (1991). Belief Revision in a Framework for Default Reasoning, in Fuhrmann, A.

and Morreau, M. (eds.), The Logic of Theory Change, Workshop, Konstanz, Germany,

October 13–15, 1989. Springer, Berlin, 206–222.

Brewka, G. and Eiter, T. (1999). Preferred Answer Sets for Extended Logic Programs. Arti-

ficial Intelligence 109: 297–356.

Brink, D. O. (1994). Moral Conflict and Its Structure. Philosophical Review 103: 215–247.

Fehige, C. (1994). The Limit Assumption in Deontic (and Prohairetic) Logic, in Meggle, G.

and Wessels, U. (eds.), Analyomen 1: Proceedings of the 1st Conference ‘‘Perspectives in

Analytical Philosophy’’, Saarbrucken 1991. de Gruyter, Berlin, 42–56.

Gardenfors, P. (1984). Epistemic Importance and Minimal Changes of Belief. Australasian

Journal of Philosophy 62: 136–157.

Goble, L. (2005). A Logic for Deontic Dilemmas, Journal of Applied Logic: 961–983.

Hage, J. C. (1991). Monological Reason Based Reasoning, in Breuker, J. A., De Mulder, R. V.,

and Hage, J. C., Legal Knowledge Based Systems: Model-based Legal Reasoning (Pro-

ceedings of the Fourth International Conference on Legal Knowledge Based Systems JURIX

1991, Lelystad, December 1991), 77–91.

Hage, J. C. (1996). A Theory of Legal Reasoning and a Logic to Match. Artificial Intelligence

and Law 4: 199–273.

Hansen, J. (2001). Sets, Sentences, and Some Logics about Imperatives. Fundamenta Infor-

maticae 48: 205–226.

Hansen, J. (2004) Problems and Results for Logics about Imperatives, Journal of Applied

logic, 36–61.

Hansen, J. (2005) Conflicting Imperatives and Dyadic Deontic Logic, Journal of Applied

Logic, 484–511.

Hansson, B. (1969). An Analysis of Some Deontic Logics. Nous 3: 373–398, reprinted in

Hilpinen, R. (1971). Deontic Logic: Introductory and Systematic Readings. Reidel:

Dordrecht, 121–147.

Hare, R. M. (1981). Moral Thinking. Clarendon Press: Oxford.

Horty, J. F. (1997). Nonmonotonic Foundations for Deontic Logic, in Nute, D. (eds.),

Defeasible Deontic Logic. Kluwer, Dordrecht, 17–44.

Horty, J. F. (2003). Reasoning with Moral Conflicts. Nous 37: 557–605.

Iwin, A. A. (1972). Grundprobleme der deontischen Logik: in Wessel, H. (eds.), Quantoren–

Modalitaten–Paradoxien. VEB Deutscher Verlag der Wissenschaften, Berlin, 402–522.

Kanger, S. (1957). New Foundations for Ethical Theory: Part 1’’, duplic., 42 p., reprinted in

Hilpinen, R. (1971). Deontic Logic: Introductory and Systematic Readings. Reidel:

Dordrecht, 36–58.

Kraus, S., Lehmann, D. and Magidor, M. (1990). Nonmonotonic Reasoning, Preferential

Models and Cumulative Logics. Artificial Intelligence 44: 167–207.

Lewis, D. (1981). Ordering Semantics and Premise Semantics for Counterfactuals. Journal of

Philosophical Logic 10: 217–234.

Makinson, D. and van der Torre, L. (2000). Input/Output Logics. Journal of Philosophical

Logic 29: 383–408.



Makinson, D. and van der Torre, L. (2001). Constraints for Input/Output Logics. Journal of

Philosophical Logic 30: 155–185.

Marcus, R. B. (1980). Moral Dilemmas and Consistency. Journal of Philosophy 77: 121–136.

McNamara, P. (1995). The Confinement Problem: How to Terminate Your Mom With Her

Trust. Analysis 55: 310–313.

Nebel, B. (1991). Belief Revision and Default Reasoning: Syntax-Based Approaches, in Allen,

J. A., Fikes, R. and Sandewall, E. (eds.), Principles of Knowledge Representation and

Reasoning: Proceedings of the Second International Conference, KR ’91, Cambridge, MA,

April 1991. Morgan Kaufmann, San Mateo, 417–428.

Nebel, B. (1992). Syntax-Based Approaches to Belief Revision, in Gardenfors, P. (eds.), Belief

Revision. University Press, Cambridge, 52–88.

Prakken, H. (1997). Logical Tools for Modelling Legal Argument. Kluwer: Dordrecht.

Prakken, H. and Sartor, G. (1997). Argument-based Logic Programming with Defeasible

Priorities. Journal of Applied Non-classical Logics 7: 25–75.

Rescher, N. (1964). Hypothetical Reasoning. North-Holland: Amsterdam.

Rintanen, J. (1994). Prioritized Autoepistemic Logic, in MacNish, C., Pearce, D. and Pereira,

L. M. (eds.), Logics in Artificial Intelligence, European Workshop, JELIA ’94, York, Sep-

tember 1994, Proceedings. Springer, Berlin, 232–246.

Ross, W. D. (1930). The Right and the Good. Clarendon Press: Oxford.

Rott, H. (1993). Belief Contraction in the Context of the General Theory of Rational Choice.

Journal of Symbolic Logic 58: 1426–1450.

Ryan, M. (1992). Representing Defaults as Sentences with Reduced Priority, in Nebel, B.,

Rich, C. and Swartout, W. (eds.), Principles of Knowledge Representation and Reasoning:

Proceedings of the Third International Conference, KR ’92, Cambridge, MA, October 1992.

Morgan Kaufmann, San Mateo, 649–660.

Sakama, C. and Inoue, K. (1996). Representing Priorities in Logic Programs, in Maher, M.

(eds.), Joint International Conference and Syposium on Logic Programming JICSLP 1996,

Bonn, September 1996. MIT Press, Cambridge, 82–96.

Sartor, G. (1991). Inconsistency and Legal Reasoning, in Breuker, J. A., De Mulder, R. V.,

and Hage, J. C., Legal Knowledge Based Systems: Model-based Legal Reasoning

(Proceedings of the Fourth International Conference on Legal Knowledge Based Systems

JURIX 1991, Lelystad, December 1991), 92–112.

Sartor, G. (2005) Legal Reasoning. A Cognitive Approach to the Law, vol. 5 of A Treatise of

Legal Philosophy and General Jurisprudence, Springer: Dordrecht.

Searle, J. (1980). Prima-facie Obligations, in Straaten, Z. v. (ed.), Philosophical Subjects:

Essays presented to P. F. Strawson. Clarendon Press, Oxford, 238–259.

Spohn, W. (1975). An Analysis of Hansson’s Dyadic Deontic Logic. Journal of Philosophical

Logic 4: 237–252.

von Wright, G. H. (1968). An Essay in Deontic Logic and the General Theory of Action. North

Holland: Amsterdam.

Ziemba, Z. (1971). Deontic Syllogistics. Studia Logica 28: 139–159.

JORG HANSEN34


Auton Agent Multi-Agent Syst

DOI 10.1007/s10458-007-9016-7

Prioritized conditional imperatives: problems

and a new proposal

Jörg Hansen

Springer Science+Business Media, LLC 2007

Abstract The sentences of deontic logic may be understood as describing what an agent

ought to do when faced with a given set of norms. If these norms come into conflict, the best

the agent can be expected to do is to follow a maximal subset of the norms. Intuitively, a

priority ordering of the norms can be helpful in determining the relevant sets and resolve con-

flicts, but a formal resolution mechanism has been difficult to provide. In particular, reasoning

about prioritized conditional imperatives is overshadowed by problems such as the ‘order

puzzle’ that are not satisfactorily resolved by existing approaches. The paper provides a new

proposal as to how these problems may be overcome.

Keywords Deontic logic · Default logic · Priorities · Logic of imperatives

1 Drinking and driving

Imagine you have been invited to a party. Before the event, you become the recipient of

various imperative sentences:

(1) Your mother says: if you drink anything, then don’t drive.

(2) Your best friend says: if you go to the party, then you do the driving.

(3) Some acquaintance says: if you go to the party, then have a drink with me.

Suppose that as a rule you do what your mother tells you—after all, she is the most important

person in your life. Also, the last time you went to a party your best friend did the driving, so

it really is your turn now. You can enjoy yourself without a drink, though it would be nice to

have a drink with your acquaintance—your best friend would not mind if you had one drink,

and your acquaintance does not care that you may be driving—but your mother would not

approve of such a behavior. Making up your mind,

J. Hansen (B)

Institut für Philosophie, University of Leipzig, Beethovenstraße 15, 04107 Leipzig, Germany

e-mail: [email protected]

123



(4) You go to the party.

I think that intuitively it is quite clear what you must do: obey your mother and your best

friend, and hence do the driving and not accept your acquaintance’s invitation. However, it

is not so clear what formal algorithm could explain this reasoning.

An example of a similar form was first employed in epistemic logic,1 and has been termed

the ‘order puzzle’ (cf. Horty [21]). For the epistemic version, consider the following sen-

tences:

(5) You remember from physics: if you are in a car, lightning won’t strike you.

(6) The coroner tells you: he was struck by lightning.

(7) Your neighbor says: he must have been drinking and driving.

Suppose that driving includes being in a car, that you firmly believe in what you remem-

ber from physics, that you believe that information by medical officers is normally based

on competent investigation, and that you usually don’t question your neighbor’s observa-

tions, but think that sometimes she is just speculating. It seems quite clear what happens:

you keep believing what you remember from school, and don’t doubt what the coroner told

you, but question your neighbor’s information, maybe answering: “This can’t be true, as the

authorities found he was struck by lightning, and you can’t be struck by lightning in a car.”

In both cases, the problem as to how the underlying reasoning can be formally recon-

structed seems so far unsolved. Both involve a priority ordering of the sentences involved.

While the paper discusses the imperative side of things from the angle of philosophical (deon-

tic) logic, its solution seems also relevant for the similarly structured problems of conditional

beliefs and desires and the modeling of agent reasoning in the face of such conditionals.

Sections 2 and 3 present the formal framework for the discussion of conditional imperatives

and resulting obligations. Section 4 examines various proposals as to how a priority ordering

may be used to resolve conflicts, it turns out that all of these do not solve the ‘order puzzle.’

A postulate at the beginning of Sect. 5 summarizes our intuitions in this matter, but also

delegates the solution to the problem to a proper definition of what it means that conditional

imperatives conflict in a given situation. Three such definitions are studied, of which the

third seems to solve the problems. Section 6 gives theorems of a corresponding deontic logic

and Sect. 7 points at remaining problems for the representation of conditional imperatives.

Section 8 concludes.

2 Formal preliminaries

To formally discuss problems such as the one presented above, I shall use a simple frame-

work: let I be a set of objects, they are meant to be (conditional) imperatives. Two functions

g and f associate with each imperative an antecedent and a consequent—these are sentences

from the language of a basic logic that here will be the language LP L of propositional

logic.2 g(i) may be thought of as describing the ‘grounds’, or circumstances in which the

consequent of i is to hold, and f (i) as associating the sentence that describes what must be

1 Cf. Rintanen [34] p. 234, who in turn credits Gerhard Brewka with its invention.

2 PL is based on a language LP L , defined from a set of proposition letters Prop = {p1, p2, ...}, Boolean

connectives ¬, ∧, ∨,→, ↔ and brackets (, ) as usual. The truth of a LP L -sentence (I use upper case letters

A, B, C, ...) is defined recursively using valuations v : Prop → {1, 0} (I write v |� A), starting with v |� p

iff v(p) = 1 and continuing as usual. If A ∈ LP L is true for all valuations it is called a tautology. PL is the

set of all tautologies, and used to define provability, consistency and derivability (I write Ŵ ⊢P L A) as usual.

⊤ is an arbitrary tautology, and ⊥ is ¬⊤.

123



the case if the imperative i is satisfied, its ‘deontic focus’ or ‘demand.’3 In accordance with

tradition (cf. Hofstadter and McKinsey [19]), I write A ⇒!B for an i ∈ I with g(i) = A and

f (i) = B, and !A means an unconditional imperative ⊤ ⇒!A. Note that A ⇒!B is just the

name for a conditional imperative that demands B to be made true in a situation where A is

true—it is not an object that is assigned truth values. A useful construction is the ‘materiali-

zation’ m(i) of an imperative i , which is the material implication g(i) → f (i) that may be

thought of as corresponding to a conditional imperative. For any i ∈ I and � ⊆ I , instead of

f (i), g(i), m(i), f (�), g(�) and m(�), I may use the superscripted i f, i g, im,� f,�g and

�m for better readability.

Let I be a tuple 〈I, f, g〉, let W ⊆ LP L be a set of sentences, representing ‘real world

facts,’ and � ⊆ I be a subset of the imperatives: then we define

TriggeredI(W,�) = {i ∈ �|W ⊢P L g(i)}.

So an imperative i ∈ � is triggered if its antecedent is true given W . Tradition wants it that

a conditional imperative can only be fulfilled or violated if its condition is the case.4 So I

define:

SatisfiedI(W,�) = {i ∈ �|W ⊢P L i g ∧ i f },

ViolatedI(W,�) = {i ∈ �|W ⊢P L i g ∧ ¬i f },

An imperative in SatisfiedI(W,�) [ViolatedI(W,�)] is called satisfied [violated] given the

facts W . It is of course possible that an imperative is neither satisfied nor violated given the

facts W . If an imperative is triggered, but not violated, we call the imperative satisfiable:

SatisfiableI(W,�) = {i ∈ TriggeredI(W,�)|W �P L ¬i f }.

Moreover, the following definition will play a major rôle in what follows:

ObeyableI(W,�) = {Ŵ ⊆ �|Ŵm ∪ W �P L ⊥}.

So a subset Ŵ of � is obeyable given W iff it is not the case that for some {i1, . . . , in} ⊆ Ŵ

we have W ⊢P L (ig1 ∧¬i

f1 )∨ . . .∨ (i

gn ∧¬i

fn ): otherwise we know that whatever we do, i.e.

given any maxiconsistent subset V of LP L that extends W ⊆ V , at least one imperative in Ŵ

is violated.5 We speak of a conflict of imperatives when the triggered imperatives cannot all

be satisfied given the facts W , i.e. when TriggeredI(W,�) f ∪ W ⊢P L ⊥. More generally

speaking I will also call imperatives conflicting if they are not obeyable in the given situation.

As prioritized conditional imperatives are our concern here, we let all imperatives in I be

ordered by some priority relation <⊆ I × I . The relation < is assumed to be a strict partial

order on I , i.e. < is irreflexive and transitive, and additionally we assume < to be well-

founded, i.e. infinite descending chains are excluded. For any i1, i2 ∈ I, i1 < i2 means that

i1 takes priority over i2 (ranks higher than i2, is more important than i2, etc.). A tuple 〈I, f, g〉

will be called a conditional imperative structure, and 〈I, f, g,<〉 a prioritized conditional

imperative structure. If all imperatives in I are unconditional, we may drop any reference to

the relation g in the tuples and call these basic imperative structures and prioritized imper-

ative structures respectively.

3 In analogy to Reiter’s default logic one might add a third function e that describes exceptional circumstances

in which the imperative is not to be applied. I will not address this additional complexity here.

4 Cf. Rescher [33], Sosa [38], van Fraassen [9]. Also cf. Greenspan [11]: “Oughts do not arise, it seems, until

it is too late to keep their conditions from being fulfilled.”

5 Terms differ here, e.g. Downing [8] uses the term ‘compliable’ instead of ‘obeyable.’

123



3 Deontic concepts

Given a set of imperatives, one may truly or falsely state that their addressee must, or must

not, perform some act or achieve some state of affairs according to what the addressee was

ordered to do. Regarding the ‘drinking and driving’ example, I think it is true that the agent

ought to do the driving, as this is what the second-ranking imperative, uttered by the agent’s

best friend, requires her to do, but that it would be false to say that the agent ought to drink

and drive. Statements that something ought to be done or achieved are called ‘normative’ or

‘deontic statements,’ and the ultimate goal of deontic logic is to find a logical semantics that

models the situation and defines the deontic concepts in such a way that the formal results

coincide with our natural inclinations in the matter.

3.1 Deontic operators for unconditional imperatives

For unconditional imperatives, such definitions are straightforward. Given a basic imperative

structure I = 〈I, f 〉, a monadic deontic O-operator, that formalizes ‘it ought to be that A (is

realized)’ by O A, is defined by

(td-m1) I |� O A if and only if (iff) I f ⊢P L A.

So obligation is defined in terms of what the satisfaction of all imperatives logically implies.6

With the usual truth definitions for Boolean operators, it can easily be seen that such a def-

inition produces a normal modal operator, i.e. one that is defined by the following axiom

schemes plus modus ponens:

(Ext) If ⊢P L A ↔ B, then O A ↔ O B is a theorem.

(M) O(A ∧ B) → (O A ∧ O B)

(C) (O A ∧ O B) → O(A ∧ B)

(N) O⊤

Furthermore, (td-m1) defines standard deontic logic SDL, which adds

(D) O A → ¬O¬A

iff the imperatives are assumed to be non-conflicting and so I f is PL-consistent, i.e. I f �P L ⊥.

It is immediate that in the case of conflicts, (td-m1) pronounces everything as obligatory, and

in particular defines O⊥ true, thus making the impossible obligatory. If conflicts are not

excluded, a solution is to only consider (maximal) subsets of the imperatives whose demands

are consistent and define the O-operator with respect to these (I write I �C for the set of all

‘C-remainders,’ i.e. maximal subsets Ŵ of I such that Ŵ f �P L C):

(td-m2) I |� O A iff ∀Ŵ ∈ I �⊥ : Ŵ f ⊢P L A

Quite similarly, a dyadic deontic operator O(A/C), meaning that A ought to be true given

that C is true, can be defined with respect to the maximal subsets of imperatives that do not

conflict in these circumstances:

(td-d1) I |� O(A/C) iff ∀Ŵ ∈ I �¬C : Ŵ f ⊢P L A

6 Such a definition of obligation was proposed e.g. by Alchourrón and Bulygin [1].

123



So A is obligatory given that C is true if A is what the imperatives in any ¬C-remainder

demand.7 In the case of conflicts, this definition produces a “disjunctive solution:” e.g. if

there are two imperatives !A and !B with ⊢P L C → (A → ¬B), then neither O(A/C) nor

O(B/C) but O(A ∨ B/C) is true.8

Often, we want to use the information that we have about the circumstances also for

reasoning about the obligations in these circumstances. E.g. if the set of imperatives is {!(p1∨

p2)}, ordering me to either send you a card or phone you, and I cannot send you a card, i.e.

¬p1 is true, I should be able to conclude that I should phone you, and so O(p2/¬p1) should

be true. Such ‘circumstantial reasoning’ is achieved by the following change to the truth

definition:

(td-d1+) I |� O(A/C) iff ∀Ŵ ∈ I �¬C : Ŵ f ∪ {C} ⊢P L A

So A is obligatory given C is (invariably) true iff all maximal subsets of the imperatives’

demands (the imperatives’ associated descriptive sentences) that are consistent with the cir-

cumstances C , plus C , derive A. With the usual truth conditions for Boolean operators, a

semantics that employs (td-d1+) has a sound and (weakly) complete axiom system PD that

extends the system P of Kraus et al. [22], defined by these axiom schemes

(DExt) If ⊢P L A ↔ B then O(A/C) ↔ O(B/C) is a theorem.

(DM) O(A ∧ B/C) → (O(A/C) ∧ O(B/C))

(DC) O(A/C) ∧ O(B/C) → O(A ∧ B/C)

(DN) O(⊤/C)

(ExtC) If ⊢P L C ↔ D then O(A/C) ↔ O(A/D) is a theorem.

(CCMon) O(A ∧ D/C) → O(A/C ∧ D)

(CExt) If ⊢P L C → (A ↔ B) then O(A/C) ↔ O(B/C) is a theorem.

(Or) O(A/C) ∧ O(A/D) → O(A/C ∨ D)

with the additional (restricted, dyadic) ‘deontic’ axiom scheme

(DD-R) If �P L ¬C then ⊢PD O(A/C) → ¬O(¬A/C)

(sometimes called “preservation of classical consistency”), hence the name PD.9

3.2 Deontic operators for conditional imperatives

Unlike their unconditional counterparts, conditional imperatives have been found hard to

reason about. von Wright [41] called conditional norms the “touchstone of normative logic,”

and van Fraassen [9] wrote with regard to logics for conditional imperatives: “There may

7 Though statements like O(¬C/C) are syntactically well-formed, they are thus defined false for any pos-

sible situation C—this is the same for any dyadic deontic logic since Hansson [15] and Lewis [23] (for the

motivation cf. [26] pp. 158–159). Similarly, imperatives like C ⇒!¬C are treated as violated as soon as they

are triggered by the facts. There exist meaningful natural-language imperatives like ‘close the window if it

is open,’ but I think that in these the proposition in the antecedent is different from the negation of the one

corresponding to the consequent, in that the second refers to a different point of time (‘see to it that the window

is closed some time in the near future if it is open now’), so they should not be represented by C ⇒!¬C .

8 For alternative solutions to the problem of conflicts cf. Goble [10] and Hansen [12], [13].

9 For proofs, and an additional “credulous ought” that defines O(A/C) true if the truth of A is required to

satisfy all imperatives in some ¬C-remainder, cf. Hansen [13].

123



be systematic relations governing this moral dynamics, but I can only profess ignorance of

them.”

Representing a conditional imperative as an unconditional imperative that demands a mate-

rial conditional to be made true yields undesired results. Most notorious is the problem of

contraposition: consider a set I with the only imperative !(p1 → p2), meaning e.g. ‘if it rains,

take an umbrella.’ (td-d1) makes true O(p2/p1), but also O(¬p1/¬p2), so if you cannot

take your umbrella (your wife took it) you must see to it that it does not rain, which is hardly

what the speaker meant you to do. One may think that such problems arise from the fact that

antecedents of conditional imperatives often describe states of the affairs that the agent is not

supposed to, and often cannot, control. But consider the set {!(p1 → p2), !(¬p1 → p3)},

it yields O(p2/¬p3) with (td-d1). Here, p2 is what the consequent of some imperative

demands, so it supposedly describes something the agent can control. Now let the impera-

tives be interpreted as ordering me to wear a rain coat if it rains, and my best suit if it does

not: it is clear nonsense that I am obliged to wear a raincoat given that I can’t wear my best

suit (e.g. it is in the laundry). Such problems are the reason why we use special models for

conditional imperatives that separate antecedents and consequents (conditional imperative

structures), and write p1⇒!p2 instead of !(p1 → p2). But this only delegates the problem

from the level of representation to that of semantics, where now new truth definitions must

be found.

Let I = 〈I, f, g〉 be a conditional imperative structure, and let us ignore for the moment

the further complication of possible conflicts between imperatives. Then the following seems

a natural way to define what ought to be the case in circumstances where C is assumed to be

true:

(td-cd1) I |� O(A/C) iff [TriggeredI({C}, I )] f ⊢P L A

So dyadic obligation is defined in terms what is necessary to satisfy all imperatives that are

triggered in the assumed circumstances. E.g. if I = {p1 ⇒!p2}, with its only imperative

interpreted as “if you have a cold, stay in bed,” then O(p2/p1) truly states that I must stay

in bed given that I have a cold.

Like in the unconditional case, it seems important to be able to use ‘circumstantial reason-

ing,’ i.e. employ the information about the situation not only to determine if an imperative

is triggered, but also for reasoning with its consequent. E.g. if the set of imperatives is

{p1⇒!(p2 ∨ p3)}, with its imperative interpreted as expressing “if you have a cold, either

stay in bed or wear a scarf,” one would like to obtain O(p3/p1 ∧¬p2), expressing that given

that I have a cold and don’t stay in bed, I must wear a scarf. So (td-cd1) may be changed

into

(td-cd1+) I |� O(A/C) iff [TriggeredI({C}, I )] f ∪ {C} ⊢P L A.

Though the step from (td-cd1) to (td-cd1+) seems quite reasonable, such definitions have

also been criticized for defining the assumed circumstances as obligatory. In the above exam-

ple, (td-cd1+) also makes true O(p1/p1 ∧ ¬p2), so given that you have a cold it is true that

you ought to have it. The criticism loses much of its edge in the present setting, where one

can point to the distinction between imperatives (there is no imperative that demands p1)

and ought sentences that describe what must be true given the facts and the satisfaction of all

triggered imperatives: then the truth of O(p1/p1) seems no more paradoxical than the truth

of O⊤ that is accepted in most systems of deontic logic.

123



3.3 Further modifications

In Makinson and van der Torre’s [25] more general theory of ‘input/output logic,’ (td-cd1)

is termed ‘simple-minded output,’ and (td-cd1+) is its ‘throughput version.’10 As the names

suggests, the authors also discuss more refined operations, which again might be consid-

ered for reasoning about conditional imperatives. One modification addresses the possibility

of ‘reasoning by cases’ that e.g. makes true O(p2 ∨ p4/p1 ∨ p3) for a set of imperatives

I = {p1⇒!p2, p3⇒!p4}. This may be achieved by the following definition, where LP L⊥¬C

is the set of all maximal subsets of the language LP L that are consistent with C :11

(td-cd2) I |� O(A/C) iff ∀V ∈ LP L⊥¬C : [TriggeredI(V, I )] f ⊢P L A

In the example, each set V ⊂ LP L that is maximally consistent with p1 ∨ p3 either contains

p1, then p1⇒!p2 is triggered and so p2 and also p2 ∨ p4 is implied by [TriggeredI(V, I )] f ,

or it contains ¬p1, but then it cannot also contain ¬p3 and so must contain p3, so p3⇒!p4

is triggered and therefore p4 and also p2 ∨ p4 implied, so for all sets V, p2 ∨ p4 is implied

and so O(p2 ∨ p4/p1 ∨ p3) made true.

In order to add ‘circumstantial reasoning’ to (td-cd3)—or, in Makinson and van der Torre’s

terms, for its ‘throughput version’—one might, in the vein of (td-d1+) and (td-cd1+), try

this definition:

(td-cd2−) I |� O(A/C) iff ∀V ∈ LP L⊥¬C : [TriggeredI(V, I )] f ∪ {C} ⊢P L A

But the definition seems too weak. Consider the set {p1⇒!(¬p2 ∨ p4), p3⇒!p4} and the

situation (p1 ∧ p2) ∨ p3. We would expect a reasoning as follows: in this situation, either

p1 ∧ p2 is true, so the first imperative is triggered but we cannot satisfy it by bringing about

¬p2, and so must bring about p4. Or p3 is true, then the second imperative is triggered and

we must again bring about p4. So we must bring about p4 in the given situation. But the

definition fails to make true O(p4/(p1 ∧ p2) ∨ p3). Like Makinson and van der Torre [25],

I therefore combine reasoning by cases with a stronger version of throughput:

(td-cd2+) I |� O(A/C) iff ∀V ∈ LP L⊥¬C : [TriggeredI(V, I )] f ∪ V ⊢P L A

As is easy to see, this resolves the difficulty: for {p1⇒!(¬p2 ∨ p4), p3⇒!p4}, O(p4/(p1 ∧

p2) ∨ p3) is now true, as desired. But this modification has an unwanted consequence: it

makes reasoning about conditional imperatives collapse into reasoning about consequences

of their materializations (cf. [26] p. 156):

Observation 1 By (td − cd2+), I |� O(A/C) iff m(I ) ∪ {C} ⊢P L A.

Proof For the right-to-left direction, for any imperative i ∈ I and any set V ∈ LP L⊥¬C ,

either V includes g(i), so i ∈ TriggeredI(V, I ) and therefore [TriggeredI(V, I )] f ⊢P L

g(i) → f (i), or it does not include g(i), but then it includes ¬g(i) by maximality, hence

V ⊢P L g(i) → f (i). So [TriggeredI(V, I )] f ∪ V ⊢P L g(i) → f (i). For the left-to-

right direction, if m(I ) ∪ {C} �P L A then m(I ) ∪ {C} ∪ {¬A} is consistent, so there is

a V ∈ LP L⊥¬C such that m(I ) ∪ {C} ∪ {¬A} ⊆ V . It is immediate that for each i ∈

10 If I resembles the generating set G of input/output logic, then O(A/C) means that A is an output given

the input C (Makinson and van der Torre write A ∈ out (G, {C})). Though these authors liken their generating

set G to a body of conditional norms, it should be noted that they do not themselves introduce dyadic deontic

operators.

11 Makinson and van der Torre’s [25] call the resulting operator ‘basic output,’ of which a syntactical version

was first presented by Swirydowicz [39] p. 32.

123



TriggeredI(V, I ), m(I ) ∪ V ⊢P L f (i), so if [TriggeredI(V, I )] f ∪ V ⊢P L A then m(I ) ∪

V ⊢P L A and since m(I ) ⊆ V also V ⊢P L A. Since V was consistent and included ¬A, it

cannot also derive A, and so by contraposition [TriggeredI(V, I )] f ∪ V �P L A.

But such an equivalence makes all the problems of identifying conditional imperatives with

unconditional ones that demand their materializations reappear, in particular the problem of

contraposition.12 So it seems we must choose between ‘reasoning by cases’ and ‘circumstan-

tial reasoning.’ Another modification that these authors consider is that of ‘reusable output:’

when an imperative is triggered that demands A, and A is the trigger for some imperative

A ⇒!B, then we also ought to do B. Such a modification can easily be incorporated into a

truth definition and its ‘throughput’ version:

(td-cd3) I |� O(A/C) iff [Triggered∗I({C}, I )] f ⊢P L A

(td-cd3+) I |� O(A/C) iff [Triggered∗I({C}, I )] f ∪ {C} ⊢P L A

where Triggered∗I(W, Ŵ) means the smallest subset of Ŵ ⊆ I such that for all i ∈ Ŵ, if

[Triggered∗I(W, Ŵ)] f ∪ W ⊢P L g(i) then i ∈ Triggered∗

I(W, Ŵ). Moreover, the two mod-

ifications of ‘reasoning by cases’ and ‘reusable output’ can be combined to produce the

following definition and its ‘throughput’ variant:

(td-cd4) I |� O(A/C) iff ∀V ∈ LP L⊥¬C : [Triggered∗I(V, I )] f ⊢P L A

(td-cd4+) I |� O(A/C) iff ∀V ∈ LP L⊥¬C : [Triggered∗I(V, I )] f ∪ V ⊢P L A

The topic of ‘reusable output’ is discussed under the name of ‘deontic detachment’ in the

deontic logic literature, and there is no agreement whether such a procedure is admissi-

ble (Makinson [24] p. 43 argues in favor, whereas Hansson [16] p. 155 disagrees). E.g. let

I = {!p1, p1 ⇒!p2}, and for its interpretation assume that it is imperative for the proper

execution of your job that you develop novel methods, which make you eligible for a bonus,

and that if you develop such novel methods you owe it to yourself to apply for the bonus.

Truth definitions that accept ‘deontic detachment’ make true O(p2/⊤), and so tell us that

you ought to apply for the bonus, which seems weird since it may be that you never invent

anything. However, proponents of deontic detachment may argue that in such a situation,

O(p1 ∧ p2/⊤) should hold, i.e. you ought to invent new methods and apply for the bonus,

and that the reluctance to also accept O(p2/⊤) is—like the inference from “you ought to put

on your parachute and jump” to “you ought to jump”—just a variant of Ross’ Paradox that

is usually considered harmless.

For (td-cd4) we once again obtain O(p2/¬p3) for I = {p1⇒!p2,¬p1⇒!p3}: for any

V ∈ LP L⊥p3,¬p3 ∈ V , furthermore either p1 ∈ V and so p1⇒!p2 ∈ Triggered∗I(V, I ),

or ¬p1 ∈ V , then ¬p1⇒!p3 ∈ Triggered∗I(V, I ), and since {p3} ∪ {¬p3} ⊢P L p1, again

p1⇒!p2 is in Triggered∗I(V, I ), hence [Triggered∗

I(V, I )] f ⊢P L p2 for all V ∈ LP L⊥p3.

But as we saw above, this result seems counterintuitive.13 Note that (td-cd4+) is again equiv-

alent to I |� O(A/C) iff m(I ) ∪ {C} ⊢P L A and thus to (td-cd2+) (cf. [25] observation 16;

[26], p. 156). ⊓⊔

Observation 2 By (td-cd4+), I |� O(A/C) iff m(I ) ∪ {C} ⊢P L A.

Proof Similar to the proof of observation 1. For the left-to-right direction, use that for each

i ∈ Triggered∗I(V, I ), m(I ) ∪ V ⊢P L f (i), which is immediate. ⊓⊔

12 (td-cd2−) does not fare much better: though it does not include contraposition, it again makes O(p2/¬p3)

true for I = {p1⇒!p2,¬p1⇒!p3}, which is counterintuitive.

13 With respect to their out4-operation that corresponds to (td-cd4), Makinson and van der Torre [25] speak

of a ‘ghostly contraposition’.

123



3.4 Operators for prioritized conditional imperatives

This paper focuses on prioritized conditional imperatives, and for these there is a further hur-

dle to finding the proper truth definitions for deontic concepts. Priorities are only required

if the imperatives cannot all be obeyed—otherwise there is no reason not to obey all, and

the priority ordering is not used. So the truth definitions must be able to deliver meaningful

results for possibly conflicting imperatives. The intuitive idea is to use the information in the

ordering to choose subsets of the set of imperatives under consideration that contain only

the more important imperatives and leave out less important, conflicting ones, so that the

resulting ‘preferred subset’ (or rather, subsets, since the choice may not always be uniquely

determined by the ordering) only contains imperatives that do not conflict in the given situa-

tion. More generally, let I be a prioritized conditional imperative structure 〈I, g, f,<〉, and

let � be a subset of I . Then PI(W,�) contains just the subsets of � that are thus preferred

given the world facts W . The above truth definitions can then be adapted such that they now

describe something as obligatory iff it is so with respect to all the preferred subsets of the

imperatives, i.e. they take on the following forms:

I |� O(A/C) iff ∀Ŵ ∈ PI({C}, I ) :

(td-pcd1) [TriggeredI({C}, Ŵ)] f ⊢P L A,

(td-pcd1+) [TriggeredI({C}, Ŵ)] f ∪ {C} ⊢P L A,

(td-pcd2) ∀V ∈ LP L⊥¬C : [TriggeredI(V, Ŵ)] f ⊢P L A,

(td-pcd2+) ∀V ∈ LP L⊥¬C : [TriggeredI(V, Ŵ)] f ∪ V ⊢P L A,

(td-pcd3) [Triggered∗I({C}, Ŵ)] f ⊢P L A,

(td-pcd3+) [Triggered∗I({C}, Ŵ)] f ∪ {C} ⊢P L A,

(td-pcd4) ∀V ∈ LP L⊥¬C : [Triggered∗I(V, Ŵ)] f ⊢P L A,

(td-pcd4+) ∀V ∈ LP L⊥¬C : [Triggered∗I(V, Ŵ)] f ∪ V ⊢P L A.

So e.g. (td − pcd1) defines A as obligatory if the truth of A is required to satisfy the trig-

gered imperatives in any preferred subset. Of course, the crucial and as yet missing element

is the decision procedure that determines the set PI({C}, I ) of preferred subsets. The next

section discusses several proposals to define such subsets; a new proposal is presented in the

section that follows it.

4 Identifying the preferred subsets

4.1 Brewka’s preferred subtheories

The idea that normative conflicts can be overcome by use of a priority ordering of the norms

involved dates back at least to Ross [35] and is also most prominent in von Wright’s work (cf.

[40] p. 68, 80).14 However, it has turned out to be difficult to determine the exact mechanism

by which such a resolution of conflicts can be achieved. This is true even when only uncon-

ditional imperatives are considered. Discussing various proposals for resolution of conflicts

between unconditional imperatives, I have argued in [14] that an ‘incremental’ definition

should be used for determining the relevant subsets. Based on earlier methods by Rescher

[32], such a definition was first introduced by Brewka [3] for reasoning with prioritized

14 But cf. already Watts [42] part II, ch. V, sec. III, principle 10: “Where two duties seem to stand in opposition

to each other, and we cannot practise both, the less must give way to the greater, and the omission of the less

is not sinful.”

123



defaults. For any priority relation <, the idea is to consider all the ‘full prioritizations’ ≺ of

< (strict well orders that preserve <), and then work one’s way from the top of the strict

order downwards by adding the ≺-next-higher imperative to the thus constructed ‘preferred

subtheory’ if its demand is consistent with the given facts and the demands of the imperatives

that were added before. For the present setting, the definition can be given as follows:

Definition 1 (Brewka’s preferred subtheories) Let I = 〈I, f, g,<〉 be a prioritized condi-

tional imperative structure, � be a subset of I , and W ⊆ LP L be a set of PL-sentences.

Then Ŵ ∈ P BI (W,�) iff (i) W �P L ⊥, and (ii) Ŵ is obtained from a full prioritization ≺ by

defining

Ŵi =

{

⋃

j ≺ i Ŵ j ∪ {i} if W ∪

[

⋃

j ≺ i Ŵ j ∪ {i}] f

�P L ⊥, and⋃

j ≺ i Ŵ j otherwise,

for any i ∈ �, and letting Ŵ =⋃

i∈� Ŵi .

Clause (i) ensures that for an inconsistent set of assumed ‘facts,’ no set is preferred. Some-

what roundabout, owed to the possibility of infinite ascending subchains, clause (ii) then

recursively defines a set Ŵ ∈ PBI (W,�) for each full prioritization ≺: take the ≺-first i (the

exclusion of infinite descending subchains guarantees that it exists) and if W ∪ {i f } �P L ⊥

then let Ŵi = {i}; otherwise Ŵi is left empty.15 Similarly, any ≺-later i is tested for possible

addition to the set⋃

j ≺ i Ŵ j of elements that were added in the step for a j ∈ � that occurs

≺-prior to i . Ŵ is then the union of all these sets.

To see how this definition works, consider the set I = {!(p1 ∨ p2), !¬p1, !¬p2}, with the

ranking !(p1 ∨ p2) < !¬p2 and !¬p1 < !¬p2. For an interpretation, let !(p1 ∨ p2)) be your

mother’s request that you buy cucumbers or spinach for dinner, !¬p1 be your father’s wish

that no cucumbers are bought, and !¬p2 your sister’s desire that you don’t buy any spinach.

We have two full prioritizations !(p1∨ p2) < !¬p1 < !¬p2 and !¬p1 < !(p1∨ p2) < !¬p2—

let these be termed ≺1 and ≺2, respectively. The construction for ≺1 adds the imperative

!(p1 ∨ p2) in the first step and, since no conflict with the situation arises, !¬p1 in the second

step. In the third and last step, nothing is added since !¬p2 conflicts with the demands of

the already added imperatives. For ≺2 the only difference is that the first two imperatives

are added in inverse order. Thus PBI (W, I ) = {{!(p1 ∨ p2), !¬p1}}. Using (td − pcd1) we

obtain O(¬p1 ∧ p2/⊤), which means that you have to buy spinach and not cucumbers, thus

fulfilling your parents’ requests but not your sister’s, which seems reasonable.

As I showed in [14], Brewka’s method is extremely successful for dealing with uncon-

ditional imperatives. It is provably equivalent for such imperatives to methods proposed

by Ryan [36] and Sakama and Inoue [37], and it avoids problems of other approaches by

Alchourrón and Makinson [2], Prakken [30] and Prakken and Sartor [31]. Moreover, an

equally intuitive maximization method proposed by Nebel [28], [29], that adds first a maxi-

mal set of the highest-ranking imperatives, then a maximal set of second-ranking imperatives,

etc., but for its construction requires the ordering to be based on a complete preorder, can

be shown to be embedded in Brewka’s approach for such orderings. So my aim will be to

retain Brewka’s method for the unconditional case—in fact, all proposals that follow meet

this criterion. However, when it is applied without change to conditional imperatives, the

15 As usual, the union of an empty set of sets is taken to be the empty set.

123



algorithm may lead to incorrect results. E.g. consider a set I with two equally ranking imper-

atives {p1⇒!p2,¬p1⇒!¬p2}, meaning e.g. “if you go out, wear your boots” and “if you

don’t go out, don’t wear your boots:” since the consequents contradict each other, an unmod-

ified application of Brewka’s method produces PBI ({p1}, I ) = {{p1⇒!p2}, {¬p1⇒!¬p2}},

which fails to make true O(p2/p1) by any truth definition of Sect. 3.4: the right set contains

no imperatives that are triggered by p1. So we cannot derive that you ought to wear your

boots, given that you are going out. But intuitively there is no conflict, since the obligations

arise in mutually exclusive circumstances only.

4.2 A naïve approach

A straightforward way to adopt Brewka’s method to the case of conditional imperatives is to

use not all imperatives for the construction, but only those that are triggered by the facts W ,

i.e. to use TriggeredI(W,�) instead of �:

Definition 2 (The naïve approach) Let I = 〈I, f, g,<〉 be a prioritized conditional imper-

ative structure, � be a subset of I , and W ⊆ LP L be a set of PL-sentences. Then Ŵ ∈

PnI(W,�) iff Ŵ ∈ P B

I (W, TriggeredI(W,�)).

The change resolves our earlier problems with Brewka’s method: consider again the set of

imperatives {p1⇒!p2,¬p1⇒!¬p2}, where the imperatives were interpreted as ordering me

to wear my boots when I go out, and not wear my boots when I don’t. The new defini-

tion produces PnI({p1}, I ) = {{p1 ⇒!p2}}, its only ‘preferred’ subset containing just the

one imperative that is triggered given the facts {p1}. By all truth definitions of Sect. 3.4,

O(p2/p1) is now true, so given that you go out, you ought to wear your boots, which is as

it should be.

The naïve approach is similar to Horty’s proposal in [20] in that conflicts are only removed

between imperatives that are triggered (though the exact mechanism differs from Horty’s).

When I nevertheless call it ‘naïve,’ this is because there are conceivable counterexamples to

this method. Consider the set of prioritized imperatives !p1 < p1⇒!p2 < !¬p2, and for an

interpretation suppose that your job requires you to go outside p1, that your mother, who is

concerned for your health, told you to wear a scarf p2 if you go outside, and that your friends

don’t want you to wear a scarf, whether you go outside or not. In the default situation ⊤

only the first imperative and the third are triggered, i.e. TriggeredI({⊤}, I ) = {!p1, !¬p2}.

Since their demands are consistent with each other, we obtain PnI({⊤}, I ) = {{!p1, !¬p2}},

for which all truth definitions of Sect. 3.4 make O(p1 ∧ ¬p2/⊤) true. So you ought to go

out and not wear a scarf, thus satisfying the first and the third imperative, but violating the

second-ranking imperative. But arguably, if an imperative is to be violated, it should not be

the second-ranking p1⇒!p2, but the lowest ranking !¬p2 instead.

4.3 The stepwise approach

To avoid the difficulties of the ‘naïve’ approach, it seems we must not just take into account

the imperatives that are triggered, but also those that become triggered when higher ranking

imperatives are satisfied. To this effect, the following modification may seem

reasonable:

Definition 3 (The stepwise approach) Let I = 〈I, f, g,<〉 be a prioritized conditional

imperative structure, � be a subset of I , and W ⊆ LP L be a set of PL-sentences. Then

123



Ŵ ∈ Ps(W,�) iff (i) W �PL ⊥, and (ii) Ŵ is obtained from a full prioritization ≺ by

defining

Ŵi =

{

⋃

j ≺ i Ŵ j ∪ {i} if i ∈ SatisfiableI(W ∪

[

⋃

j ≺ i Ŵ j

] f

,�), and⋃



i∈� Ŵi .

So at each step one considers what happens if the imperatives that were included so far are

satisfied, and adds the current imperative only if it is satisfiable given W and the satisfaction

of these previous imperatives. Note that satisfiability of an imperative, like its satisfaction and

violation, presupposes that the imperative is triggered. The new definition not only includes,

at each step, those imperatives that are triggered and can be satisfied given the facts and the

supposed satisfaction of the previously added imperatives: it also includes those that become

triggered when a previously added imperative is satisfied.

This modification avoids the previous difficulty: consider again the set of prioritized

imperatives !p1<p1⇒!p2<!¬p2. There is just one full prioritization, which for W = {⊤}

yields in the first step the set {!p1}, and in the second step {!p1, p1⇒!p2}, since p1⇒!p2 is

triggered when the previously added imperative !p1 is assumed to be satisfied. In the third

step, nothing is added: though the imperative !¬p2 is triggered, it cannot be satisfied together

with the previously added imperatives. So we obtain PsI({⊤}, I ) = {{!p1, p1⇒!p2}}, and

hence O(p1/⊤), but not O(p1 ∧¬p2/⊤), is defined true by all truth definitions of Sect. 3.4.

Operators that accept ‘deontic detachment’ (td-pcd3(+), 4(+)) make true O(p1 ∧ p2/⊤), so

you must go out and wear a scarf, which is as it should be.

However, a small change in the ordering shows that this definition does not suffice: let

the imperatives now be ranked p1 ⇒!p2 < !p1 < !¬p2. (For the interpretation, assume

that the conditional imperative to wear a scarf when leaving the house was uttered by some

high-ranking authority, e.g. a doctor.) Then again PsI({⊤}, I ) = {{!p1, !¬p2}}: in the first

step, nothing is added since p1⇒!p2 is neither triggered by the facts nor by the assumed

satisfaction of previously added imperatives (there are none). In the next two steps, !p1 and

!¬p2 are added, as each is consistent with the facts and the satisfaction of the previously

added imperatives. So again all truth definitions of Sect. 3.4 make true O(p1 ∧ ¬p2/⊤), i.e.

you ought to go out and not wear a scarf, satisfying the second and third ranking imperatives

at the expense of the highest ranking one. But surely, if one must violate an imperative, it

should be one of the lower-ranking ones instead.

4.4 The reconsidering approach

The merits of the stepwise approach were that it did not only consider the imperatives that are

triggered, but also those that become triggered when already added imperatives are satisfied.

Such considerations applied to those imperatives that follow in the procedure. Yet the satis-

faction of already added imperatives might also trigger higher-ranking imperatives, which by

this method are not considered again. So it seems necessary, at each step, to reconsider also

the higher-ranking imperatives. An algorithm that does that was first introduced for default

theory by Marek and Truszczynski [27] p. 72, and later employed by Brewka in [4]; it can

be reformulated for the present setting as follows:

Definition 4 (The reconsidering approach) Let I = 〈I, f, g,<〉 be a prioritized conditional


123



Ŵ ∈ PrI(W,�) iff (i) W �P L ⊥, and (ii) Ŵ is obtained from a full prioritization ≺ by

defining

Ŵi =⋃

j ≺ iŴ j ∪ min≺

[

SatisfiableI

(

W ∪[

⋃

j ≺ iŴ j

] f

,�

)

\⋃

j ≺ iŴ j

]

for i ∈ �, and letting Ŵ =⋃

i∈� Ŵi .16

The definition reconsiders at each step the whole ordering, and adds the ≺-first17 impera-

tive (due to the definition of ≺ there is just one) that has not been added previously and is

satisfiable given both the facts W and the consequents of the previously added imperatives.

To see how the definition works, consider again the example which the stepwise approach

failed, i.e. the set of prioritized imperatives p1⇒!p2 < !p1 < !¬p2. We are interested in the

preferred sets for the default circumstances ⊤, i.e. the sets in P rI({⊤}, I ). I is already fully

prioritized, so there is just one such set. Applying the algorithm, we find the minimal (highest

ranking) element in SatisfiableI({⊤}, I ) is !p1, so this element is added in the first step. In

the second step, we look for the minimal element in SatisfiableI({⊤} ∪ {!p1}f , I ), other

than the previously added !p1. It is p1⇒!p2, since the assumed satisfaction of all previously

added imperatives triggers it, and its consequent can be true together with {⊤} ∪ {p1}. So

p1⇒!p2 is added in this step. In the remaining third step, nothing is added: !¬p2 is not in

SatisfiableI({⊤}∪{!p1, p1⇒!p2}f , I ), and all other elements in this set have been previously

added. So P rI({⊤}, I ) = {{!p1, p1⇒!p2}}. Now all truth definitions of Sect. 3.4 make true

O(p1/⊤), but not O(p1 ∧ ¬!p2/⊤), and operators that accept ‘deontic detachment’ make

true O(p1 ∧ p2/⊤). So, in the given interpretation, you must go out (as your job requires)

and wear a scarf (as the doctor ordered you to do in case you go out), which is as it should

be.

However, again problems remain. Let the imperatives now be prioritized in the order

p1 ⇒!p2 < !¬p2 < !p1. Let p1 ⇒!p2 stand for the doctor’s order to wear a scarf when

going outside, let !¬p2 stand for your friends’ expectation that you don’t wear a scarf, and let

!p1 represent your sister’s wish that you leave the house. Construct the set in P rI({⊤}, I )—

since I remains fully prioritized, there is again just one such set. The minimal element

in SatisfiableI({⊤}, I ) is !¬p2, and so is added in the first step. The minimal element in

SatisfiableI({⊤} ∪ {!¬p2}f , I ), other than !¬p2, is !p1 which therefore gets added in the

second step. Nothing is added in the remaining step: !¬p2 and !p1 have already been added,

and p1 ⇒!p2 is not in SatisfiableI({⊤} ∪ {!¬p2, !p1}f , I ): though it is triggered by the

assumed satisfaction of !p1, its consequent is contradicted by the assumed satisfaction of

!¬p2. So PrI({⊤}, I ) = {{!p1, !¬p2}}. Hence all truth definitions of Sect. 3.4 again make

true O(p1 ∧¬p2/⊤), so you ought to go out without a scarf, again satisfying the second and

third ranking imperatives at the expense of the first, which seems the wrong solution.

4.5 A fixpoint approach

To eliminate cases in which the ‘reconsidering approach’ still adds imperatives that can

only be satisfied at the expense of violating a higher-ranking one, a ‘fixpoint’ approach was

first proposed for default reasoning by Brewka and Eiter [5]. It tests each set that may be

considered as preferred to see if it really includes all the elements that should be included:

16 Note that in ‘Ŵi , ’ i is used just as an index—it does not mean that i is considered for addition at this step,

and may be added at an earlier or later step (or not at all).

17 For any ordering < on some set Ŵ, min<Ŵ = {i ∈ Ŵ|∀i ′ ∈ Ŵ : if i ′ �= i, then i ′ ≮ i}, and max<Ŵ = {i ∈

Ŵ|∀i ′ ∈ Ŵ : if i ′ �= i, then i ≮ i ′}, as usual.

123



imperatives that are triggered given the facts and the assumed satisfaction of all imperatives

in the set, and would be added by Brewka’s [3] original method that adds the higher ranking

imperatives first. The procedure translates as follows:

Definition 5 (The fixpoint approach) Let I = 〈I, f, g,<〉 be a prioritized conditional


Ŵ ∈ PfI (W,�) iff Ŵ ∈ P B

I (W, TriggeredI(W ∪ Ŵ f ,�)).

To see how this definition works, consider the above set of prioritized imperatives p1 ⇒

!p2 < !¬p2 < !p1. It is immediate that the set {!p1, !¬p2} cannot be in P fI({⊤}, I ):

if we assume all imperatives in this set to be satisfied, then all imperatives are triggered,

i.e. TriggeredI({⊤} ∪ {!p1, !¬p2}f , I ) = I . By Brewka’s original method, PB

I (W, I ) =

{{p1⇒!p2, !p1}}: < is already fully prioritized, and for this full prioritization the method

adds p1⇒!p2 in the first step, !¬p2 cannot be added in the second step since its consequent

contradicts the consequent of the previously added p1 ⇒!p2, and in the third step !p1 is

added. So since the considered set {!p1, !¬p2} is not in PBI (W, I ), it is not a ‘fixpoint.’

Rather, as may be checked, the only ‘fixpoint’ in P fI({⊤}, I ) is {p1⇒!p2, !p1}. Then all

truth definitions of Sect. 3.4 make true O(p1/⊤), but no longer O(p1 ∧¬p2/⊤). Moreover,

truth definitions that allow ‘deontic detachment’ make true O(p1 ∧ p2/⊤). In the given

interpretation this means that you must leave the house at your sisters request and wear a

scarf, as the doctor ordered you to do in case you go out.

Though the construction now no longer makes true O(p1 ∧ ¬p2/⊤), its solution for

the example, that determines the set {p1⇒!p2, !p1} as the fixpoint of the set of prioritized

imperatives p1 ⇒!p2 < !¬p2 < !p1, seems questionable. Though this now includes the

doctor’s order, you now have no obligation anymore to satisfy the imperative that is second

ranking, i.e. your friends’ request that you don’t wear a scarf; truth definitions that accept

‘deontic detachment’ even oblige you to violate it by wearing a scarf. Now consider the

situation without the third ranking imperative !p1: it can easily be verified that for a set

I = {p1 ⇒!p2, !¬p2} the only fixpoint in P fI({⊤}, I ) is {!¬p2}. So for the reduced set,

(td − pcd1) makes true O(¬p2/⊤), i.e. you ought to obey your friends’ wish. That the satis-

faction of this higher ranking imperative !¬p2 should no longer be obligatory when a lower

ranking imperative !p1 is added, seems hard to explain. If the ranking is taken seriously, I

think one should still satisfy the higher ranking imperatives, regardless of what lower ranking

imperatives are added.

For another problem consider the set of prioritized imperatives p1 ⇒!p2 <!(p1 ∧¬p2) <

!p3. For an interpretation, let the first imperative be again the doctor’s order to wear a scarf

in case you go out, the second one be your friends’ request to go out and not wear a scarf,

and the third be the wish of your aunt that you write her a letter. It is easily proved that the set

has no fixpoint, and so there is also none that contains !p3, hence all truth definitions make

O(p3/⊤) false, so you do not even have to write to your aunt. But even if the presence of

a higher ranking conditional imperative and a lower ranking imperative to violate it poses a

problem (why should it? after all, the lower ranking imperative is outranked), it is hard to see

why the subject should be left off the hook for all other, completely unrelated obligations.18

18 The lack of fixpoints is a well-known problem of such definitions (cf. e.g. Caminada and Sakama [7]).

Another approach to conditional imperatives by Makinson in [24] has trouble resolving the same example:

for the default circumstances ⊤ it produces the set {!(p1 ∧ ¬p2), !p3}. p1⇒!p2 is not considered, since its

only ‘label’ (roughly: a conjunction of the circumstances, the imperatives’ antecedents that would trigger∗ it,

and its consequent) is inconsistent (it is ⊤ ∧ (p1 ∧ ¬p2) ∧ p2). But why should the agent not be free to obey

p1⇒!p2, and not violate it by satisfying !(p1 ∧ ¬p2)?

123



4.6 Discussion

For a discussion of our results so far, let us return to the ‘drinking and driving’ example from

the introduction. Let the three imperatives:

(1) Your mother says: if you drink anything, then don’t drive.

(2) Your best friend says: if you go to the party, then you do the driving.

(3) Some acquaintance says: if you go to the party, then have a drink with me.

be represented by the set of prioritized imperatives p1 ⇒¬p2 < p3⇒p2 < p3⇒p1. Let the

set of facts be {p3}, i.e. you go to the party. Brewka’s original method is not tailored to

be directly employed on conditional imperatives, as it ignores the antecedents altogether.

The next three approaches, the naïve, the stepwise and the reconsidering ones, produce

PnI({p3}, I ) = Ps

I({p3}, I ) = PrI({p3}, I ) = {{p3⇒!p2, p3⇒!p1}}, which by all truth

definitions of Sect. 3.4 makes true O(p1 ∧ p2/p3), so you ought to drink and drive. The

fixpoint approach produces P fI({p3}, I ) = {{p1⇒!¬p2, p3⇒!p1}}, so all truth definitions

make true O(p1/p3), which means you ought to drink. Truth definitions with ‘deontic detach-

ment’ additionally make true O(p1 ∧ ¬p2/p3), so you ought to drink and not drive. But

being obliged to drink runs counter to our intuitions for the ‘drinking and driving’ example.

So we have to look for a different solution.

Before we do that, I will, however, question again our intuition in this matter. Horty [21]

has recently used a structurally identical example to argue for just the opposite, that the solu-

tion by the fixpoint approach is correct. His example is that of three commands, uttered by

a colonel, a major and a captain to a soldier, Corporal O’Reilly. The Colonel, who does not

like it too warm in the cabin, orders O’Reilly to open the window whenever the heat is turned

on. The Major, who is a conservationist, wants O’Reilly to keep the window closed during

the winter. And the Captain, who does not like it to be cold, orders O’Reilly to turn the heat

on during the winter. O’Reilly is trying to figure out what to do. The intended representation

is again the prioritized conditional imperative structure employed above for the ‘drinking

and driving’ example, where p1 now means that the heat is turned on, p2 means that the

window is closed, and p3 means that it is winter. We saw that the fixpoint approach yields

the preferred subset {p1⇒!¬p2, p3⇒!p1}, making true O(p1/p3) for all truth definitions,

and O(p1 ∧¬p2/p3) for truth definitions that accept ‘deontic detachment,’ so O’Reilly must

turn on the heat and then open the window, and thus violate the Major’s order. Horty argues

as follows in support of this choice:

“O’Reilly’s job is to obey the orders he has been given exactly as they have been

issued. If he fails to obey an order issued by an officer without an acceptable excuse,

he will be court-martialed. And, let us suppose, there is only one acceptable excuse for

failing to obey such an order: that obeying the order would, in the situation, involve

disobeying an order issued by an officer of equal or higher rank. (...) So given the set

of commands that O’Reilly has been issued, can he, in fact, avoid court-martial? Yes

he can, by (...) obeying the orders issued by the Captain and the Colonel (...). O’Reilly

fails to obey the Major’s order, but he has an excuse: obeying the Major’s order would

involve disobeying an order issued by the Colonel.”

Horty’s principle seems quite acceptable: for each order issued to the agent, the agent may

ask herself if obeying the order would involve disobeying an order of a higher ranking officer

(then she is excused), and otherwise follow it. The result is a set of imperatives where each

imperative is either obeyed, or disobeyed but the disobedience excused. When I nevertheless

think the argument is not correct, it is because I think it confuses the status quo and the status

123



quo posterior. Obeying the Major’s order does not, in the initial situation, involve disobeying

the Colonel’s order. Only once O’Reilly follows the Captain’s order and turns on the heat, it

is true that he must obey the Colonel, open the window and thus violate the Major’s order. But

this does not mean that he should follow the Captain’s order in the first place—as by doing

so he brings about a situation in which he is forced, by a higher ranking order, to violate a

command from another higher ranking officer. Quite to the contrary, I think that being forced

to violate a higher ranking order when obeying a lower ranking one is a case where following

the lower one ‘involves’ such a violation, and so the only order the agent is excused from

obeying is the lowest ranking command.

Consider finally this variant: suppose that if I am attacked by a man, I must fight him (to

defend my life, my family etc.). Furthermore, suppose I have pacifist ideals which include

that I must not fight the man. Now you tell me to provoke him, which in the given situation

means that he will attack me. Let self-defense rank higher than my ideals, which in turn rank

higher than your request. Should I do as you request? By the reasoning advocated by Horty,

there is nothing wrong with it: I satisfy your request, defend myself as I must, and though I

violate my ideals, I can point out to myself that the requirement to fight back took priority.

But I think if I really do follow your advice, I would feel bad. I think this would not just be

some irrational regret for having to violate, as I must, my ideals, but true guilt for having

been tempted into doing something I should not have done, namely provoking the man: it

caused the situation that made me violate my ideals. So I think our intuitions in the ‘drinking

and driving’ example and the other cases have been correct.

5 New strategies and a new proposal

In the face of the difficulties encountered so far, it seems necessary to address the issue of

finding an appropriate mechanism for a resolution of conflicts between prioritized conditional

imperatives in a more systematic manner. So far intuition has guided us mainly as to what

imperatives should be included in a subset of ‘preferred imperatives.’ I think the following

postulate sums up the intuitions that have so far influenced the proposal and rejection of

solutions:

Postulate Any imperative should be considered relevant (included) as long as it is not

violated or, in the given situation, conflicts with other imperatives that are also considered

relevant (included) and do not rank lower.

The postulate makes clear that the need to satisfy a lower ranking imperative cannot serve

as an excuse to violate an imperative of higher priority. However, this postulate delegates

the answer of how the set of ‘preferred imperatives’ should be constructed to the answer of

another question: when do conditional imperatives conflict in a given situation? There appear

to be several possible answers to this question, which lead to different solutions.

5.1 Subsets with consistent extensions

For the definition of conflicts between conditional imperatives one might recur to a definition

of conflicts between desires as proposed by Broersen et al. [6]. Their idea, translated to our

setting, is that for any set of facts W and set of conditional imperatives � there is a smallest set

E(W,�) ⊆ LP L such that (i) W ⊆ E(W,�) and (ii) for any i ∈ �, if E(W,�) ⊢P L g(i)

then f (i) ∈ E(W,�). The set of imperatives � is then defined as conflicting given the facts

W if the thus constructed extension E(W,�) of the facts is inconsistent, i.e. E(W,�) ⊢P L ⊥

123



(cf. [6] def. 5.5). This ‘test’ for conflicts may then be employed within a variant of Brewka’s

original method.

Definition 6 (Preferred Subsets with Consistent Extensions) Let I = 〈I, f, g,<〉 be a pri-

oritized conditional imperative structure, � be a subset of I , and W ⊆ LP L be a set of

PL-sentences, and let E(W,�) be defined as described above. Then Ŵ ∈ PeI(W,�) iff (i)

W �P L ⊥, and (ii) Ŵ is obtained from a full prioritization ≺ by defining

Ŵi =

{⋃

j ≺ i Ŵ j ∪ {i} if E(W,⋃

j ≺ i Ŵ j ∪ {i}) �P L ⊥, and⋃



i∈� Ŵi .

To see how this definition works, consider the ‘drinking and driving’ example, where the set

of prioritized imperatives is p1 ⇒!¬p2 < p3 ⇒!p2 < p3 ⇒!p1 and the situation W = {p3}.

There is only one full prioritization which is identical with <. In the first step, p1 ⇒!¬p2 is

added, since E({p3}, {p1 ⇒!¬p2}) = {p3} which is consistent. In the second step, p3 ⇒!p2

is added, as E({p3}, {p1 ⇒!¬p2, p3 ⇒!p2}) = {p3, p2}, which is again consistent. In

the third step, p3 ⇒!p1 is rejected, since E({p3}, {p1 ⇒!¬p2, p3 ⇒!p2, p3 ⇒!p1}) =

{p3, p2, p1,¬p2}, which is inconsistent. So we have PeI({p3}, I ) = {{p1 ⇒!¬p2, p3 ⇒

!p2}}, making true O(p2/p3) for all truth definitions of Sect. 3.4, so given that I go to the

party I must do the driving, which is as it should be.

However, there is a problem for the test using consistent extensions, as for some truth def-

initions it delivers sets of imperatives that are clearly conflicting, in the sense that they make

O(⊥/C) true for a consistent fact C : Consider the set of facts W = {p1 ∨ p2} and the set of

imperatives I = {p1 ⇒!p3, p2 ⇒!p3, !¬p3}. We have E(W, I ) = {p1 ∨ p2,¬p3}, which

is consistent, and so all imperatives in I are added to the preferred subset, regardless of their

ordering. But for any truth definition that allows for ‘reasoning by cases’ (td-pcd2(+),4(+))

we then have both O(p3/p1 ∨ p2) and O(¬p3/p1 ∨ p2), and so also O(⊥/p1 ∨ p2).

This is simply because the construction of extensions does not take care of reasoning by

cases, i.e. it does not add p3 to the extension in case we both have p1 ∨ p2 in W and

{p1 ⇒!p3, p2 ⇒!p3} ⊆ I . Perhaps the definition of consistent extensions can be amended,

but to avoid delivering preferred sets of imperatives that make true O(⊥/C) for some truth

definition and some consistent fact C suggests a different solution that is explored in the next

section.

5.2 Deontically tailored preferred subsets

In the unconditional case, the reason to move from definition (td-m1) to (td-m2) was that

when there are conflicts between imperatives, the former makes true the monadic deontic

formula O⊥, i.e. the agent ought to do the logically impossible. This result was avoided by

considering only maximal sets of imperatives with demands that are collectively consistent,

i.e. sets that do not make O⊥ true. When faced with the question what dyadic deontic for-

mula should not be true when conflicts are resolved for arbitrary situations C , the formula

O(¬C/C) appears to be the dyadic equivalent: a mechanism for conflict resolution should

not result in telling the agent to change the supposed, unalterable facts.19 So to define the set

PI({C}, I ) required by the truth definitions (td-pcd1(+) − 4+), we can modify Brewka’s

19 This test is identical to the one used by Makinson and van der Torre [26] p. 158/159 to determine ‘consis-

tency of output’ (cf. also for arguments why O(¬C/C) should be used, i.e. for their setting, the ‘output’ should

be consistent with the ‘input,’ rather than the formula O(⊥/C) and thus consistency of output simpliciter).

123



original method in such a way that it tests, at each step, for each of the constructed subsets, if

the corresponding truth-definition (td-cd1(+) − 4+) does not make O(¬C/C) true for this

set.20 Formally:

Definition 7 (Deontically Tailored Preferred Subsets) Let I = 〈I, f, g,<〉 be a prioritized

conditional imperative structure, and C ∈ LPL describe the given situation. Let (td-pcd∗) be

any of the truth definitions (td-pcd1(+) − 4(+)). Then Ŵ is in the set P∗I({C}, I ) employed

by this truth definition iff (i) {C} �P L ⊥, and (ii) Ŵ is obtained from a full prioritization ≺

by defining

Ŵi =

{⋃

j ≺ i Ŵ j ∪ {i} if 〈⋃

j ≺ i Ŵ j ∪ {i}, f, g〉 � O(¬C/C)by(td-cd∗),⋃


for any i ∈ I , and letting Ŵ =⋃

i∈I Ŵi .

By this construction, each of the preferred subsets contains a maximal number of the imper-

atives such that they do not make true O(¬C/C) for the situation C and the truth definition

that is employed, and so the resulting truth definition likewise avoids this truth. Such a con-

struction of the preferred subsets might be considered ‘tailored’ to the truth definition in

question, and any remaining deficiencies might be seen as stemming from the employed

truth definition. But this being so, the method reveals a strong bias towards truth definitions

that accept ‘deontic detachment’, and in particular truth definitions (td-pcd2+, 3(+), 4(+)):

Consider the set of imperatives I = {!p1, p1 ⇒!p2, !¬p2} with the ranking !p1 <

p1⇒!p2 < !¬p2, that was used to refute the ‘naïve approach.’ As can be easily checked,

P∗I({⊤}, I ) = {I } for all truth definitions (td-pcd1, 1+, 2). So by all these truth definitions,

O(p1 ∧ ¬p2/⊤) is true. So they commit us to violating the second-ranking imperative,

whereas intuitively, the third-ranking imperative should be violated instead. By contrast,

all truth definitions (td-pcd3(+), 4(+)), that employ reusable output, and of course likewise

(td-pcd2+) that is equivalent to (td-pcd4+), handle all given examples exactly as it was sug-

gested they should. In particular, consider the ordered imperatives p1⇒!p2 < !¬p2 < !p1,

that were used to refute both the ‘reconsidering’ and the ‘fixpoint’ approaches: for ∗ =

2+, 3(+), 4(+), P∗I({⊤}, I ) is {{p1⇒!p2, !¬p2}}, making O(¬p2/⊤) true by all these truth

definitions, which thus commit us to satisfying the second ranking imperative, and not to

violating it in favor of satisfying the third ranking imperative as these approaches did. The

‘drinking and driving’ example is also handled correctly: the set of prioritized imperatives

p1 ⇒!¬p2< p3 ⇒!p2< p3 ⇒!p1 produces, for the situation p3, the set P∗I({p3}, I ) =

{{p1⇒!¬p2, p3⇒!p2}}. So the third ranking imperative, that commits the agent to drinking

and thus, by observation of the highest ranking imperative, prevents the agent from driving,

is disregarded. Instead, the truth definitions make true O(p2/p3), so the agent must do the

driving if she goes to the party, as her best friend asked her to.

Is this the solution, then? Some uneasiness remains as to the quick way with which def-

initions (td-pcd1, 1+, 2) are discharged as insufficient. Why should it not be possible to

maintain, as these definitions do, that conditional imperatives only produce an obligation

if they are factually triggered, while at the same time maintaining that the above examples

should not be resolved the way they are? The purpose of a truth definition for the deontic

O-operator is to find a formal notion of ‘ought’ that reflects ordinary reasoning, and our

intuitions on that matter may differ from our ideas as to what may constitute a good choice

from a possibly conflicting set of prioritized conditional imperatives. I will now make a new

proposal how to construct the ‘preferable subsets’, that keeps the positive results without

20 The preferred subsets are thus a choice from the ‘maxfamilies’ defined in [26].

123



committing us to prefer any of the truth definitions of Sect. 3.4 by virtue of their handling of

prioritized imperatives alone.

5.3 Preferred maximally obeyable subsets

What made Brewka’s approach so successful is that it maximizes the number of higher rank-

ing imperatives in the preferred subsets of a given set of unconditional imperatives: for each

‘rank,’ a maximal number of imperatives are added that can be without making the set’s

demands inconsistent in the given situation. As was shown, Brewka’s approach cannot be

directly applied to conditional imperatives, since it makes no sense to test the demands of

imperatives for inconsistencies if these imperatives may not be triggered in the same circum-

stances. Just considering triggered imperatives is also not enough, as was demonstrated for

the ‘naïve approach.’ But if the maximization method is to include imperatives that are not

(yet) triggered, then we must look for something else than inconsistency of demands to take

the role of a threshold criterion for the maximization process.

To do so we should ask ourselves why, for the unconditional case, the aim was to find a

maximal set of imperatives with demands that are collectively consistent with the situation.

I think that by doing so we intend to give the agent directives that can be safely followed.

While in the unconditional case this means that the agent can satisfy all the chosen imper-

atives, the situation is different for conditional imperatives: here an agent can also obey

imperatives without necessarily satisfying their demands. If you tell me to visit you in case I

go to Luxembourg next month, I can safely arrange to spend all of next month at home and

still do nothing wrong. If we think not so much of imperatives, but of legal regulations, then

I can obviously be a law-abiding citizen by simply failing to trigger any legal norm (even

though this might imply living alone on an island): whether I do that or boldly trigger some

of the regulations’ antecedents and then satisfy those I have triggered seems not a question of

logic, but of individual choice. So I think the threshold criterion to be used should be that of

obeyability: we should maximize the set of imperatives the agent can obey, and only disregard

an imperative if its addition to the set means that at least one of the added imperatives must

(now) be violated, given the facts.21

For a given set of conditional imperatives � and a set of factual truths W , the subsets of

imperatives that can be obeyed are described by ObeyableI(W,�), i.e. they are those subsets

Ŵ ⊆ � such that W ∪Ŵm �P L ⊥. To maximize not by collective consistency of demands, but

by collective obeyability, Brewka’s original approach can therefore be changed as follows:

Definition 8 (Preferred Maximally Obeyable Subsets) Let I = 〈I, f, g,<〉 be a prioritized

conditional imperative structure, � be a subset of I , and W ⊆ LP L be a set of PL-sentences.

Then Ŵ ∈ PoI(W,�) iff (i) W �P L ⊥, and (ii) Ŵ is obtained from a full prioritization ≺ by

defining

Ŵi =

{⋃

j ≺ i Ŵ j ∪ {i} if⋃

j ≺ i Ŵ j ∪ {i} ∈ ObeyableI(W,�), and⋃



i∈� Ŵi .

The change from Brewka’s original definition is only minute: we test not the demands of the

imperatives for consistency, but their materializations. Note that this is a conservative exten-

sion of Brewka’s method, since for any unconditional imperative i we have ⊢P L f (i) ↔

21 While Hansson, in [16] p. 200, also advocates a move from ‘consistency’ to ‘obeyability,’ what is meant

there is rather the step from (td-m2) to (td-d1).

123



m(i). As can easily be seen, the new construction solves all of the previously considered

difficulties, regardless of the chosen truth definition for the deontic O-operator:

• To refute a direct application of Brewka’s original method, we used the set I = {p1⇒p2,

¬p1⇒¬p2} with no ranking imposed. m(I ) is consistent and so PoI({p1}, I ) = {I },

making O(p2/p1) true for all definitions of Sect. 3.4 So you ought to wear your boots in

case you go out, as it should be.

• To refute the ‘naïve approach,’ we used the set of prioritized imperatives p1 < p1⇒p2 <

¬p2. Since < is already fully prioritized, the construction produces just one maximally

obeyable subset, which is {!p1, p1⇒!p2}, as its two imperatives get added in the first two

steps, and nothing is added in the third since m(I ) is inconsistent. All truth definitions

make true O(p1/⊤), none makes true the non-intuitive formula O(p1 ∧ ¬p2/⊤), and

definitions that accept ‘deontic detachment’ make true O(p1 ∧ p2/⊤). So you must go

out and wear a scarf, which is as it should be.

• To refute the stepwise approach the ordering of the imperatives was changed into p1⇒p2 <

p1 < ¬p2. Still PoI(⊤}, I ) = {{!p1, p1⇒!p2}}, so the sentences made true by the truth

definitions of Sect. 3.4 likewise do not change, and in particular the non-intuitive formula

O(p1 ∧ ¬p2/⊤) is still false, and definitions that accept ‘deontic detachment’ make true

O(p1 ∧ p2/⊤), so again you must go out and wear a scarf, which is as it should be.

• To refute the reconsidering and the fixpoint approaches the ordering of the imperatives

was again changed into p1⇒!p2 < !¬p2 < !p1. Now PoI(⊤}, I ) = {{p1⇒!p2, !¬p2}}.

All truth definitions make true O(¬p2/⊤) but not O(p1/⊤) so the agent must satisfy the

second ranking imperative, but not the third ranking imperative, which otherwise would

include violating the highest ranking imperative, which is as it should be.

• Troublesome for the fixpoint approach was also the set of prioritized imperatives p1⇒

!p2 < !(p1 ∧¬p2) < !p3: no fixpoint could be made out and so the approach produced no

preferred subset, making everything obligatory. The preferred maximally obeyable subset

is {p1⇒!p2, !p3}, eliminating the second ranking imperative that demands a violation of

the first, and making O(p3/⊤) true for all truth definitions, which again is as it should be.

• Finally, consider the ‘drinking and driving’ example: the set of prioritized imperatives

p1⇒!¬p2 < p3⇒!p2 < p3⇒!p1 produces, for the situation p3, the set of preferred max-

imally obeyable subsets PoI({p3}, I ) = {{p1⇒!¬p2, p3⇒!p2}}, making true O(p2/p3)

for all truth definitions of Sect. 3.4, so given that I go to the party I must do the driving,

which is as it should be.

As could be seen, all truth definitions now produce the ‘right’ results in the examples

used. Moreover, since all truth definitions refer to the same preferred subsets PoI({C}, I ), it

is possible to index the O-operators according to the truth definition employed, and e.g. state

truths like O1(A/C)∧ O3(B/C) → O4(A ∧ B/C), meaning that if, for any maximal set of

imperatives that I can obey in the situation C , imperatives are triggered that demand A, and

that if I satisfy all such triggered imperatives, I will have to do B, then obeying a maximal

number of imperatives includes having to do A ∧ B. It may well be that natural language

‘ought-statements’ are ambiguous in the face of conditional demands, the discussion in Sect.

3 suggested this. If maximal obeyability is accepted as the threshold criterion that limits what

norms an agent can be expected to conform to in a given situation, then definition 8 leaves

the philosophical logician with maximal freedom as to what deontic operator is chosen.

123



6 Theorems

Truth definitions (td-pcd1(+) − 4(+)) define when a sentence of the form O(A/C) is true

or false with respect to a prioritized conditional structure I and a situation C . So I briefly

consider what sentences are theorems, i.e. hold for all such structures, given the usual truth

definitions for Boolean operators. It is immediate that for all truth definitions, (DExt), (DM),

(DC), (DN) and (DD-R) are theorems (cf. Sect. 3.1). (DD-R) states that there cannot be both

an obligation to bring about A and one to bring about ¬A unless the situation C is logically

impossible, so our truth definitions succeed in eliminating conflicts. All these theorems are

‘monadic’ in the sense that C is kept fixed; in fact, they are the C-relative equivalents of

standard deontic logic SDL. More interesting are theorems (known from the study of non-

monotonic reasoning) that describe relations between obligations in different circumstances.

Obviously we have

(ExtC) If ⊢P L C ↔ D then O(A/C) ↔ O(A/D) is a theorem

for all truth definitions, i.e. for equivalent situations C , the obligations do not change. As

long as truth definitions are not sensitive to conflicts, e.g. for (td − cd(+) − 4(+)), we have

‘strengthening of the antecedent,’ i.e. for these definitions

(SA) O(A/C) → O(A/C ∧ D)

holds. When only maximally obeyable subsets are considered, i.e. for truth definitions (td-

pcd1(+) − 4(+)), both (SA) and the weaker ‘rational monotonicity’ theorem

(RM) ¬O(¬D/C) ∧ O(A/C) → O(A/C ∧ D)

are refuted e.g. by a set I = {!(p1 ∧ p2), !(p1 ∧ ¬p2), p2 ⇒¬p1} of equally ranking imper-

atives: though O(p1/⊤) is true and O(¬p2/⊤) false, O(p1/p2) is false. However, for all

definitions of Sect. 3.4, ‘(conjunctive) cautious monotonicity’

(CCMon) O(A ∧ D/C) → O(A/C ∧ D)

holds, which states that if you should to two things and you do one of them, you still have

the other one left.22 Moreover, truth definitions (td-pcd1+, 2+, 3+, 4+) validate the ‘circum-

stantial extensionality’ rule

(CExt) If ⊢P L C → (A ↔ B) then O(A/C) ↔ O(B/C) is a theorem

that corresponds to ‘circumstantial reasoning.’ All definitions that accept ‘reasoning by

cases,’ i.e. (td-pcd2, 2+, 4, 4+), make

(Or) O(A/C) ∧ O(A/D) → O(A/C ∨ D)

a theorem. Note that (CExt) and (Or) derive

(Cond) O(A/C ∧ D) → O(D → A/C),

which in turn derives (Or) in the presence of (DC), and that by adding (CExt) and (Or) we

obtain again the system PD (cf. Sect. 3). Finally, all definitions with ‘deontic detachment,’

i.e. (td-pcd3, 3+, 4, 4+), make

(Cut) O(A/C ∧ D) ∧ O(D/C) → O(A/C)

22 This is Hansson’s [15] theorem (19).

123



a theorem. (Cut) is derivable given (Cond) (use Cond on the first conjunct O(A/C ∧ D) to

obtain O(D → A/C), agglomerate with O(D/C), and from O(D ∧ (D → A)/C) derive

O(A/C)), which syntactically mirrors the semantic equivalence of definitions (td-pcd2+)

and (td-pcd4+). Theoremhood of all of the above for semantics that employ the respective

truth definitions is easily proved and left to the reader (cf. Hensen [13] and [14] as well as

Makinson and van der Torre [25] for the general outline). Makinson and van der Torre’s results

also suggest that these theorems axiomatically define complete systems of deontic logic with

respect to semantics that employ the respective truth definitions (td-pcd1(+) − 4(+)), but this

remains a conjecture that further study must corroborate.23

7 Back to the beginning: questions of representation

One might wonder if it is always adequate to represent a natural language conditional imper-

ative ‘if … then bring about that ww’ by use of a set I containing an imperative i with a

g(i) that formalizes ‘…’ and a f (i) that formalizes ‘ww.’ This is because there is a second

possibility: represent the natural language conditional imperative by an unconditional imper-

ative !(g(i) → f (i)). We saw in Sect. 3 that this is not generally adequate. But that does

not mean that such a representation is not sometimes what is required. Consider the crucial

imperatives in the previous examples: perhaps what your mother meant was simply ‘don’t

drink and drive;’ perhaps what the doctor meant was ‘don’t go out without a scarf;’ perhaps

the Colonel meant to tell O’Reilly not to do both, turn the heat on and keep the window

closed; perhaps self-defense required me to see to it that I am not attacked without fighting

back. These interpretations seem not wholly unreasonable, and if they are adequate, then the

best representation would be by an imperative !(g(i) → f (i)) instead of g(i) ⇒! f (i).

What then are the conditions that make a representation by an unconditional imperative

adequate? One test may be to ask: ‘Would bringing about the absence of the antecedent

condition count as satisfaction of the imperative?’ Would not drinking, not going out, not

turning on the heat, making the man not attack, count as properly reacting to the imperatives

in question? It should be if what the imperatives demand is a material conditional, since then

the conditional imperatives in question are equivalent to telling the agent ‘either don’t drink

or don’t drive, its your decision,’ ‘either don’t go out, or wear a scarf,’ ‘either don’t turn on

the heat, or open the window,’ etc. Another test would be to examine if contraposition is

acceptable. Can we say that your mother wanted you not to drink if you are going to drive,

that the doctor wanted you to stay inside if you are not going to wear a scarf, that the Colonel

wanted O’Reilly to turn off the heat if the window is closed, that self-defense requires you

to make the man not attack if you are not going to fight back? If the proper representation is

by imperatives that demand a material conditional, then the answers should be affirmative. I

do not think these are easy questions, however, and leave them to the reader to discuss and

answer at his or her own discretion. But it is easy to see that, had we chosen to represent the

crucial imperatives in the above examples (including the ‘drinking and driving’ problem) by

unconditional imperatives that demand a material implication to be realized, then all of the

discussed methods would have resolved these examples.

23 For (td-pcd2+, 4+), completeness of PD is immediate from the results in [13], [14].

123



8 Conclusion

Reasoning about obligations when faced with different and possibly conflicting imperatives

is a part of everyday life. To avoid conflicts, these imperatives may be ordered by priority

and then observed according to their respective ranks. The ‘drinking and driving’ case in the

introduction presented an example of such natural reasoning. Providing a formal account is,

however, additionally complicated by the fact that there are various and mutually exclusive

intuitions about what belongs to the right definition of an ‘obligation in the face of conditional

imperatives,’ i.e. the definition of a deontic O-operator. Based on similar definitions of oper-

ators by Makinson and van der Torre [25], [26] for their ‘input/output logic’, but leaving

the choice of the ‘right’ operator to the reader, I presented several proposals in Sect. 3 for

definitions of a dyadic O-operator, namely (td-pcd1(+)− 4(+)). These were dependent on

a choice of ‘preferred subsets’ among a given set of prioritized conditional imperatives. A

particularly successful method to identify such subsets, but applying to unconditional imper-

atives only, was Brewka’s [3] definition of ‘preferred subtheories’ within a theory. In Sect. 4

I discussed various approaches that extend this method to conditional imperatives, but these

failed to produce satisfactory results for a number of given examples. In Sect. 5 I proposed

that the maximality criterion used to construct the preferred subsets should be the avoidance

of conflict. A recent approach to this task by ‘consistent extensions’ was found to be biased

towards definitions of obligation that do not accept reasoning by cases; another, that aims

to avoid the truth of O(¬C/C) for possible circumstances C , produced satisfactory results

only for truth definitions that accept deontic detachment. I then argued that the solution is

to adapt Brewka’s method in such a way that it constructs, instead of maximal subsets of

imperatives that are collectively satisfiable by an agent, maximally obeyable subsets of the

imperatives, in the sense that the facts do not derive that some imperative of the set must be

violated. I showed that this new proposal provides adequate solutions to all of the examples,

and in particular the ‘drinking and driving’ example is resolved in a satisfactory fashion for

all of the discussed deontic operators.

Acknowledgments I am grateful to Lou Goble, John F. Horty, David Makinson and Leon van der Torre

for helpful comments and discussions. Also, I thank four anonymous referees for pointed remarks and useful

suggestions. A preliminary version of this paper was presented to the participants in the seminar “Normative

Multi-agent Systems (NorMAS’07)” held in Dagstuhl, Germany, from 18 to 23 March 2007.

References

1. Alchourrón, C. E., & Bulygin, E. (1981). The expressive conception of norms. In [18], 95–124.

2. Alchourrón, C. E., & Makinson, D. (1981). Hierarchies of regulations and their logic. In R. Hilpinen

(Ed.), New Studies in Deontic Logic (pp. 125–148). Dordrecht: Reidel.

3. Brewka, G. (1989). Preferred subtheories: An extended logical framework for default reasoning. In

N. S. Sridharan (Ed.), Proceedings of the eleventh international joint conference on artificial intelligence

IJCAI-89, Detroit, Michigan, USA, August 20–25, 1989 (pp. 1043–1048). San Mateo, Calif.: Kaufmann.

4. Brewka, G. (1994). Reasoning about priorities in default logic. In B. Hayes-Roth & R. E. Korf (Eds.),

Proceedings of the 12th national conference on artificial intelligence, Seattle, WA, July 31st–August 4th,

1994 (Vol. 2 pp. 940–945). Menlo Park: AAAI Press.

5. Brewka, G., & Eiter, T. (1999). Preferred answer sets for extended logic programs. Artificial Intelligence,

109, 297–356.

6. Broersen, J., Dastani, M., & van der Torre, L. (2002). Realistic desires. Journal of Applied Non-Classical

Logics, 12, 287–308.

7. Caminada, M., & Sakama, C. (2006). On the existence of answer sets in normal extended logic programs.

In G. Brewka, S. Coradeschi, A. Perini, & P. Traverso (Eds.), Proceedings of the 17th European con-

123



ference on artificial intelligence (ECAI 2006), August 29–September 1, 2006, Riva del Garda, Italy (pp.

741–742). Amsterdam: IOS Press.

8. Downing, P. (1959). Opposite conditionals and deontic logic. Mind, 63, 491–502.

9. van Fraassen, B. (1973). Values and the heart’s command. Journal of Philosophy, 70, 5–19.

10. Goble, L. (2005). A logic for deontic dilemmas. Journal of Applied Logic, 3, 461–483.

11. Greenspan, P. (1975). Conditional oughts and hypothetical imperatives. Journal of Philosophy, 72, 259–

276.

12. Hansen, J. (2004). Problems and results for logics about imperatives. Journal of Applied Logic, 2, 39–61.

13. Hansen, J. (2005). Conflicting imperatives and dyadic deontic logic. Journal of Applied Logic, 3, 484–511.

14. Hansen, J. (2006). Deontic logics for prioritized imperatives. Artificial Intelligence and Law, 14, 1–34.

15. Hansson, B. (1969). An analysis of some deontic logics. Nôus, 3, 373–398. Reprinted in [17], 121–147.

16. Hansson, S. O. (2001). The structure of values and norms. Cambridge: Cambridge University Press.

17. Hilpinen, R. (Ed.) (1971). Deontic logic: Introductory and systematic readings. Dordrecht: Reidel.

18. Hilpinen, R. (Ed.) (1981). New studies in deontic logic. Dordrecht: Reidel.

19. Hofstadter, A., & McKinsey, J. C. C. (1938). On the logic of imperatives. Philosophy of Science, 6,

446–457.

20. Horty, J. F. (2003). Reasoning with moral conflicts. Noûs, 37, 557–605.

21. Horty, J. F. (2007). Defaults with priorities. Journal of Philosophical Logic, 36, 367–413.

22. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumu-

lative logics. Artificial Intelligence, 44, 167–207.

23. Lewis, D. (1974). Semantic analysis for dyadic deontic logics. In S. Stenlund (Ed.), Logical theory and

semantic analysis (pp. 1–14). Dordrecht: Reidel.

24. Makinson, D. (1999). On a fundamental problem of deontic logic. In P. McNamara & H. Prakken (Eds.),

Norms, logics and information systems (pp. 29–53). Amsterdam: IOS.

25. Makinson, D., & van der Torre, L. (2000). Input/output logics. Journal of Philosophical Logic, 29, 383–

408.

26. Makinson, D., & van der Torre, L. (2001). Constraints for Input/output logics. Journal of Philosophical

Logic, 30, 155–185.

27. Marek, V. W., & Truszczynski, M. (1993). Nonmonotonic logic context-dependent reasoning. Berlin:

Springer.

28. Nebel, B. (1991). Belief revision and default reasoning: Syntax-based approaches. In J. A. Allen, R.

Fikes & E. Sandewall (Eds.), Principles of knowledge representation and reasoning: Proceedings of the

second international conference, KR ’91, Cambridge, MA, April 1991 (pp. 417–428). San Mateo: Morgan

Kaufmann.

29. Nebel, B. (1992). Syntax-based approaches to belief revision. In: P. Gärdenfors (Ed.), Belief revision (pp.

52–88). Cambridge: Cambridge University Press.

30. Prakken, H. (1997). Logical tools for modelling legal argument. Dordrecht: Kluwer.

31. Prakken, H., & Sartor, G. (1997). Argument-based logic programming with defeasible priorities. Journal

of Applied Non-classical Logics, 7, 25–75.

32. Rescher, N. (1964). Hypothetical reasoning. Amsterdam: North-Holland.

33. Rescher, N. (1966). The Logic of commands. London: Routledge & Kegan Paul.

34. Rintanen, J. (1994). Prioritized autoepistemic logic. In: C. MacNish, D. Pearce, & L. M. Pereira (Eds.),

Logics in artificial intelligence, European Workshop, JELIA ’94, York, September 1994, Proceedings (pp.

232–246). Berlin: Springer.

35. Ross, W. D. (1930). The right and the good. Oxford: Clarendon Press.

36. Ryan, M. (1992). Representing defaults as sentences with reduced priority. In B. Nebel, C. Rich, &

W. Swartout (Eds.), Principles of knowledge representation and reasoning: Proceedings of the third

international conference, KR ’92, Cambridge, MA, October 1992. (pp. 649–660). San Mateo: Morgan:

Kaufmann.

37. Sakama, C., & Inoue, K. (1996). Representing priorities in logic programs. In: M. Maher (Ed.), Joint

international conference and symposium on logic programming JICSLP 1996, Bonn, September 1996

(pp. 82–96). Cambridge: MIT Press.

38. Sosa, E. (1996). The logic of imperatives. Theoria, 32, 224–235.

39. Swirydowicz, K. (1994). Normative consequence relation and consequence operations on the language

of dyadic deontic logic. Theoria, 60, 27–47.

40. von Wright, G. H. (1968). An essay in deontic logic and the general theory of action. Amsterdam: North

Holland.

41. von Wright, G. H. (1984). Bedingungsnormen, ein Prüfstein für die Normenlogik. In W. Krawietz,

H. Schelsky, O. Weinberger, & G. Winkler (Eds.), Theorie der Normen (pp. 447–456). Berlin: Dunc-

ker & Humblot.

123



42. Watts, I. (1725). Logick: or, the right use of reason in the inquiry after truth, with a variety of rules to

guard against error in the affairs of religion and human life, as well as in the sciences. London: for John

Clark, Richard Hett et al.

123

Zusammenfassung

zur DissertationImperatives and Deontic Logic:On the Semantic Foundations of Deontic Logic

eingereicht an der Fakultat fur Sozialwissenschaften und Philosophie der Uni-versitat Leipzig

von Herrn M. A. Jorg Hansen

angefertigt am Institut fur Philosophie

offentlich verteidigt am 18. November 2008

Im Jahr 1671 wies Gottfried Wilhelm Leibniz auf die Ahnlichkeit der”juristi-

schen“ (deontischen) Modalitaten des Gebotenen, Erlaubten und Verbotenenmit den

”naturlichen“ (alethischen) Modalitaten des Notwendigen, Mogli-

chen und Unmoglichen hin und stellte die These auf, daß alle bekannten

”Komplikationen, Transpositionen und Oppositionen“ zwischen diesen auf

jene ubertragen werden konnen. Mit seinem Aufsatz”Deontic Logic“ stellte

Georg Henrik von Wright 1951 den ersten Modalkalkul vor, der dieser Be-hauptung Rechnung tragt. Sein zum Standardsystem der deontischen Logikweiterentwickeltes Axiomensystem ist ein normaler Modalkalkul, in welchemfreilich das zusatzliche Axiom (T) 2A→ A der alethischen Modallogik nichtgilt, sondern stattdessen das Axiom (D) OA → PA: Was geboten ist, ist –

274

ANHANG: ZUSAMMENFASSUNG 275

anders als das Notwendige – zwar nicht zwingend auch der Fall, aber zumin-dest ist das, was geboten ist, zugleich auch erlaubt. (D) ist in diesem Systemaquivalent zu ¬(OA∧O¬A): Wenn A geboten ist, kann nicht zugleich non-Ageboten sein. Eine interessante weitere Beziehung zwischen der deontischenLogik und der alethischen Modallogik wurde 1957 von Alan Ross Andersonaufgezeigt: Die deontische Logik kann auf die alethische Modallogik reduziertwerden, wenn OA als 2(¬A→ S) definiert wird – A ist geboten, wenn non-Anotwendig S nach sich zieht, wobei die Konstante S fur das Eintreten einerSanktion steht. Unterstellt man, daß ¬S moglich ist, also das Eintreten einerSanktion vermieden werden kann, so wird das Standardsystem der deonti-schen Logik mittels einer solchen Definition zu einem Fragment des (um dieKonstante S bereicherten) alethischen Modalkalkuls.

Wie die Ausdrucke der alethischen Modallogik werden auch die Aus-drucke der deontischen Logik uberwiegend mithilfe einer

”Mogliche-Welten-

Semantik“ interpretiert: OA,”der mit A beschriebene Zustand ist geboten“,

ist wahr, wenn A in allen Idealwelten zu unserer aktuellen Welt wahr ist. Mit-hilfe einer logischen Semantik, deren Modelle eine Menge moglicher Welten,eine zwischen ihnen bestehenden Idealitatsrelation, sowie eine Wahrheits-funktion angeben, die den Zustand in jeder Welt beschreibt, kann sodanndie Gultigkeit aller Satze und die Vollstandigkeit des Standardsystem derdeontischen Logik und seiner Abwandlungen bewiesen werden.

Die deontische Logik wird bis heute gelegentlich als eine”Logik der Nor-

men“ charakterisiert. Diese Darstellung ist freilich falsch, denn Normen kon-nen weder wahr noch falsch sein, sie wollen die Welt nicht beschreiben, son-dern etwas in ihr bewirken. Nur wahre oder falsche Satze konnen aber Ein-setzunginstanzen fur OA und PA etwa in dem obigen Schema (D) sein, denndie durch den Pfeil

”→“ausgedruckte

”wenn ... dann“ Beziehung ist wahr-

heitsfunktional: Der Satz ist falsch genau dann wenn A geboten ist, aber Agleichwohl nicht erlaubt ist (weil z.B. ¬A ebenfalls geboten ist). Wenn OAdeshalb keine Norm sein kann, sondern ein beschreibender Satz ist, so geltendie normalen Regeln: OA, also

”A ist geboten“, ist genau dann wahr wenn

A geboten ist, wenn es also eine Norm gibt, die A gebietet. Die deontischeLogik versteht daher ihre Ausdrucke ublicherweise als wahre oder falsche Be-hauptungen daruber, was gemaß einem (meist nicht naher beschriebenen)Normsystem geboten, verboten oder erlaubt ist.

Wenn sich aber die Wahrheit der Satze der deontischen Logik nach demrichtet, was gemaß tatsachlich bestehender Normen als geboten, verbotenoder erlaubt anzusehen ist, muß sie die Eigenheiten bestehender Normsy-


steme abbilden konnen. Hierzu gehort zunachst die Begleitung von Primar-normen durch Sekundarnormen (oder

”contrary-to-duty imperatives“), die

Pflichten fur solche Falle statuieren, in denen die Primarnormen verletztwerden. Solche Sekundarpflichten sind etwa: sich entschuldigen, den Scha-den wiedergutmachen, eine Strafe zahlen usw. Hierzu gehort aber auch dieMoglichkeit eines Normenkonflikts: Es mag, durch einen Fehler des Gesetz-gebers oder den Irrtum eines Richters, zwei oder mehr Normen geben, dieGegensatzliches verlangen. Haufig gibt es aber in Fallen von Normenkonflik-ten zugleich Losungen mithilfe von Prioritatsregelungen: die Norm des Ver-fassungsrechts hat Vorrang vor der des normalen Gesetzesrecht, die spatereNorm vor der fruheren, die speziellere vor der allgemeinen, diejenige, die be-deutende Rechtsguter schutzt, hat Vorrang vor bloßen Ordnungsregeln, usw.

Sowohl axiomatische wie semantische Ansatze der traditionellen deonti-schen Logik geraten bei der Bewaltigung dieser Besonderheiten in Schwierig-keiten. Das Standardsystem der deontischen Logik hat sich zur Beschreibungvon Sekundarnormen als unbrauchbar erwiesen und wurde durch Systememit zweistelligen deontischen Operatoren ersetzt: In der sogenannten dyadi-schen deontischen Logik bedeutet O(A/B):

”A ist geboten in der Situation

B“. In der logischen Semantik wurden die Idealwelten durch eine Praferenz-relation zwischen moglichen Welten ersetzt: O(A/B) ist nun wahr, wenn Ain den besten aller moglichen B-Welten der Fall ist. Noch schwieriger istdie Darstellung von Normenkonflikten: wenn es moglich ist, daß sowohl OAwie auch O¬A wahr sind, ist (D) als Axiom widerlegt. Aber auch die ubri-gen Axiome des Standardsystems mussen modifiziert werden, damit aus OAund O¬A nicht OB folgt (sogenannte

”deontische Explosion“). Semantisch

reichen die Abhilfevorschlage vom Einsatz minimaler Modelle (sog. Nach-barschaftsemantiken, bei denen nicht eine, sonderen mehrere Mengen vonWelten gegenuber einer anderen

”ideal“ sind) bis zu Multipraferenzseman-

tiken, bei denen nicht eine, sondern mehrere Praferenzrelationen die jeweilsbesten aus einer gegebenen Menge von Welten auswahlen. Noch schwieri-ger ist die Rekonstruktion von Konfliktlosungsstrategien mittels Prioritaten:Hier mußten konsequenterweise Praferenzrelationen zwischen Mengen idealerWelten (

”beste Nachbarschaften“) oder zwischen anderen Praferenzen zum

Einsatz kommen, selbst wenn dies meines Wissens noch nirgendwo versuchtworden ist. Alle diese Semantiken haben jedenfalls den Nachteil, daß sie nichtsonderlich intuitiv sind.

In der Behandlung von Beispielen hat es sich als besonders unvorteilhafterwiesen, daß es in der Sprache der deontischen Logik unmoglich ist, unmittel-


bar auf bestimmte Normen Bezug zu nehmen. So wurde etwa vorgeschlagen,in dem Fall, daß sowohl O(A∧B) als auch O(¬B ∧C) wahr sind, zwar keineKumulierung der Gebotsinhalte zuzulassen, denn O(A ∧ B ∧ ¬B ∧ C) wareunerfullbar, wohl aber die disjunktive Erfullung beider Gebote zu fordern,also O((A ∧ B) ∨ (¬B ∧ C)). Damit wurde der Normadressat im Falle einesKonflikts zur Erfullung wenigstens eines Gebots angehalten. Doch die SatzeO(A ∧ B) und O(¬B ∧ C) mussen nicht deshalb wahr sein, weil zwei be-stimmte Normen genau A∧B und ¬B ∧C fordern: Vielleicht sind die Satzedeshalb wahr, weil zwei Normen A∧B∧¬C und ¬A∧¬B∧C fordern. Dannstellt O((A ∧ B) ∨ (¬B ∧ C)) allein das widersinnige Gebot dar, wenigstenseine Norm zu verletzen. Ebenso ist angedacht worden, unter Berucksichti-gung von Normenkonflikten eine Kumulation zweier Gebote OA und OB zuO(A∧B) gleichwohl dann zuzulassen, wenn A∧B konsistent ist (sogenannte

”konsistente Agglomeration“). Doch OA und OB mogen deshalb wahr sein,

weil zwei Normen A∧C und B∧¬C fordern – es macht dann keinen Sinn, vonden Normadressaten sowohl die Realisierung von A als auch B zu fordern,wenn doch nur eine dieser beiden Normen erfullt werden kann: Das Tun vonentweder A oder B ware uberflussig.

In dieser Situation ist es kein Wunder, wenn sogar in Standardwerken zurdeontischen Logik aus Gebots- und Erlaubnissatzen klammheimlich Impera-tive oder

”deontische Normen“ werden, etwa gesagt wird, daß ¬A die

”deon-

tische Norm“ OA verletze. Daß dies unzulassig ist, habe ich oben festgestellt.Und so war auch meine eigene Ausdrucksweise im vorigen Absatz irrefuhrend,denn OA ist kein Gebot, sondern ein Satz, daß etwas Bestimmtes gebotenist, er kann nicht verletzt oder erfullt, sondern nur wahr oder falsch sein;er ist bestenfalls eine Pramisse. David Makinson sah sich deshalb wahrenddes Workshops ∆EON’98 in Bologna 1998 zu der Feststellung veranlaßt, dieArbeiten zur deontischen Logik erweckten vielerorts den Eindruck, als habeder Autor noch nie von einer Unterscheidung zwischen Normen und deon-tischen Satzen uber Normen gehort. Die deontische Logik sei in Einzelteilezerfallen: in zahlreiche Axiomatisierungen, und in noch mehr unterschiedli-che Mogliche-Welten-Semantiken mit ihren unzahligen Detailabweichungen.Er forderte dazu auf, die deontische Logik zu

”rekonstruieren“ : als eine Lo-

gik, die mit Normen befaßt ist, aber respektiert, daß Normen weder wahrnoch falsch sein konnen.

Diesem Ruf folgend habe ich seither in einer Reihe von Forschungsarbeitenuntersucht, wie sich mithilfe einer

”imperativistischen Semantik“ die deonti-

schen Operatoren neu definieren lassen, und welche alten oder neuen Systeme


der deontischen Logik hierzu vollstandig alle gultigen Satze enthalten. DasGrundkonzept ist denkbar einfach: Sei I eine Menge (von Imperativen) undsei f eine Funktion, die jedem Objekt dieser Menge (jedem Imperativ) einenSatz aus einer formalisierten Sprache (etwa der Aussagenlogik) zuordnet. DieVorstellung ist, daß dieser Satz beschreibt, was der Fall ist, wenn der Impera-tiv erfullt ist, und was nicht der Fall ist, wenn der Imperativ verletzt ist. Daßes fur jeden Imperativ einen solchen Satz geben muß, ist erkennbar unstreitig,denn anderenfalls konnte der Adressat des Imperativs nicht verstehen, wasvon ihm verlangt wird. Wir konnen nun z.B. Operatoren des Typs OA defi-nieren, die wahr sind, wenn A genau einen Imperativ erfullt, wenn A mehrereImperative erfullt, wenn A notwendig zur Erfullung eines Imperativs, meh-rerer Imperative oder aller Imperative ist. Wir konnen in einer bestimmtenSituation dasjenige als geboten beschreiben, was zur Erfullung aller Impera-tive erforderlich ist, die in dieser Situation noch erfullt werden konnen; wirkonnen Konflikte zwischen den Imperativen zulassen und fragen, was not-wendig ist, um eine oder alle maximalen Mengen von nicht miteinander inKonflikt stehenden Imperative zu erfullen; wir konnen Prioritatsbeziehun-gen zwischen den Imperativen darstellen und untersuchen, auf welche Weisediese zur Losung mancher oder aller Konflikte beitragen konnen. Schließlichkonnen wir bedingte Imperative abbilden, indem wir etwa eine zweite Funk-tion g jedem Imperativ einen

”Ausloser“ zuordnen lassen, einen Satz, der

beschreibt, unter welcher Bedingung der Imperativ zu erfullen ist (und ver-letzt werden kann). O(A/B) mag dann etwa in einer bestimmten Situation Bdasjenige als geboten beschreiben, was zur Erfullung aller in dieser Situation

”ausgeloster“ Imperative erforderlich ist.

Es erweist sich sodann, daß das Standardsystem der deontischen Logikeiner Definition entspricht, welche dasjenige als geboten beschreibt, was zurErfullung eines oder mehrerer Imperative erforderlich ist, wenn zugleich un-terstellt wird, daß die Menge der Imperative nicht leer ist und die Imperativenicht miteinander in Konflikt stehen. Die beiden Hauptsysteme der dyadi-schen deontischen Logik entsprechen solchen Definitionen, die dasjenige ineiner bestimmten Situation als geboten beschreiben, was zur Erfullung allerverbliebenen, in dieser Situation noch nicht verletzten Imperative erforder-lich ist, wenn wir unterstellen, daß eine totale Prioritatsrelation etwa nacheiner von Gerhard Brewka entwickelten Methode alle fur diese Situation ent-stehenden Normenkonflikte auflost. Auch die Andersonsche Reduktion derdeontischen Logik auf die alethische Modallogik laßt sich mittels der neuenSemantik rekonstruieren: Seiner Sanktionskonstante S entspricht dabei die


Feststellung, daß mindestens ein Imperativ verletzt ist. Das Projekt einerRekonstruktion der traditionellen deontischen Logik ist damit abgeschlos-sen. Zugleich treten neben die traditionellen Systeme jedoch weitere. Hierzugehoren etwa solche, deren zugehorige Semantik nicht die Auflosung alleretwaigen Normenkonflikte verlangt, diese entsprechen im wesentlichen nicht-monotonen Logiken, wie sie etwa von Kraus, Lehmann und Magidor odervon Alexander Bochman entworfen wurden.

Meine Arbeit beschaftigt sich zunachst mit der Frage, ob es nicht eine

”Logik der Imperative“ gibt. Gabe es sie, wurde eine Rekonstruktion der

deontischen Logik als Logik von Satzen daruber, was gemaß einer Menge vonangenommenen Imperativen geboten und erlaubt ist, denkbar einfach sein:Wir konnten eine solche Menge von Imperativen nach deren eigenen logischenGesetzmaßigkeiten abschließen, und die Satze der deontischen Logik wurdendann nur noch spiegelbildlich beschreiben, welche Imperative in einem derartabgeschlossenen

”System“ existieren oder nicht existieren. Die Diskussion in

Kapitel 1 ergibt jedoch, daß die Existenz unmittelbarer logischer Beziehun-gen zwischen Imperativen nicht unterstellt werden kann. Hierfur liefere ich einArgument in der Tradition der

”Ordinary Language Philosophy“, und damit

ist zugleich klargestellt, daß mein Ausgangspunkt die Position eines”Impe-

rativologischen Skeptizismus“ ist. Im Kapitel 2 erlautere ich die spater zu re-konstruierenden traditionellen Systeme der deontischen Logik und beschrei-be dann die Schwierigkeiten, die zu Makinsons Aufforderung zu ihrer nor-menbezogenen Rekonstruktion gefuhrt haben. Kapitel 3 wendet sich sodannder

”imperativistischen“ oder

”normativen Tradition“ der deontischen Logik

zu, denn einige Autoren haben bereits in Abweichung von der herkommli-chen Mogliche-Welten-Semantik versucht, die deontischen Operatoren mit-tels explizit angegebener Normsysteme oder Imperativmengen zu definieren.Sodann erlautere ich die zur Rekonstruktion der traditionellen deontischenSysteme verwendete Semantik und beschreibe zwei wichtige Grundvoraus-setzungen: die Unabhangigkeit und die Untrennbarkeit der in der logischenSemantik modellierten Imperative. Die oben bereits kurz erlauterten Ergeb-nisse meiner Forschungsarbeiten zu dieser Semantik werden anschließend zu-sammengefaßt. Abschließend wende ich mich den sogenannten

”Paradoxien

der deontischen Logik“ zu und untersuche, wie der neue semantische Ansatzmit diesen Paradoxien umgehen kann.

In gewisser Weise bedeutet eine”imperativistische Semantik“ einen Pa-

radigmenwechsel fur die deontische Logik. Verstand sich die deontische Logikzuvor, etwa im Spatwerk Georg Henrik von Wrights, als Ratgeber eines ra-


tionalen Gesetzgebers, so steht im Vordergrund der vorgeschlagenen Seman-tik fur die deontische Logik das rationale Normsubjekt: Wie laßt sich mitNormen umgehen, wenn diese Normenkonflikte enthalten konnen, wenn wirin vom Gesetzgeber unvorhergesehene dilemmatische Situationen geraten,wenn das schrittweise entwickelte Normsystem Vorschriften enthalt, die eineinziger rationaler Gesetzgeber moglicherweise nicht erlassen hatte? KonnenKonflikte isoliert werden, laßt sich sagen, was gleichwohl noch geboten bleibt,und wie sehen die Mechanismen aus, mit denen rationale Normsubjekte Nor-menkonflikte mittels Prioritatsbeziehungen tatsachlich losen? Wenn man sowill, kann man dies auch als

”positivistische deontische Logik“ bezeichnen:

Eine Rationalitat des Gesetzgebers wird nicht vorausgesetzt, wohl aber einuberlegendes Normsubjekt. Neben solchen grundsatzlichen Erwagungen zuunserem Projekt wird im Schlußteil auf fortbestehende Schwierigkeiten hinge-wiesen. Gebotsausdrucke lassen sich in Bezug auf konditionale Imperative inverschiedener Weise definieren, konnen z.B. die sogenannte Wahlfeststellungzulassen (wenn A∨B der Fall ist, und A und B beide Imperative

”auslosen“,

die nur zu erfullen sind, wenn C wahr ist, ist dann bereits C geboten?), odereinen normativen Kettenschluß (wenn A der Fall ist, und einen Imperativ

”auslost“, der seinerseits zu seiner Erfullung die Wahrheit von B erfordert,

ist dann auch geboten, was ein Imperativ verlangt, der durch B”ausgelost“

wird?), und schließlich umstandebezogenes Argumentieren zulassen (wennA ∧ B der Fall ist, und A einen Imperativ

”auslost“, zu dessen Erfullung

¬B ∨ C erforderlich ist, ist dann C geboten?). Diese Argumentationswei-sen konnen jedoch nur in bestimmten Kombinationen zugelassen werden,und fuhren in anderen Kombinationen zu Paradoxien: Welches ist also die

”richtige“ Gebotsdefinition fur konditionale Imperative? Erlaubende Normen

sind bislang vernachlassigt worden: Sie sollten sinnvollerweise nicht einfach

”neben“ Imperative gestellt werden, sondern das bislang Gebotene so modi-

fizieren, daß etwas moglicherweise vormals Verbotenes nun als erlaubt gilt.Dies und anderes, wie etwa die Reprasentation genereller Normen und vonNormen, die die Erzeugung von Normen regeln, stellen weitere Herausforde-rungen fur die Entwicklung einer imperativ- bzw. normbezogenen Semantikfur die deontische Logik dar.

Bibliographische Beschreibung

Hansen, JorgImperatives and Deontic Logic: On the Semantic Foundations ofDeontic Logic

Universitat Leipzig, Diss., 120 S., 260 Lit.

Referat:

Die Satze der deontischen Logik werden traditionell mithilfe der”Mogliche-

Welten-Semantik“ interpretiert. Andererseits versteht die deontische Logik,in Abgrenzung von einer

”Normenlogik“, ihre Satze als wahre oder falsche

Aussagen daruber, was gemaß einem meist nicht naher beschriebenen Sy-stem von Normen geboten, verboten oder erlaubt ist. Dieser Erklarung ih-res Gegenstands folgend haben einige Autoren versucht, die Wahrheit oderFalschheit der Satze der deontischen Logik nicht mit Rucksicht auf moglicheWelten zu interpretieren, sondern mittels einer explizit angegebenen Norm-menge oder Menge von Imperativen, und was zu ihrer Erfullung erforderlichist. Es kann dann gezeigt werden, daß sich im Sinne dieser

”imperativisti-

schen Tradition“ der deontischen Logik die traditionellen axiomatischen Sy-steme der monadischen und dyadischen Logik

”rekonstruieren“ lassen. Zu-

gleich ermoglicht dieser logisch-semantische Ansatz die Konstruktion ande-rer Systeme, die etwa Konflikte zwischen Imperativen zulassen oder solcheKonflikte mittels Vorrangbeziehungen zwischen den Imperativen ganz oderteilweise losen. Die Arbeit erlautert die Grundidee dieser

”imperativistischen

Semantik“, ihre Anwendungen sowie die bereits erzielten Resultate.

281

Lebenslauf

09.09.1967 Geburt in Cochem an der Mosel

01/1985 – 12/1985 Bowen State High School, Bowen, N.Q., Australien

21.06.1988 Abitur

10/1988 – 03/1991 Studium an der Universitat Bielefeld: Rechtswissen-schaft (Staatsexamen), Philosophie (Magister).

05/1989 – 03/1991 Hilfskraft am Lehrst. f. Offentliches Recht (Dr. Wer-ner Heun) u. f. Europarecht (Prof. Dr. Meinhard Hilf)

04/1991 – 06/1997 Studium an der Universitat Hamburg: Rechtswissen-schaft (Staatsexamen), Philosophie (Magister).

05/1991 – 09/1993 Hilfskraft am Seminar fur Europarecht (Prof. Dr.Meinhard Hilf)

08/1992 – 10/1999 Mitarbeiter der Verbraucher-Zentrale Hamburg e.V.

05.08.1994 Erste Juristische Staatsprufung

04/1995 – 09/1995 Tutor fur Logik am Philosophischen Seminar (Dr. AliBehboud)

10/1995 – 03/1996 Tutor fur Logik am Philosophischen Seminar (Prof.Dr. Werner Diederich)

02/1997 – 06/1999 Rechtsreferendar am HansOLG

02.06.1997 Magister Artium der Universitat Hamburg, Fachbe-reich Philosophie und Sozialwissenschaften

04.06.1999 Zweite Juristische Staatsprufung

14.09.1999 Zulassung als Rechtsanwalt

09/1999 – Rechtsanwalt, Kanzlei Hansen & Varwig, Eisenach

10/1999 – Promotionsstudium an der Universitat Leipzig

282

ANHANG: LEBENSLAUF 283

10/2000 – 03/2001 Hilfskraft am Institut fur Logik und Wissenschafts-theorie (Dr. Peter Steinacker)

07/2001 – 12/2001 Geschaftsfuhrer Bachhaus Eisenach gGmbH

12/2005 – Geschaftsf. u. Direktor Bachhaus Eisenach gGmbH

21.11.2007 Fachanwalt f. Arbeitsrecht

Publikationen

”Paradoxes of Commitment“, in: Meggle, G. (Hrsg.), Actions, Norms, Values.

Discussions with Georg Henrik von Wright, De Gruyter: Berlin, New York,1999, 255–263.

”On Relations between Aqvist’s Deontic System G and Van Eck’s Deontic

Temporal Logic“, in: McNamara, P. und Prakken, P. (Hrsg.), Norms, Logicsand Information Systems, IOS Press: Amsterdam, 1999, 127–144.

”Sets, Sentences, and Some Logics about Imperatives“,Fundamenta Informa-

ticae, 48, 2001, 205–226.

”Problems and Results for Logics about Imperatives“, Journal of Applied

Logic, 2, 2004, 39-61.

”Conflicting Imperatives and Dyadic Deontic Logic“, Journal of Applied Lo-

gic, 3, 2005, 484–511.

”Deontic Logics for Prioritized Imperatives“, Artificial Intelligence and Law,

14, 2006, 1–34.

”The Paradoxes of Deontic Logic: Alive and Kicking“, Theoria, 72, 2006,

221–232.

”Prioritized Conditional Imperatives: Problems and a New Proposal“, Auto-

nomous Agents and Multi-Agent Systems, 2007, 17, 2008, 11–35.

Mit Gabriella Pigozzi und Leon van der Torre:

”Ten Philosophical Problems in Deontic Logic“, in: Boello, G., van der Torre,

L. und Verhagen, H., Normative Multi-Agent Systems, Dagstuhl SeminarProceedings 07122, Internationales Begegnungs- und Forschungszentrum furInformatik (IBFI), Schloss Dagstuhl, 2007. Online verfugbar unterhttp://drops.dagstuhl.de/opus/volltexte/2007/941.

284

http://drops.dagstuhl.de/opus/volltexte/2007/941

Vortrage

10.01.1998”On Relations between Aqvist’s Deontic System G and Van Eck’s

Deontic Temporal Logic“ (Workshop ∆EON’98 in Bologna, Italien)

22.01.2000”Sets, Sentences, and Some Logics about Imperatives“ (Workshop

∆EON’00 in Toulouse, Frankreich)

22.05.2002”Problems and Results for Logics about Imperatives“ (Workshop

∆EON’02 in London, England)

28.05.2004”Conflicting Imperatives and Dyadic Deontic Logic“ (Workshop

∆EON’04 in Madeira, Portugal)

27.05.2005”Deontic Logics for Prioritized Imperatives“ (Arbeitsgruppe

”Law

and Logic“ i.R.d. Weltkongresses der Internationalen Vereinigungfur Rechts- und Sozialphilosophie IVR 2005, Granada, Spanien)

20.03.2007”Prioritized Conditional Imperatives: Problems and a New Propo-

sal“ (Seminar NorMAS’07, Internationales Begegnungs- und For-schungszentrum fur Informatik IBFI, Schloß Dagstuhl)

20.03.2007”Ten Philosophical Problems in Deontic Logic“ (mit Gabriella Pi-

gozzi und Leon van der Torre, Seminar NorMAS’07, InternationalesBegegnungs- und Forschungszentrum fur Informatik IBFI, SchloßDagstuhl)

05.08.2007”Conditional Imperatives and Dyadic Deontic Logic“ (Arbeitsgrup-

pe”Legal Argumentation and Legal Logic“ i.R.d. Weltkongresses

der Internationalen Vereinigung fur Rechts- und SozialphilosophieIVR 2007, Krakau, Polen)

285

ANHANG: VORTRAGE 286

28.04.2008”Ten Problems in Deontic Logic“ (mit Gabriella Pigozzi, Institut

d’histoire et de philosophie des sciences et des techniques, Sorbonne,Universite de Paris)

04.-15.08.2008”Deontic Logic in Computer Science“ (Seminar mit Leon van der

Torre, ESSLI 2008, 20th European Summer School in Logic, Lan-guage and Information, Hamburg)

Versicherung

Hiermit versichere ich, daß ich die vorliegende Arbeit ohne unzulassige HilfeDritter und ohne Benutzung anderer als der angegebenen Hilfsmittel ange-fertigt habe; die aus fremden Quellen direkt oder indirekt ubernommenenGedanken sind als solche kenntlich gemacht.

Bei der Auswahl und Auswertung des Materials sowie bei der Herstellungdes Manuskriptes habe ich Unterstutzungsleistungen von folgenden Personenerhalten: dem englischsprachigen Korrekturleser Herrn Jason F. Ortmann.Weitere Personen waren an der geistigen Herstellung der Arbeit nicht be-teiligt. Insbesondere habe ich nicht die Hilfe eines Promotionsberaters inAnspruch genommen. Dritte haben von mir weder unmittelbar noch mittel-bar geldwerte Leistungen fur Arbeiten erhalten, die im Zusammenhang mitdem Inhalt der vorgelegten Dissertation stehen.

Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oderahnlicher Form einer anderen Prufungsbehorde vorgelegt und ist auch nochnicht veroffentlicht worden.

287

Imperatives and Deontic Logic

Documents

Transcript of Imperatives and Deontic Logic