Dynamic Semantics, Imperative Logic and Propositional Attitudes

8/14/2019 Dynamic Semantics, Imperative Logic and Propositional Attitudes

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Dynamic Semantics,Imperative Logic and

Propositional Attitudes

Berislav Zarnic

UPPPUPPSALA PRINTS AND PREPRINTS IN PHILOSOPHY

2002 NUMBER 1


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Contents

1 One move eliminative semantics 21.1 Some remarks . . . . . . . . . . . . . . . . . . . . 2

1.2 Semantic notions . . . . . . . . . . . . . . . . . . 5

2 Simple update system for imperatives 9

2.1 Three moves language L! for imperative logic 122.2 Reduction of uncertainty in the practical setting 15

3 Prima facie consequence 18

3.1 Nonpersistent sentences and preferred model . . 20

4 Negation of imperatives and notion of change 244.1 Extended language L!! and refined models . 27

4.1.1 Explanation of semantic moves: ! and

! . . . . . . . . . . . . . . . . . . . . . 294.1.2 Conditional imperative . . . . . . . . . . . 32

5 Semantics of propositional attitudes reports 355.1 Characterization of intentional states . . . . . . . 375.2 Validity of rationalizations . . . . . . . . . . . . . 39

5.2.1 Two modes of validity for rationalizations 395.3 Application: anatomy of excuse . . . . . . . . . . 42

6 Appendix1 47

1 Appendix by Berislav Zarnic and Damir Vukicevic

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1 One move eliminative seman-ticsVeltmans simple update system. The main features of the

system are given in the following citation.

Let W be the powerset of the set A of atomicsentences. is an information state iff W; 0,the minimal state, is the information state given byW; 1, the absurd state is the information state givenby the empty set;[...]

[p] = {w W | p w}

[] = [] [ ] = [] [] [ ] = [] [] [might ] = if [] 6= 1 [might ] = 1 if [] = 1[...] If [] 6= 1, is acceptable in . If [] 6=

1, is not acceptable in and if [] = , isaccepted in . [...] sequence of sentences 1; ...; n

is consistent iff there is an information state suchthat [1] ... [n] 6= 1. [32] p.228.

1.1 Some remarks

In the dynamic semantical framework one is not supposed tothink of meaning only as a relation between a sentence and theworld (including possible ones). Rather meaning is conceived asresult of an interpretation process. In this case, interpretationis understood as elimination process. In a metaphor, interpreterentertains different candidate pictures of yet unknown situation,sentences are picture fragments, and interpretation consists ofcomparing picture fragments with candidate pictures and re-

moving the nonfitting ones. The number of remaining picturesmeasure the amount of information, the more pictures remain

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the less information a text conveys. The residual pictures createa context for the next step in interpretation process. The truthvalue of a sentence may be unsettled at a particular stage if thecontext contains both pictures having and pictures lacking thefragment. Sentence being true in a context means that the con-

text is not empty and that the sentence does not have a powerto eliminate any picture from it.

One-move system. The simple update system is basicallya one move system with basic instruction [] = kkW,where kkW is a set of valuations v W verifying in thestandard sense.

Context dependent intension. A sentence gets its mean-ing in the context created by other sentences. A context depen-dent intension [] of sentence is the intersection of its contextfree intension kkW and a context .

Semantic value. Sentences are state or context transitions.Relative to a context a sentence may has one among foursemantic values. A sentence is either () accepted and accept-able, or () not accepted and acceptable, or () not acceptedand not acceptable, or () accepted and not acceptable in a

given context.

V(, ) =

if [] = [] 6= 1 if [] 6= [] 6= 1 if [] 6= [] = 1 if [] = [] = 1

Update paths. We call a sentence basic if its meaning is

defi

ned only in terms of the basic intersective action (e.g. p

q isbasic since [p q] = kp qkW). The meaning of a complexsentence is defined in terms of executability of basic actions (e.g.might p is complex sentence).

Define meaning potential of a sentence in the simple up-date system dyn relative to the set of contexts as a relationkkdyn = {h,

0i | [] = 0}, where = 2W.

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transition type kkdyn success basic test

[] =

h, 0i 2 |

V (, ) = V (, 0) =

+ + +

[] =

h, 0i 2 |

V (, ) = V (, 0) =

+ + -

[] =

h, 0i 2 |V (, ) = V (, 0) =

- + -

[] =

h, 0i 2 |

V (, ) = V (, 0) =

- + +

Since the simple update system is eliminative, there are onlyfour transition types. If one is interested in meaning relationsrather then in modelling of cognitive dynamics, he may disre-

gard 12 remaining transition types. E.g. revision transitiontype [] requires some worlds to be restored.

Distinction between basic and test sentences is visible intheir transition types. Basic sentences allow of all four types,while tests allow only of two. Tautology is identity relation:k>kdyn = [>] [>] =

h, 0i 2 | = 0

, while con-

tradiction is all-to-one relation: kkdyn = [] [] =h,

0

i 2

| 0

=

.

pqp

q

-

pq

p

q

p

q

-

pq

p

-

pq

q

-

pq

q

-

pq

q

pq

-

p

q

p

-

pq

p

p

q

-

absurd

1

Contexts in which

might p is accepted

Sentence p changes

context

Update paths for sentence p in the set of contexts = 2W.

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A comparison with ukasiewicz three-valued sys-tem. The eliminative semantics provides a simple solutionfor the problem of unknown truth-value. In order to compareeliminative system with ukasiewicz three-valued system wecan connect dynamic values with static ones: dynamic value

corresponds to 1, to 12 , and (i.e. value in a nonab-surd state and value in the absurd state, respectively) corre-spond to 0. Counterintuitive principle ( )is valid in ukasiewicz logic, but it is not valid in the simpleupdate system. If both p and p are epistemically possible,then surely p p is not epistemically possible. In ukasiewiczsystem, ifV(p) = 12 and V(p) =

12 , then V(p p) =

12 and

V( (p p)) = 1. In the simple update system, ifV(p, ) = and V(p, ) = , then V(p p, ) = and V( (p p) , ) =.

static

1 12 0

1 1 12 012

12

12 0

0 0 0 0

1 012

12

0 1

1 112 10 0

dynamic1

?

might

1.2 Semantic notions

Consequence relations. No adding metaphor of valid conclu-sion is explicated as no removal case. Within this framework onecan define different variants of no removal consequence, three

1 indicates that a particular combination of semantic values is notpossible. ? indicates that value cannot be calculated for the general case;in some cases = and in others = .

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variants are discussed in [32] (reproduced here in a slightly mod-ified notation and denoted by 0ut, ut, tt). It is important tonote that using dynamic semantics one can define other mean-ing relations that resemble consequence relation, one of them isadded here, namely uu[5].

1. 0-update-to-test consequence: 1; ...; n 0ut iff

0 [1] ... [n] = 0 [1] ... [n] []

2. Update-to-test consequence: 1,..., n ut iff

: [1] ... [n] = [1] ... [n] []

3. Test-to-test consequence: 1

; ...; n

tt iff

: [1] = ... = [n] [1] [] = ... = [n] []

4. Update-to-update consequence: 1; ...; n uu

: [1] ... [n] 6= 1 [1] ... [n] [] 6= 1

pq

p

q

-

pqp

q

p

q

-

pq

p

-

pq

q

-

pq

q

-

pq

q

pq

-

p

q

p

-

pq

p

p

q

-

Absurd

state

1

Successful update

Unsuccessful update

Update-to-test consequence:

Conclusion must

loop

pq

q

p

pq

pq

q

q

p

Some update paths for text p q; q;p. Sentence p addsnothing in the contexts created by preceding two sentences. If

the starting point is not 0, then some information has alreadybeen accepted.

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First two notions of validity, 0utand ut may be usefulin the imperative logic for semantic phenomena that exhibitorder sensitivity and defeasibility. They do not have structuralproperties of classical consequence relation [5] [21](table belowshows some properties).

reflexivity permutation monotonicity

...,C,... C

X, P1, P2, Y C

X, P2, P1, Y C

X, Y |= CX,P,Y C

0-update-to-test - - -Update-to-test - - only leftTest-to-test + + +

Given these weaker notions of validity, one may hope todiscover cases of weak inclusion of meaning. Weak inclusionmay be lost by changing the order of sentences or by adding newones. Such a weaker notion of validity was implicitly present innumber of works on practical logic (e.g. [14], [19], [24], [33]).

Another semantic notion that will be used here (see 5.1 be-low) is the notion of dynamic coherence of a text. It requires

existence of a nonabsurd state in which every sentence from thetext is accepted. Dynamic coherence corresponds to the staticnotion of satisfiability. Dynamic consistency, on the other hand,points to a transition path and it may be realized by an inco-herent text.

Coherence. A sequence of sentences 1; ...; n is coherent [20]iff : 6= 1 [1] = ... = [n].

Example 1 (Order sensitivity.) It might be raining...It isnot raining is a consistent and incoherent text, reversing theorder gives an inconsistent text. It is impossible to open thiswindow...You should open this window is an inconsistent text,reversing the order gives consistent, yet incoherent text.

Example 2 (Defeasibility.) (i) Make people laugh! (ii) If

you tell that joke, you will make them laugh. So, (iii) maybeyou should tell the joke. seems to be a valid argument. The

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conclusion is defeated by additional premises: (iv) If I tellthat joke, I will ruin my wifes good mood. (v) Let it be the casethat my wifes good mood is sustained!

The last example can be analyzed in the framework of Geachs

characterization of defeasibility of practical inference..

But even if a conclusion is validly drawn from theacceptable premises, we are not obliged to accept itif those premises are incomplete... (Geach [19] p.77.)

If we follow his line of thought throughout the Example 2,

then a person who accepts (i) and (ii) and who thinks that (iii)is their consequence nevertheless is not obliged to accept (iii)if he considers the premises to be incomplete since (iv) and(v) are missing.

Example 3 (Nonreflexivity) Make people laugh! If you tellthat joke, you will make them laugh. Maybe you should tell the

joke... If you tell that joke, you will ruin your wifes good mood.

Sustain wifes good mood! So, maybe you should tell thejoke. does not seem to be a valid argument.

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2 Simple update system for imper-

atives

Moods and modal elements.

Imagine a picture representing a boxer in a par-ticular stance. Now, this picture can be used to tellsomeone how he should stand, should hold himself;or how a particular man did stand in such-and-suchsituation; and so on. One might (using the language

of chemistry) call that picture a proposition-radical.[35] 23.

Speaking in terms of the picture-metaphor: the picture, orrather - picture fragment means something, but its meaningis unsaturated until it has been used in a certain way. Theactive part, use is modal element1 of the sentence, the passiveits sentence-radical. M is a saturated expression having

sentence-radical and modal element M. The sentence-radicalmay be used for different semantic actions, and the way of itsuse is determined by the modal element of sentence.

The dynamic semantics provides a natural way of thinkingabout meaning of natural language sentences: different types ofsemantic actions correspond to different moods. The number

of conceivable types of semantic actions exceeds the number ofnatural language moods. Still the standard threefold divisioninto indicatives, imperatives and interrogatives may (and per-haps should) be sustained. There are two obvious ways to doso. First, one may divide the class of generated states in such a

1 This terminology is used in [30], but not in the identical sense. WhileStenius takes modal element in the sense of mood indicator, here sentencemoods are taken to be a category of modal elements.

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Figure 1 Modal element, sentence radical and semantic action.

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way to define a characteristic property for each mood2. Second,one may define sentence mood in terms of inter-mood and intra-mood meaning relations between sentences having contradictorysentence radicals3.

Two galleries metaphor. It seems that we can apply

eliminative semantics approach in imperative logic, mutatis mu-tandis. Let us start with the metaphor of two galleries, one con-taining, as already stated, alternative pictures of yet unknownactual situation, the other gallery containing pictures of yet un-known ought-to-be situation. When both galleries contain allalternative pictures, then no information has flown in. If all thepictures are visible in upper, ought-to-be or goal gallery thenthere is nothing that should be done. If there is a strong dis-crepancy in the content of the galleries (i.e. when they have nocommon element), the system should change the actual situa-tion.

Four moves system. While the simple elimination se-mantics is a one (re)move system, the imperative logic requiresat least a four moves system. On the indicative side, besidessentences that give picture fragments for unraveling facts, one

needs also sentences for expressing regularities, since they re-strict the range of possible present and future situations. Onthe imperative side, the idea that imperatives solely act upongoal gallery seems to be straightforward. Still, if we take im-peratives to be instructions for changing or sustaining a type ofsituation, then change to means change actual situationinto situation, and sustain situation means do not changeactual situation into situation. These double semantic

moves, the asymmetric and symmetric one, seem to be more ac-curate account of natural language imperative semantics whencompared to act-on-goals-only approach. There is another mo-tivation for this kind of imperative semantics: in connecting

2 In the system developed here the generated classes are not disjoint.Rather, the class of states I generated by indicative sentences is a supersetof the class of states M generated by imperative sentences.

3 For imperatives and indicatives it holds that if a text Ma;Mb is

consistent, then modal elements Ma and Mb do not belong to the samemood category.

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imperative logic with belief-desire logic one may characterizeagent motivational state by the imperative the agent accepts(see Chapter 5). The asymmetric imperative can characterizethe state of desire, since anyone...who desires, desires...what isnot present (Plato, Symposium, 200A-201A) .

Suggestions. If there is a degree of soundness in this im-perative variant of eliminative semantics, then test sentences asphenomena visible through eliminative semantic lenses shouldbecome visible if present. It seems that the answer is positive:suggestion expressing sentences seem to be imperative sentencesin a test-mode. The suggestions are complex test sentences. Inthe formal treatment, they test acceptability of certain impera-tives and they also include a test whether a model belongs to thecategory of models generated by application of imperative sen-tences. The latter test reflects the fact that suggestions in thetypical case propose a relative goal for acceptance or rejection.

2.1 Three moves language L! for im-

perative logic

Syntax. If , are propositions in the standard language LPof propositional logic, then , , ! are sentences of the lan-guage L! of practical update logic. Nothing else is in L!. Asequence of sentences 1, ..., n in L! is called a text

4.Semantics. Sentences are interpreted as functions from

to

. The set

= {h

,

i |

W,

W} is a family of states(i.e. model variations or contexts). The set W = PD is the setof situations i.e. valuations over the finite set D of all propo-sitional letters in the part of language LP under consideration.In the class distinguished elements and subclasses are:

0 = hW, Wi, initial or minimal state

4 Bold Greek letters stand for sentences in L! .

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the class offinal states F = {h, i | = = } in-cluding absurd state 1 F defined as 1 = h, i.

Sentences.Preliminary definitions. Truth set of sentence radical

LP in W: kkW

= {wW | w }. Truth set of sentence radical LP in X W: kk

X = kkW X

h, i [] = hkk , kki

h, i [] = h, kki

h, i [!] = hkk , kki

Text. [1; ...; n] = ( [1] ...) [n] = [1] ... [n], wherei L!

The repertoire of semantic actions for imperativelogic. The basic moves are: (1) information update abouthard facts which cannot be changed in any case (determin-istic rule), (2) information about the current situation whereone believes oneself to be (fact), (3) information about the de-sired non-actual state (goal and fact). The moves consist in

removing points from the set of future and desired situationsor removing points from the set of situations which are indis-tinguishable with respect to their actuality.

The semantic impact of a deterministic rule cannot be iden-tified with the impact of a factual sentence. Deterministic rulefixes that what cannot be the case neither now or in the future.Usually it is expressed by conditional if is the case, then will be

the case. Therefore, sentence-radical of a rule is pro-jected on both sets of candidate situations. On the other hand,information on facts is projected on the set leaving only thosesituations which are indistinguishable regarding their actuality.

Instructions to change situation in a certain way is projectedboth to the set of situations that are indistinguishable withrespect to their desirability and to set . Imperatives have dou-ble semantic impact. Change to thus consists of projection

kk

and kk

: it sets a goal and reports on facts. Thiscombined goal-fact approach seems to justified if one assumes

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a semantic basis for pragmatics. If a speaker suggests a goalshe believes to be impossible or s/he commands that the actualstate is to be brought about, then the logic of suggestions andcommands is violated. When the Imperator is not sure wether is the case, s/he should say Change to if is not the case.

If one does not choose mixed, goal-and-fact semantics for im-peratives, then s/he must endorse the idea that an imperative! commands two actions: first, an epistemic one consisting inchecking whether is the case, and the second, conditional oneconsisting in seeing to it that (if is not the case). On thisapproach, but not on ours, a conditional ! is identicalwith !.

Semantic notions. Variants of ordinary semantic notionsof eliminative semantics (see page 5).

Acceptance and acceptability A sentence L! is ac-cepted in a state iff [] = . A sentence L! isacceptable in a state iff [] / F.

Update-to-test consequence relation

1; ...; n ut iff : [1] ... [n] = [1] ... [n] []

Zero-update-to-test consequence relation

1; ...; n 0ut iff 0 [1] ... [n] = 0 [1] ... [n] []

Consistency and coherence. A sequence of sentences in L!1; ...; n is consistent iff : [1] ... [n] / F, and co-

herent iff /F : [1] = ... = [n]

Example 4 Goal: Pluto () is inside; Fact: Fido () is in-side; Deterministic rule: Pluto will not be inside unless Fidois outside. What should be done? Goal: Pluto is inside andFido is not inside.

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Goal: Fido is not

inside

Goal: Fido is not

inside

Goal:Plutoisinside

Ru

le:Tw

odog

scann

ot

bein

thesame

spot

Fact:Fidoisinside

Reduction of uncertainty regarding goals (white area) andfacts (grey area) may lead towards an emergence of a new

relative goal.

2.2 Reduction of uncertainty in the prac-

tical setting

A way of explicating the notion of practical reasoning can begiven in terms of reduction of uncertainty. In theoretical rea-soning an agent discovers the actual situation (i.e. the situationbelieved to be actual) by eliminating its possible descriptions.Parallel to this popular metaphor of information growth as un-certainity reduction, practical reasoning may be viewed as re-

duction of uncertainty with respect to facts and goals. In a

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successful case, practical uncertainty reduction is an informa-tional process in which determination of the actual situation andcovering rules enables the determination of the minimal, feasibleand permitted change needed for realization of an original goal.In a typical case reduction of motivational uncertainty evolves

through the sequence: initial state 0 imperative= extended motiva-tional state indicative= maximal motivational state. In an extendedmotivational state some goals are acceptable and not acceptedi.e. they are rejectable (see Definition 8 and Proposition 10).On the other hand, maximal motivational states are completein the sense that for any sentence in the part of language L!under consideration it holds that the sentence is either accepted

or not acceptable (see Definition 7 and Proposition 4). A max-imal motivational state is a point of motivational certainty inso far it has been settled which situation is the actual one andwhich situation ought to be brought about. If there is morethan one candidate for the goal situation, it is possible that oneof them could turn out not to be a desired one. The informationthat the situation believed to be actual has changed leads to afinal state. Within eliminative approach the notion of satisfac-

tion should be modelled as a historical entity having a geneticdefinition5.

Some technical remarks Any maximal state can be reachedin two steps i.e. by a two sentence text (Proposition 5).For any text there is a decision procedure which deter-mines whether given amount of information and dutiesleads to maximal state (Proposition 6).

The interaction between theoretical and practical reasoningis reflected in the possibility that a indicative-sentence updateon a motivational state in which goal is accepted may leadtowards a state that verifies goal (where is not equivalentto ). In literature two types of relations between goals havedrawn attention: the relation between a goal and its subgoals,

5

The notion of satisfaction will not be discussed here since its requiresadditional structure.

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={,W and W}={,W and W}L! / 0-accessible states:

or=L! / 0-accessible states:

or=

Final states: = or=Final states: = or=Motivational states: =Motivational states: =

Maximal states:

==1Maximal states:

==1

Figure 2 Update paths.

and the relation between an end and its means. Subgoal isusually understood as having the property of being entailed bygoal: if is a goal and is its subgoal, then |= . On theother hand, ends and means are logically unrelated goals: if isan end and is a means to it, then 2 and 2 . It seemsthat a distinction between original and derived (relative) goal

may be introduced with all reason: on that account subgoalsand means are derived or relative goals.

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3 Prima facie consequenceNo adding metaphor. Suppose that a process of eliminationby text T has created context C in which sentence makes nochange. Shall we say that is a consequence ofT since adds(removes) nothing? The answer is positive only provided thatnothing else besides T has been accepted. How can we be surethat in this process of elimination only sentences from T havetaken part? If all the models from the family are present at theinitial point of elimination, the context created by T will be thelargest. One may call this context a minimal one: its informa-tional content is the least in the class of contexts that verify

T. Minimal context can be useful in formal semantics if thereis a meaning relation that exists between a sentence and a textin the minimal context. A meaning relation of that kind seemsto be a ground for the notion ofprima facie consequence. Theconclusion that cannot be detached from its grounds must be ofa special kind. For if it is a model eliminating sentence, then itwill hold in any context that verifies premises: if it can not re-move models from the largest class, it surely will not be able to

remove the models from any of its subclasses. So undetachableconclusion must be a noneliminative sentence. It seems thatin natural language we encounter such sentences, which reporton the gallery condition instead of removing pictures from it.They are introspective sentences reporting on a context (state,model) status.

In next paragraphs we will discuss test-sentences in impera-tive mood1. They exhibit nonpersistence and in the framework

of eliminative semantics they qualify as plausible candidates forthe role of defeasible conclusion.

Extending language to L!. For the purpose of accom-modating test-sentences we will extend the language.

Syntax. A string of symbols is a sentence of the languageof L! iff L!, or = or = ! or = ! ,

1 An idea similar to the here presented modalities within moods was

hinted by Castaeda for deontic logic: [...] statemental modal operatorsapply to deontic assertables p.39.[8]

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and therefore, it can be further refined by additional sentences3.That state of mind can be changed only with respect to relativegoals and uncertain facts. In that sense some accepted sentencecan be rejected later (see Propositions 8, 9 and Lemma 7).

3.1 Nonpersistent sentences and preferredmodel

It has been noted by several authors in philosophy of action andin philosophical logic that the notion of validity in practical logic

is not classical one. Usually nonmonotonicity is recognized asa specific feature of consequence relation in practical logic. Ex-pression prima facie is used either to indicate that additionalpremises may defeat the conclusion (Wallace [33]) or that con-clusion can not be detached from its premises (Davidson [14]).

The idea that there can be some kind of valid inference withdefeasible conclusion can be modeled as a premises-conclusionmeaning relation holding in the minimal model of the premises.

In the set of states T for a sequence of sentences T there isa minimal state minTsuch that for any state 0 T, 0 6= there is a sentence accepted in 0 and not accepted in minT.Assume that prima facie consequence means (relation of)conclusion being acceptable in the light of premises T but pos-sibly rejectable by an extended text a T, where denotesoperation of arbitrary positioning . Since minT = 0 [T], itfollows that prima facie consequence relation can be explicatedby notions of rejectable sentence (p. 19) and a consequence re-lation defined over the minimal state (p. 14).The informationalcontent of the minimal state in the class of models for of thetext T is the smallest content (see Propositions 8 and 9).

Reconciliation. The conflicting intuitions on the validforms of imperative-indicative inferences seem to be reconcil-

3 A state is accessible if there is an update path leading to it. Our

attention is restricted here only to 0-accessible states; other states have adifferent genealogy pertaining to a richer language.

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able to an degree. Typically, if a schema is taken to be invalidby an author, and valid by another, then the premises are in-complete (in the sense of Proposition 6) while the conclusion isa rejectable sentence accepted in the minimal state.

Example 5 (Practical syllogism) Disagreement with respectto validity in imperative and belief-desire logic.

Formulations of the allegedly valid principle:

(1) S genuinely wants p to be the case for its ownsake. (2) Only if S does X will p be the case. (1) and(2) constitute prima facie grounds for (C) S should

do X. (Wallace [33])If X believes [knows] that (p implies q), then that

X intendsp implies that X intends q. (Castaeda [9])One wants to attain x. Unless y is done, x will

not be attained. Therefore y must be done. (VonWright [37])

Systems with explicit criteria4:

Valid?Segerberg [27] (!p [p] q) !q noBelnap, Perloff, H orty [4][23] ([ dstit : p] (p q)) [ stit : q] noCros s [13] (4(p) (p q)) 4q noChellas [11] (!p (p q)) !q yesK enny [24] (Fiat(p) Est(p q) Fiat(q) noFulda [18] ((p s) (p q)) (q s) noL! !p; (p q) ut!q no

L! !p; (p q) 0ut ! q yes.

Different results arise from different semantics. From ourstandpoint, the premises are incomplete in the sense that they

4 Main features of formal semantic systems are given in the remark on

the page??.

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do not necessarily lead to the maximal state (in this case addi-tion ofq would suffice) thus giving grounds for acceptance ofdefeasible conclusion ! q (which can be ruled out by q).

Example 6

Valid?Segerberg [27] (!(p q) p) !q noBelnap, Perloff, H o rty [4] [23] ([ dstit : p q] p)) [ dstit : q] noCros s [13] (4(p q) p)) 4q noCros s [13] (4(p q) p)) 4q yesChel las [11] (! (p q) p) !q yes

K enny [24] (Fiat(p q) Est(p) Fiat(q) yesFulda [18] (((p q) s) p) (q s) yesL! ! (p q) ; p ut!q yes

L! ! (p q) ; p 0ut ! q yes

In this example the premises determine the maximal state.Still the validity declarations do not converge. It could be usefulto point out semantic peculiarities which give rise to negative

results.Context independency. The authority commands Close

the door and open the window in a situation which is believedboth by the authority and the addressee as the situation wherethe door is closed (and each of them knows what the other be-lieves). It is redundant then to issue the further command Openthe window. If it is nevertheless issued, it will not change or addanything to the initial command. In the sequence (i) Close thedoor and open the window! (ii) The door is closed. Therefore,(iii) open the window! the last sentence (iii) does not seem tobe contextually independent. It seems natural to have alongsidewith the contextually independent reading5, i.e. Do anythingto bring it about that the window is open! (as suggested inSegerberg [27]), a contextual dependent one Make the minimal

5 Counterexample is an action hw, vi such that w kpk and v kp qk

which does not belong to a command set: kp qk w but hw, vi /{hw, yi : y kp qk}, therefore, it is possible that kqk / w.

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change of actual state that will bring about that the windowis open. In contextual dependent reading the imperative (iii)does not change the initial goal. Similar arguments apply toSTIT logic where the possibility that some agent does not havea choice to open the window makes this subgoal inference form

invalid.Relative notion of satisfaction. One translation for !(p

q), p Therefore, !q in MLD (Cross [13]) is a valid schema

(4(p q) p) 4p

4 reads desire in the sense of incompatible goal-belief discrep-ancy and p reads agent is satisfied that p. The implicit ideathat an agent is satisfied by realization of a subgoal seems coun-

terintuitive. One would rather say that agent wants to preservea subgoal if it happens that the actual situation belongs to thesame situation type. The other reading

(4(p q) p) 4p

gives an invalid inference. Motivational extension can be con-ceived of as a process in which the goal remains fixed whilesubgoals change, but the assertion that every realization of a

subgoal leads to the state of satisfaction (Cross, 1997) does notseem acceptable. The motivational force of a desire is exhaustedby the realization of its content, and only that final point of mo-tivation extension is a state of satisfaction, not an intermediatemotivational state in which an agent desires to sustain some-thing.

Remark 1 Main semantic features of several systems.

[13] k!k (w, t) = 1kk (w0, t) = 1

for every w0 such that Rt (w, w0)

[23] M,m/h [ dstit : ]h0 : h0 Choicem (h) M,m/h

0

andh00 : h00 H(m) M,m/h00 2

[13] VM,w(4) = >w0

wR1w

0 VM,w0() = >

andw00 wR2w

00 VM,w00() = [27] xrequires! {hx, yi : y kk} x

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4 Negation of imperatives and no-tion of changeThe task of dynamic semantics is to provide a connection be-

tween a structure that verifies a sentence and operations thatproduce verifying structural variation. In the simple updatesystem an imperative is verified by a pair h, i of disjoint sets, = and an asymmetric action is connected with im-peratives in order to generate required structure. To update astructure with a sentence means to produce a minimal modifica-tion of the structure needed for the verification of the sentence.Following that line of thought, negation of a sentence can beunderstood as an operation that will prevent any successful1modification by that sentence. In that sense, negation changesa structure in a way that makes accepting of negated sentenceimpossible. There are different ways of making an imperativeunacceptable. According to the semantics for imperatives aschange directives the imperative ! is not acceptable in any stateh, i such that kk = or kk = . Since fulfillment of

one condition suffi

ces, indicative sentence could be regardedas a variant of a dynamic negation (see2! below). In general

dynamic negation in a eliminative update system (forward look-ing system) can be characterized by the following proposition.

Proposition 1 : [] [] F.

It could be interesting to examine several sentential instruc-tions that will satisfy the proposition.

1. h, ih1!

i= hkk , i Dont change to what-

ever the actual situation may be.

2. h, ih2!

i= h, kki = h, i [] It is the case

that .

1 Successful means not landing into a final state.

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3. h, ih3!

i= hkk , kki = h, i [!] Change

to .

4. h, ih4!

i= hkk , kki (i) Preserve ,

Do not change , Do not change to , Sustain

5. h, ih5!

i= hkk , kki (i) Preserve , Do

not change , Do not change to

We will assume that negation of an imperative is an imper-

ative too, ruling out2!. The opposition ! vs.

3!(or !) is

not a good candidate for the role of negated change instruction

if it is intuitively acceptable that the negated change amountsto no change. The opposition ! vs.

5! is not a good candi-

date either in so far both commands share the same goal con-

tent. The remaining cases1! and

4! give good candidates.

The one-sided goal negation has special attractiveness since,together with -type sentences, it can make all states reachable. On the other hand, within the proposed semantics

for imperatives as change instructions negation1

turns out tobe indeterminate between a variant of a conditional imperative

h, ih1.1 !

i=

hkk , i if h, i [] = h, i

h, i otherwise

and a command without informational content

h, i h1.2 !i = hkk

, i

We opt for4! as a plausible candidate for negated imperative

since4! and ! impose different imperative alternatives for

the same actual situation. Still, the identical symmetric seman-tics given to the negations 4 and 5and regularity expressingindicatives ( and , respectively) shows that the formalstructure must be refined since it is obvious that e.g. and4

! do not mean the same.

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GOAL

p

p

p

p

Producep!

Destr

oyp

!

Susta

inp!

Supp

ressp

!

Change !

Dont change !

ACTUAL

Figure 1 Negated and affirmed imperative: alternatives for the

same situation.

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4.1 Extended language L!! and refined

models

In order to accommodate negated imperatives, more refinedmodels will be introduced (i.e. ordered triples of sets of val-uations). The resulting system will be a four moves system.

Semantics. Family of models

= {h, , i | , , W} .

0 = hW,W,Wi, initial or minimal model

the class offinal models F = {h, , i | = = }including absurd model 1 F defined as 1 = h, , i

the class of nonfinal models S characterized by 6=

the class of nonfinal models M S characterized by =

4.1.0.0.1 Sentences

Definition 1 (Truth set) kkX = X kkW

Basic sentences.

necessary

h, , i [] = hkk , kk , kki

actually

h, , i [] = h, kk , i

change to

h, , i [!] = hkk , kk , i

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sustain (do not change to )

h, , ih

! i

=

hkk , kk , i if kk 6=

1 otherwise

Defi

ned, complex sentences.

possibly

h

i

=

if [] 6= 11 otherwise

not necessary (possibly not )

h

i

= h

i

not actually (actually not )

)

= [)]

might

h

i

=

if [] / F1 otherwise

might be good to produce

h

! i

=

if [!] / F1 otherwise

might be good to sustain

h

! i

=

if ! / F1 otherwise

maybe should be brought about

h ! )i =

if

M and [!] / F and

[!] F1 otherwise

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maybe should be preserved

h

! )i

=

if

( M or S) and

h

! i

/ F and

h ! i F1 otherwise

Remark 2 We call !0, ! , ! , ! , ! , ! modal

elements of imperative mood, and , , , , modal elements of indicative mood.

ChangetoA isinside

Actu

allyBi

sinside

DonotchangetoB isnotinside

Change to

A and B are

inside

AB

A

B

B

A A B

AB

A

A

B

B

B

A A B

A B A

B

B

A

A B AA

B

B

A

A B

B

A

A B

B

A

A B A

B

B

A A B

A B AA

B

B

A A B

4.1.1 Explanation of semantic moves: ! and !

The sentences having form (i) Maybe should be broughtabout and (ii) Maybe should be sustained could be regarded

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as kind of imperatives in so far as their sentence radical denotesthe type of an acceptable (yet defeasible) goal situation. Usingsentences of the kind one usually express practical suggestions2.The sentence forms (i) and (ii) seem to be a complex test func-tions. The Speaker, having incomplete knowledge of Hearers

reasons and supposing an inclination for a suggested goal onthe Hearers side, proposes a relative goal for Hearer. A not rig-orous extraction of semantic parts of suggestion sentences couldgive us the following decomposition. For (i): on Speakers opin-ion there is something x Hearer already wants + is related tothat x so that Hearer does not want to have and Hearer maywant to have . For (ii): there is something x Hearer alreadywants it+ is related to that x so that Hearer does not want tolose and Hearer may want to keep .

Proposition 2 : [!] = [!] F

Proposition 3 : h

! i

= h

! i

F

Assume that there are only 2 permitted moves in giving thesemantics for complex sentences: 1. use basic sentences, 2. usedistinguished elements and subclasses of the family of models.

We want to delineate two classes of states, SC for suggestedchanges and SS for suggested preservations.

(i) SC =n

h, , i | kkW kkW 6= o

and

(ii) SS = nh, , i | kkW kkW 6= o(i) Maybe should be brought about+ stands for All worlds are...+/- stands for Some worlds are and some worlds are not...- stands for No world is...@ marks states in which produce can not be accepted3

2 suggest v. 2. to propose (a plan or theory) for acceptance or rejection(The Oxford Dictionary for Modern English)

3 All goal situations are -situations or no actual situation is a -situation.

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* marks states in which produce can be acceptedM marks states which are necessarily M-type statesdouble lines mark the classes of models that satisfy the con-

ditions for ! being acceptedGoal situation Actual situation [!] F [!] / F

worlds worlds+ + @+ +/- M @ *+ - M @ *

+/- - @ *- - @

(ii) Maybe should be preserved

+ stands for All worlds are...+/- stands for Some worlds are and some worlds are not...- stands for No world is...# marks states in which sustain can not be accepted4

$ marks states in which sustain can be acceptedM marks states which are necessarily M-type statesS marks states which are usually S-type statesdouble lines mark the classes of models that usually satisfy

the conditions for ! being acceptedGoal

situationActual

situationh

! i

F h

! i

/ F

worlds worlds

+ + S* # $+ +/- M # $+ - M #

+/- + # $- + #

Remark 3 See appendix for a way to prove whether a class ofstates is a subset of M-type states (an example covering secondrow in the table above is given in Proposition 17).

4 All goal situations are -situations or all actual situations are -situations.

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ChangetoAisinside

Actually

Bisins

ide

Ch

angetoBisnotinside

Change toA is

insideand B is not

inside

AB

A

B

B

A AB

AB A

B

B

A

AB A

B

B

A AB

Maybe Bshould

stay inside

A

B

B

A

Nonperisistentsentence

is defeasible.

Dotted arrow denotes the nullifying of nonpersistent sentenceby additional sentence. Also note that a nonpersistent

imperative sentence may be accepted even when suggestedgoal is not a relative one.

4.1.2 Conditional imperative

Conditional imperative anchors a type of a goal situation to aparticular kind of actual situations. The simplest specimen is

!

The simplest solution for the semantics of a conditional im-

perative within here proposed modelling would be to treat it assimple test function.

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[ !] =

[!] if [] =

otherwise

We do not want semantics in which conditional imperativemay be forgotten if the antecedent is not true (desired be-

havior is depicted in the picture below). On the contrary, thesemantics of the conditional imperative should restrict the pos-sible evolution of a model even when the indicative antecedentis not true. In order to meet that demand, the semantics of aconditional imperative must somehow encapsulate the semanticimpact of different sequences of sentences in order to restrict thespace of evolution paths.

0!

!

!

!

!

NONFIN

ALSTATE

S,T,F

!

Part of evolution tree for conditional imperatives.

Unfortunately the proposed semantics of ordered triples ofsets of valuations can not fullfil that demand5. The solutionrequires different modelling like the one given in the example 7.

5 If 0 [!; ; ] = h,,i, then h,,i [!] = h,,i

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Example 7

=

hP, Ri | R P P, P W, W = 2 set of propositional letters

hP, Ri [!] = hP, {(w, v) R | w v 2 }i

hP, Ri [] = hP, {(w, v) R | v }i

hP, Rih

! i

= hP, {(w, v) R | w 2 v 2 }i

Operation d : hPi, Rii d hPj, Rji = hPi Pj, Ri Rji

The desired behavior is obtained by

[ !] =

[!] if [] = [] [!] d [] otherwise

should hold. The latter requires kkW . On the other hand, if0 [ !; ; !] = h0,0,i, then h0,0,i / F should hold. Thelatter requires that 0 kkW 6= . Since indicative texts ; and cannot eliminate points from goal set, it follows that 0 [!] =

h00

,00

,i should satisfy contradictory conditions i.e. 00

kkW and 00

kkW 6= .

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5 Semantics of propositional atti-tudes reportsWe use term propositional attitude report to denote a class

of sentences in which a psychological intentional verb is used.The term psychological -intentional verb will be used in a in-formal sense and therefore we introduce it by way of examples:to desire, to think, to believe, to hope are examples, andnon-examples are: to touch, to taste, to suffer. We adoptthe tradition which treats propositional attitude reports as sen-tences (that can be rephrased as) having a verb of the kindfollowed by an embedded sentence describing a state of affairsthat is the object of the attitude denoted by the verb , e.g. Shedesires that her art would have an emotional impact.

Two-part anatomy and direction of fit. In analyz-ing intentional states, i.e. propositional attitudes, and sentencemoods authors usually distinguish two components.

mental states

Husserl noesis noemaRussell attitude propositionAnscombe direction offit content

sentences

Hare tropic phrasticStenius modal element sentence radicalKenny mood indicator descriptive content

This psychological - linguistic parallelism in anatomy is fur-ther emphasized by another similarity. Imperatives and desiresare such that the world must fit with them, while declarativesand beliefs are such that they must fit the world. This twofoldparallelism motivates an idea of establishing connection betweenmental states and sentences.

Dynamic semantics. It is not necessary to identify thechanging models of dynamic semantics with the internal states

of an computing system (be it a machine or a pre-linguisticchild). We can treat dynamic semantics as just another kind

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of model theory which uses variations of Kripkean, Tarskianand Kripkean-Tarskian models to account for some semanticphenomena, like order sensitivity, non-classical consequence re-lation and nonpersistent sentences. Nevertheless, the idea ofconnecting models and intentional states seems to be natural

in that framework: models are states of (an ideal) system andmodel evolution paths are histories of such states.

Representation free approach. By accepting the ideathat the model changes are triggered by sentences we are notforced to adopt a variant of representationalistic metaphor ac-cording to which the sentences are components in internal func-tioning of a system. The metaphor was prominent in historyof ideas (e.g. inner voice, cultural imperative, categorical im-perative, language of thought, to name a few), but it is weakin number of points which have been discussed in the litera-ture. Instead we adopt a non-realist position: by using propo-sitional attitude predicates we adopt an interpretational stance(intentional stance, Dennett [16]) in which we exploit inferentialrelations between mood designated sentences (modified versionof Davidsons measurement theoretic approach to the semantics

of propositional attitude reports [15]). The position advocatedhere differs from the Davidsons position in so far as we takeentities to be mood designated sentences and not mood freepropositions.

Just as in measuring weight we need a collectionof entities which have structure in which we can re-flect the relations between weighty objects, so in at-

tributing states of belief (and other propositional at-titudes) we need a collection of entities related in theways that will allow us to keep track of the relevantproperties and relations among the various psycho-logical states. ( Davidson, What is present to themind in [15] p. 60.)

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5.1 Characterization of intentional states

The approach to the propositional attitude report semantics,which we want to explore here, relies on two basic theoreticalassumptions. Intentional states may be (i) theoretically identi-fied with models of some sort and (ii) characterized by semanticrelations between models and mood designated sentences. Onthis approach, the informational content of sentence type

[mental predicate P] ([agent a] , [proposition ])

is identical with the informational content of the sentence there

is a model such that agents state is identical with it and belongs to the class of models satisfying update condition withrespect to a mood M designated sentence(s) M.

agent a is in a state s.t. { | ...update condition...}(i) a wants to change to [!] =

(ii) a wants to sustain h

! i

=

(iii) a believes that is the case [] = (iv) a believes that might be the case

h

i=

(v) a believes that is necessary the case [] =

(vi) a opposes that will be produced h

! i

=

(vii) a opposes that will be sustained h

! i

=

The states verifying suggested changes or preservations of

the situation type seem to be useful for modelling of a weakersense of desire.

[...] we may now distinguish what we might call vari-ous levels of want or desire, beginning at the lowest level:(i) He prefers A to not-A. (ii) He opposes not-A. (iii)He favors A. (iv) He favors A and opposes not-A. [...].The terms want and desire, in their ordinary use, maybe used to refer to any of these various levels of want

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or desire. They are generally used in such a way thatthe objects of want and desire are restricted to objectsthat are future [...]and to objects such that the personswho wants or desires them does not believe them to beimpossible. (Chisholm [12] 623)

Remark 4 Cross conceptualizes state (i) as a state of desire inthe sense of incompatible goal-belief discrepancy and state (ii)as a state of satisfaction. Our position is quite different. SeeSection 3.1 for a discussion. Note that on Chisholmian readingopposing that means desiring that in a weaker sense.

Semantic holism and rationality assumption. The

semantic holism involved in use of mental vocabulary is ex-plained by the assumption that propositional attitudes charac-terize mental states of an agent. Propositional attitude reportsdo not single out an intentional state, rather they characterizea class of states by a necessary condition. The use of men-tal vocabulary imposes rationality assumption on descriptum[14][16].

Any effort at increasing the accuracy and power of atheory of behavior forces us to bring more and more ofthe whole system of the agents beliefs and motives di-rectly into account. But in inferring this system from theevidence, we necessarily impose conditions of coherence,rationality, and consistency. (Davidson [14] p. 231.)

One of the ways the rationality assumption is introduced isreflected in coherence condition: only coherent text can charac-terize a non-final mental state.

Example 8 The text It might be raining...It is not raining isdynamically or sequentially consistent. Still it cannot be used forcharacterization of an intentional state due to its incoherence.

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5.2 Validity of rationalizations

Founded and founding systems. The studies in the logical form of intentional explanations and in meaning relations be-tween sentences in different logical moods may be given a unifiedtreatment by distinguishing founded and founding system. Thefounding inferential relations are meaning relations betweensentences in imperative and declarative logical moods. The va-lidity of rationalizations formulated within intentional languagedepends on the validity of corresponding inference formulatedin the language using logical moods.

5.2.1 Two modes of validity for rationalizations

Rationalization and rationality. First we introduce a crite-rion of validity for intentional explanations (also called: ratio-nalizations or reason explanations. A rationalization is valid iffit exemplifies a norm of rationality.

Two modes of rationality. We make a distinction be-

tween two modes of rationality. Horizontal rationality is inter-nal relation between intentional states. Vertical rationality isa external relation between an intentional state and the realworld (see [38]). Two modes of rationality correspond to thetwo Davidsonian principles of interpretation. Principle of Co-herence corresponds to the horizontal mode, Principle of Corre-spondence to the vertical mode.

The process of separating meaning and opinion in-vokes two key principles which must be applicable if aspeaker is interpretable: the Principle of Coherence andthe Principle of Correspondence. The Principle of Co-herence prompts the interpreter to discover a degree oflogical consistency in the thought of the speaker; the prin-ciple of Correspondence prompts the interpreter to takethe speaker to be responding to the same features of the

world that he (the interpreter) would be responding un-der similar circumstances. Both principles can be (and

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have been) called the principles of charity: one principleendows the speaker with the modicum of logic, the otherendows him with a degree of what interpreter takes tobe true belief about the world. Successful interpretationnecessarily invests the person interpreted with the basic

rationality. It follows from the nature of correct interpre-tation that an interpersonal standard of consistency andcorrespondence to the facts applies to both the speakerand the speakers interpreter, to their utterances and totheir beliefs. (Davidson [15] p.211)

Horizontal mode.

Definition 2 The sequence

P1(a, 1),...,Pn(a, n), therefore, (should) P

n+1(a, n+1)

exemplifies a prima facie norm of horizontal rationality iff

M1(1); ...; Mn(n) |=0ut M

n+1(n+1)

and 0

M1(1); ...; Mn(n)

/ F

where Mi is a modal element corresponding to psychologicalpredicate Pi .



n+1(a, n+1)

exemplifies a norm of horizontal rationality iff

M1(1); ...; Mn(n) ut M

n+1(n+1)

andM1(1); ...; M

n(n) is coherent

where Mi is a modal element corresponding to psychological

predicate Pi .

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Example 9 The statement The desire that p and the beliefthat p only if q can rationalize the desire that q, but cannotrationalize the belief that q should be conceived as abbreviation

for The desire that p and the belief that p only if q rationalizethe desire that q only if !p; (p q) !q, but cannot rationalize

the belief that q since !p; (p q) 2 q.

The definitions given above are simplified in order to pointout the connection between belief-desire logic and imperativelogic. In order to make the definitions applicable one should ex-tract sentence radical and consequently treat intentional-psychologicalpredicates generously. For example, the intentional predicate insentence a believes that it might be snowing is believes that

it might be the case. The other option in which complete sen-tences rather then their radicals are taken to be the objects ofa propositional attitude seems more natural.

Example 10 a believes that it is necessary that it is cold if itsnows. She believes that it might be snowing. Therefore, she(should believe) believes that it might be cold is not formalized

as B(a, (s c)), B(a, s) B(a, c), but rather asB(a, s c), B(a, s) B(a, c).

Vertical modeThe definition of validity in vertical mode requires additional

notions. The intentional states are rational if connected to thereal world in appropriate way. It is not enough for a belief tobe justified by another beliefs, both its grounds and the belief

itself should be true. The same goes for desires: if one desiresthe impossible, she should revise her beliefs. For example, whileit may be horizontally rational to want to open the windowbelieved to be closed, it is always vertically irrational to wantto open it if it is already open.

Definition 4 w is a model of the real situation . Rw is the setof objectively (really) possible continuations of the real situation,

Rw = {v | v is really possible atw} .

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Definition 5 Theoretical model h, , i of an intentional stateis (possibly -incomplete) veridical state iff

v : v v Rw

andw



n+1(a, n+1)

exemplifies a norm of vertical rationality iff (i) it exemplifies anorm of horizontal rationality and (ii) every / F such that

M1(1); ...; Mn(n)

= is a veridical model (where Mi is

a modal element corresponding to psychological predicate Pi).

5.3 Application: anatomy of excuse

Making an excuse shows how two kinds of semantic relationsare being used in a language game. In the typical case, Excusershows that his intentional action would not have been preformedif his state of mind was a veridical one. In virtue of having ahorizontal and vertical semantic dimensions, the language ofintentionality makes it possible for a horizontally rational in-tentional state to be vertically irrational.

At a dull party, Husband wanted to make people laugh bytelling a joke. Unfortunately, his telling a joke spoiled Wifesgood mood. Husband apologizes: I did not know that the jokewould spoil your mood. In that way he shows that his desire totell a joke was horizontally prima facie rational, but irrationalin the vertical sense.

Full blown excuse:a) realized horizontal part:Wants!(Husband, The people

are laughing), Believes( Husband, If Husband tells that joke,

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the people will laugh), 1, therefore, prima facieWants!(Husband,Husband tells that joke),

b) failed vertical part: Wants!(Husband, The people arelaughing), Believes(Husband, If Husband tells that joke, thepeople will laugh), Wants!(Husband, Wife is in a good mood),

Believes(Husband, If Husband tells that joke, Wife will not bein a good mood), therefore, Wants!( Husband, Husband does nottell the joke).

1

Wants

!

(Husband, Wife is in a good mood)reads Husband wants tosustain Wifes good mood.

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Initial state, 0-state. L=People are laughing, J=Husband tellsthe joke, M=Wife is in a good mood. 1={L,J,M}, 2={L, J},

3={L, M}, 4={L}, 5={J, M}, 6={J}, 7={M}, 8={}.

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Defeasible conclusion holding in a non-veridical state(situation 1 is not really possible situation, 1/Rw).

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Vertically waranted conclusion: Dont tell that joke!.

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6 Appendix1

Remark 5 Test sentences will be left out of consideration inthe lemmata and propositions regarding properties of accessiblestates. Nevertheless, the results apply to test sentences in virtueof the fact that their application may generate only one change,namely change to the absurd state, 1. The results are given forL! and L!, still most of them apply to L!! mutatismutandis.

Definition 7 h, i is maximal state iff || = || = 1 and = .

Proposition 4 is maximal state iff M and LP

: h

! i

= [!] =

Proof. Define function nf : 2W LP which delivers adisjunctive normal form for a set of worlds X as nf(X) =WwX

Vlw l

V

l /w l

. We have to prove that || =|| = 1 and = , iff h, i M and LP :

h, ih

! i

= h, i [!] = h, i. Going from leftto right, suppose for reductio that h, i

h!

i= h, i

h, i [!] 6= h, i. Then, either wv : w v w /kkW v / kkW or wv : w v w / kkW v kkW. Therefore, either || 6= 1 or || 6= 1. Contradiction.Going from right to left, suppose for reductio that || 6= 1 or|| 6= 1 or 6= . If || = 0 or || = 0, then h, i F.

Contradiction. If || > 1, thenw : w {w} 6=

h, ih

! nf({w})i

= h, i h, i [!nf({w})] = h{w} , i .

Therefore, some acceptable sentence is not accepted. Contra-diction. If || > 1 then

w : w {w} 6=

1 Appendix by Berislav Zarnic and Damir Vukicevic

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h, ih

! nf({w} )i

= h, i

h, i [!nf({w} )] = h, {w}i .

Contradiction. If 6= 0, then h, i / M. Contradiction.

Definition 8 is extended motivational state iff LP

: h

! i

= [!] 6=

Proposition 5 Any maximal state can be reached in two steps.

Use function nf : 2W LP which delivers a disjunctive nor-mal form for a set of worlds X i.e. nf(X) =

WwX

Vlw l

V

l/w l

.Let h0, 0i be a maximal state and 0 , 0 . Text

nf(0) ; !nf(0) is an instance of a complete two-sentence textsince h, i [nf(0)] [!nf(0)] = h0, 0i

Proposition 6 It is decidable whether a given text can reach amaximal state.

For each sentence from a text s1; s2; ...; sn extract sentenceradical or its negation, and form a conjunction according to thefollowing recipe:

n =^

i{1,...,n}[(si=!)(si=)]

andn =

^i{1,...,n}[(si=)(si=)(si=!)]

.

Text s1; s2; ...; sn may reach a maximal state iffknk

W

= 1 knk

W

= 1 knkW 6= knk

W

Definition 9 0/L!-accessibility relationR0/L! L!

L! :

R0/L! =

R0/L! = =-

, 0

|s

1...s

ns

n+1...s

m: s

1,...,s

m L

!

0 [s1; ...; sn] = [sn+1; ...; sm] = 0

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Lemma 7 Let h, i = 0 [T]. Then for any model h0, 0i suchthat h0, 0i [T] = h0, 0i and h0, 0i 6= h, i it holds that0 or 0 .

Proof. Let T = s1; s2; ...; sn. Use the recipe from Propo-

sition 6 to obtain n and n for text T. Given that 0 [T] =Dknk

W , knkWE

, 0 knkW , 0 knk

W, lemma follows.

Proposition 8 Each model for a text T is accessible from itsminimal model.

Proof. Let h0, 0i be a nonfinal, 0/L!-accessible model for

a text T and minT

= h, i = 0 [T], thenh, i

nf

0 0

nf

0

=-

0, 0

.

Text nf(0 0) ; nf(0) is an instance of a text that reachesh0, 0i. Therefore,

-minT, h0, 0i

R0/L! .

Proposition 9 For each nonminimal model h0, 0i for a text

T there is a text T

which is not accepted in the minimal modelh, i = 0 [T].

Proof. Denote by kTk0/L! a set of nonfinal states in which

T is accepted. If0 or 0 , then 0 [T] [nf(0 0)] [nf(0)] 6=0 [T]. Therefore, T; nf(0 0) ; nf(0) = T

Proposition 10 A sentence , accepted in a model h, i, is

rejectable iff: (i) = and kk

6= kk

6= , or (ii) = ! andkk 6= kk 6= (kk 6= kk 6= ),

or (iii) = ! and = kk 6= kk 6= ((kk = kk 6= ) (kk 6= kk = )).

Proof. Direct calculation.

Lemma 11 (Elimination lemma) For any , , 0, 0 such

that hh, i , h0

, 0

ii R0/L! : 0

and 0

. If h, i 6=h0, 0i, then 0 or 0 .

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Proof. Trivial.

Lemma 12 Let T = s1; ...; sn, where si = or si = ,h, i [T] = h0, 0i, . Then 0 0.

Proof. Induction.

Lemma 13 Leth, i [T] = h0, 0i, = . Then00 = .

Proof. Use Lemma 11

Lemma 14 Leth, i [s1; ...; sn] = h0, 0i and at least one sen-

tence si is ! for some LP. Then 0 0 = .

Proof. It is obvious that for any and that kkkk = and that property is preserved by Lemma 13.

Theorem 15 The state = h, i is 0/L!-accessible if andonly if = or .

Proof. Going from right to left. First, assume = .

Then 0 [ (nf( ))] [! (nf())] = h, i .

Second, assume . Then

0 [ (nf())] [ (nf())] = h, i .

Going from left to right, we shall prove that a state is not0/L!-accessible if 6= and * . Let * and

0 [s1; ...; sn] = h, i . It implies that there is a sentence si suchthat si 6= and si 6= . Since there are only three typesof basic sentences, then it must be the case that si =!. ButLemma 14 and the fact 6= imply that there is no suchsentence in s1; ...; sn. Contradiction.

Proposition 16 h, i M iff : LP h, i [!] =h, i

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Proof. Going from left to right. For reductio assume : LP h, i [!] = h, i. But, h, i [!nf()] = h, i .Contradiction. The other direction trivially follows from kkkk = .

Proposition 17 Let kkW and kkW 6= and *kkW. Then h, i M.

Proof. For reductio suppose 6= . Use 15 to obtain acontradiction.

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[Acknowledgement.] The paper was completed during theauthors research visit at Department of Philosophy, Universityof Uppsala. The visit was funded by Swedish Institute researchgrant.

Dynamic Semantics, Imperative Logic and Propositional Attitudes

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