J Cantwell Conditionals in Reasoning 2009

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Synthese (2009) 171:4775DOI 10.1007/s11229-008-9379-6

Conditionals in reasoning

John Cantwell

Received: 23 October 2006 / Accepted: 21 July 2008 / Published online: 27 August 2008 Springer Science+Business Media B.V. 2008

Abstract The paper presents a non-monotonic inference relation on a language

containing a conditional that satisfies the Ramsey Test. The logic is a weakening of

classical logic and preserves many of the paradoxes of implication associated with the

material implication. It is argued, however, that once one makes the proper distinction

between supposing that something is the case and accepting that it is the case, these

paradoxes cease to be counterintuitive. A representation theorem is provided where

conditionals are given a non-bivalent semantics and epistemic states are representedvia preferential models.

Keywords Conditionals Non-monotonic logic Ramsey test

Reasoning is a mental activity involving mental states and mental acts. Part of the sub-

ject matter of logic is to provide a theory of what constitutes correctreasoning. Such

theories are often presented by rules of inference, where an inference is a particular

kind of transition from one mental state to another and the rules governing inference

characterize the legitimate transitions. The manner in which such rules are presented

often leaves implicit both the nature of the transition and the normative status of the

rule. For instance, the following is a standard rule presented in a standard way:

A&B

A.

I am here less interested in the rule following aspect than I am in the underlying norm

that justifies the rule. For this particular rule I would suggest that the underlying norm

is: It is a requirement of rationality that if one accepts A&B, then one accepts A.


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Instead of the above vertical notation I will use the following horizontal notation

to express that A is accepted by an agent j :

|j A.

With this new notation the norm corresponding to the above rule could be stated asfollows:

For any agent x, it is a requirement of rationality that if |x A&B, then |x A.

This, in turn, will be abbreviated to:

If | A&B, then | A.

Not all rules of inference have this format. The dominant logics, like classical and

intuitionistic logic, contain a number of rules of inference that govern the interaction

between what one supposes to be the case and what one accepts to be the case (Gentzen1969; Prawitz 1965). So one finds the rule (sometimes called disjunction elimination)

that here can be formulated as follows IfC is accepted on the supposition that A, and

C is accepted on the supposition that B, then C is accepted on the supposition that

A B. Or, letting A |j B stand for B is accepted by j when supposing that A:

For any agent x, it is a requirement of rationality that if A |x C and B |x C,

then A B |x C.

In abbreviated form:

If A | C and B | C, then A B | C.Keep in mind that the left-hand side and right-hand side of the wobbly turnstile

represent two different kinds of mental states. The act ofsupposing that A or assum-

ing that A (I use the terms interchangeably) is a mental act that puts one in the mental

state where one supposes or assumes that A (the same word is used to describe both

the act and the state that results from the act). To suppose that A and to accept that A

are two different things (e.g. see Levi 1996). If one supposes that A then one accepts

that A (while one reasons under the supposition), but the converse need not hold.

For instance, I believe, accept, that it will rain this evening (the clouds look very

heavy), but this belief does not have the status of an assumption. This shows up becauseI can coherently (without accepting any contradiction) suppose, for the sake of the argu-

ment, that it will not rain this evening. The assumption that it will not rain, contradicts

my belief that it will rain, but upon supposing that it will not rain I no longer, for the

duration of the argument, accept that it will rain. My belief has been bracketed by

an assumption to the contrary. Assumptions can be formed and given up at will, but

while they are in place, they are not undermined by any further assumptions. Thus if I

first assume that it will rain and then, without canceling this first assumption, assume

that it will not rain, then I am committed to accepting a contradiction.

Suppositions allow us to reason with and explore possibilities that are not supportedby any available evidence, indeed that may contradict the available evidence. Thus

th h I k thi th th t ith th d th b tl did it I
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reasoning under this supposition, accept that the butler did it. Or I can assume that it

wont rain even though the heavy clouds provide sufficient evidence for me to believe

(but not to be indefeasibly certain) that it will rain.

The fact that suppositions can undermine acceptances makes our reasoning non-

monotonic in that reasoning does not (and should not) in general satisfy the requirementofmonotonicity:

If A | C, then A, B | C.

As the classical consequence relation is monotonic this disqualifies it as the basis

for a general theory of correct reasoning.1 However, classical logic is (a strong candi-

date to be) the correct theory ofpurely suppositional reasoning: a theory of the kind

of reasoning that proceeds only from what one assumes to be the case and what one

accepts on the basis of these assumptions, without involving any proposition that one

merely (and defeasibly) believes to be the case.2

The correct general theory of correctreasoninga theory which allows that assumptions can undermine or bracket what

one otherwise acceptsmust be sought within the domain of non-monotonic logics.

Non-monotonic logic is often associated with ampliative reasoning: default logic,

circumscription logic, inductive logic can all be seen as forms of reasoning govern-

ing more or less plausible ways of extending ones beliefs beyond the available evi-

dence, beyond but yet consistent with what is required by rationality. By contrast the

non-monotonicity in reasoning that has been discussed above arises from the fact that

beliefs may be defeated by assumptions and is not of the ampliative kind, it is not

reasoning intended to reach beyond what one already has reason to believe on theassumptions made. If anything it is of the opposite kind: the non-monotonicity arises

because not every belief can legitimately be introduced at every stage of reasoning,

some beliefs are temporarily blocked by assumptions to the contrary.3

Yet another form of non-monotonicity arises from the dynamics of information.

Acquiring new evidence can make you surrender old beliefs. The non-monotonicity

in reasoning that I have in mind is not of this kind either. While the adding and canceling

of assumptions in reasoning gives rise to a dynamics, it is not the dynamics of learning:

to suppose that something is the case is quite different from learning that it is the case.

Still, the different forms of non-monotonicity are structurally similar and are not

always easy to keep apart. So for instance Isaac Levi has argued that some aspects of the

influential AGM (after Alchourrn etal. 1985) framework for information dynamics

1 A general theory should not only cover reasoning from what one assumes or believes to be the case

theoretical reasoningit should also cover reasoning about what one wants and intends to dopractical

reasoning. In this paper only theoretical reasoning is studied. Broome (2001; 2002) outlines an account of

practical reasoning.

2 Axiomatic presentations of logic provide a list of claims that are to be indefeasibly accepted. So for

instance, in some presentations of classical logic A A is put forward as an axiom, which means that one

should accept it regardless of what one otherwise assumes.

3 There is, of course, a sense in which this kind of reasoning is ampliative. It may be required of a particular
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(belief revision) are best viewed as an analysis not of belief revision, but of supposi-

tional reasoning. Likewise the preferential non-monotonic logic of KLM (after Klaus

et al. 1990), often presented as a logic for ampliative reasoning, can be re-interpreted

as a theory of reasoning along the lines sketched above. The connection between them

can easily be seen if we change representation (Grdenfors and Makinson 1991). Ifwe let Kj represent the set of propositions accepted by the agent j , and Kj A be the

set of propositions accepted by j under the supposition that A, then |j A if and only

if A Kj , and A |j B if and only if B Kj A. I prefer to use the wobbly turnstile

| rather than the set notation as the former, for historical reasons, emphasizes the

connection to logic and, through it, to reasoning.

The distinction between accepting that A and supposing that A is by now well-

established. But it is not always appreciated how profound the consequences are for

the proper understanding of even the most ancient and well-established rules of infer-

ence. Many classical rules of inference can be interpreted in several different ways.Take an apparently simple rule like the disjunctive syllogism which (trying to formu-

late it in a neutral way) says that B can be inferred from the premises A B and A. In

the present setting this loose statement of the disjunctive syllogism can be interpreted

in four different ways:

DS1 A B, A | B.

DS2 If| A B and | A, then | B.

DS3 If| A B, then A | B.

DS4 If| A, then A B | B.

The key here is whether we are to understand the premises A B and A as

assumptions (DS1), or as claims that one (perhaps defeasibly) accepts (DS2), or as

some combination of assumptions and acceptance (DS3, DS4). It is far from obvious

that all these formulations are equivalent. Indeed, I claim that not only are they not all

equivalent, but that two of them are false: they do not reflect requirements of rationality.

The first two, DS1 and DS2, I think are correct, but it is easy to find counterexamples

to DS3 and DS4. For instance, say that I believe that the butler did it (committed

the murder). Then I accept that either the butler or the gardener did it. But upon the

supposition that the butler didnt do it I am not required to accept that the gardenerdid it. For upon the assumption that the butler didnt do it I bracket the belief that the

butler did it, and with it the (dependent) belief that either the butler or the gardener

did it. Thus DS3 is false. Similarly, even though I might believe that the gardener did

it, this doesnt mean that on the assumption that either the gardener didnt do it or my

dead grand-mother did it, I am required to accept that my dead grand-mother did it:

the possibility that my dead grand-mother did it is not a live option. Thus DS4 is false.

Enter conditionals.4 The Ramsey Test for conditionals, an essential ingredient in

the analysis of conditionals, holds that there is a close-knit connection between the

4 The conditionals I have in mind are conditionals in the past or present tense: If Jane accepted the bet, she
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conditionals that one accepts, and what one accepts on the basis of an assumption. It

says (letting A B correspond to the indicative conditional If A, then B):

| A B if and only if A | B.

We can immediately note that the ancient rule modus ponens suffers the same ambi-

guity as the disjunctive syllogism. Trying to formulate it in a neutral way it says that

B can be inferred from the premises A and A B. In the present setting this loose

statement can be interpreted in four different ways:

MP1 A, A B | B.

MP2 If| A and | A B, then | B.

MP3 If| A, then A B | B.

MP4 If| A B, then A | B.

Again the key here is whether we are to understand the premises A and A B

as assumptions (MP1), or as claims that one (perhaps defeasibly) accepts (MP2), or

as some combination of assumptions and acceptance (MP3, MP4). Each are distinct

non-equivalent formulations of modus ponens and, as it happens, I think two of them

are requirements on reasoning and two are not. The two correct principles are MP1

and MP4 (the latter is just one direction of the Ramsey Test), while MP2 and MP3 are

both subject to counterexamples.

Here is a counterexample to MP3. I believe that Oswald killed Kennedy, but upon

assuming that if Oswald killed Kennedy, then my grand-mother was a co-conspirator,I would bracket the belief that Oswald killed Kennedy and so I would not conclude

that my grand-mother was a co-conspirator to the murder. For I believe that if Oswald

killed Kennedy only if my grand-mother was a co-conspirator, then Oswald didnt kill

Kennedy.

Van McGee has presented a nice counter-example to MP2:

Opinion polls taken just before the 1980 election showed the Republican Ron-

ald Reagan decisively ahead of the Democrat Jimmy Carter, with the other

Republican in the race, John Anderson, a distant third. Those apprised of the

poll results believed, with good reason:If a Republican wins the election,

then if its not Reagan who wins it will be Anderson.

A Republican will win the election.

Yet they did not have reason to believe

If its not Reagan who wins, it will be Anderson. (McGee 1985, p. 462)

The literature on conditionals, on their relationship to suppositional reasoning, on

how they interact with the Ramsey Test in non-monotonic reasoning, and on the prob-

lems one encounters along the way is by now considerable (see Bennett 2003 for a

general overview, and Arl-Costa 2007 for an overview of the main frameworks for

dealing with conditional logic). I want to focus briefly on two recurring issues in the
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conditionals) discussed here have no semantics. In the most radical form it is held

that there is no sense in which these conditionals can be taken to express propositions

that are true or false, and that the acceptance of a conditional is no more than the

expression of a state of mind. Ramsey is a precursor:

Many sentences express cognitive attitudes without being propositions; and the

difference between saying yes or no to them is not the difference between saying

yes or no to a proposition. F. P. Ramsey,Law and Causality (Ramsey 1929/1990).

One need not actually endorse such expressivism to see the virtues of a non-seman-

tic analysis. It is surely a worthwhile ambition to state the requirements of rationality

that govern reasoning without appealing to any semantic properties of the language.

Indeed there is an important tradition, inferentialism, which holds that once we have

provided a theory of the norms governing the use of the various connectives we have

also provided a theory of meaning for those connectives (e.g. Brandom 1994; Dummett1997). Such a view, however, need not be irreconcilable with the view that the connec-

tives have truth conditions. Soundness and completeness proofs are standard ways of

linking the normative perspective with the semantic-descriptive perspective, regardless

of which perspective one thinks has conceptual priority.

A considerable amount of the formal work in the area has focused on epistemic

models for the logic of conditionals, models that do not assume that conditionals have

a semantics.5 This in itself need not be seen as a form of expressivism. It should already

be clear that a theory of correct reasoning cannot be given in purely semantical terms.

The mental state one is in when one believes that A has the same propositional contentas the mental state one is in when one assumes that A, but the normative requirements

on reasoning distinguish between the two mental states, thus the normative require-

ments on reasoning take more than propositional content into consideration.6 Express-

ivists take the need for epistemic (rather than semantic) analyses to be grounded in the

fact that conditionals lack semantics, and they often invoke an array of impossibility

resultsthe best known due to Lewis (1976)that suggest that conditionals do not

behave like standard truth-carrying constructions.

I think these semantic sceptics are right up to a point, but that they overstate their

case. There are good reasons for holding that a conditional (of the kind consideredhere) with a false antecedent lacks truth value, here I agree with the expressivists.

But there are also good reasons to hold that a conditional with a true antecedent and

a false consequent is false, just as there are good reasons to hold that a conditional

with a true antecedent and a true consequent is true. That is, I am inclined to think

that conditionals express partial propositions, propositions that can be true or false,

but that also can lack truth value. This is not a new idea (it goes back at least to Quine

5 There are those who hold that conditionals describe the epistemic state of the speaker or describe the

epistemic state of whoever assesses a conditional, in effect relativizing the truth conditions for conditionalsto some agent (e.g. Stalnaker 1975; Weatherson 2001). Space does not allow for an in-depth discussion of

these views here, although it should be noted that a difference in view on the semantics basis for conditionals
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1950), indeed some expressivists (e.g. Bennett 2003 and Edgington 1995) are willing

to concede the point, but do not regard this as anything that can be put to use. I think

they are wrong.

The second issue is closely connected with the first: the difficulty of giving a general

treatment of embedded conditionals, that is, of conditionals that occur in the scopeof some other connective or in the scope of some other conditional. It is commonly

conceded that a conditional like A (B C) is equivalent to (A&B) C. But

apart from this special case there is little agreement on how to proceed. There are

different suggestions on how to deal with negated conditionals, and many hold that

conditionals cannot even meaningfully occur as left-hand embedded, embedded in

the antecedent of another conditional (as in (A B) C). Seen from the per-

spective of suppositional reasoning this would be a disaster. For it seems clear that

we can coherently assume that a conditional A B holds and reason from such an

assumption to a conclusion C. But if we have A B | C, then the Ramsey Testentails | (A B) C.

There have been attempts at enriching standard models for representing epistemic

states in order to analyze left-hand embedded conditionals. But some of these are too

domain specific to serve as a general analysis, and more general frameworks (such

as Hansson 1992 and Arl-Costa 1995; 1999) are so general in character that they

provide little guidance on how we are to proceed to extract a reasonable theory of

conditionals.

In this paper I take classical logic as a starting point and I assume that it to a large

extent captures the actual norms invoked in purely suppositional reasoning. It is mystarting point for investigating the norms involved in reasoning that mixes beliefs and

assumptions. The intuitionist or relevance logician would choose a different start-

ing point and hence arrive at a different theory. My choice of starting point should

not be taken as unconditional acceptance of classical logic (to the contrary: it has

problems). However, some of the general lessons emphasized herethe importance

of keeping beliefs and assumptions apart, about the non-monotonicity induced by

the fact that assumptions undermine beliefs, and the specific problems introduced by

conditionalswould remain valid in a variety of starting points. In Section 2 I present

and argue for a non-bivalent semantics for conditionals. Section 3 presents a prefer-

ential model for epistemic states and provides a representation theorem. Section 4,

finally, discusses one of the problems with taking classical logic as a starting point.

1 Inference relations

1.1 Categorical inference relations

First a language without conditionals will be considered:

Definition 1 The categorical language L0 consists of a countable set of propositional
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Definition 2 A Categorical Inference Relation is a relation | (finite) sets of sentences

of L0 to sentences of L0 satisfying (, A | B abbreviates {A} | B):

&Acc | A&B if and only if | A and | B.

I If | A, then | A B and | B A.

E If, A | C and , B | C, then , A B | C.

I If, A | , then | A.

E If, A | , then | A.

I If | A&A, then | .

E If | , then | B, for any B.

Together with the structural constraints:

Reflexivity , A | A.

Cut If | A and , A | B, then | B.

Cautious Monotony If | A and | B, then , A | B.The requirements that characterize categorical inference relations provide a natural

generalization of the classical requirements on reasoning on the basis of assumptions

and axioms alone. Note that we get classical logic by replacing (actually: strengthen-

ing) Cautious Monotony by:

Monotony If | B, then , A | B.

1.2 Conditional inference relations

Definition 3 The language L consists of a countable set of propositional atoms, closed

under the connectives , and & (so that if A and B are sentences of L , then so are

A, A B and A&B). A B is defined: (A&B).

A sentence in L is categorical if it contains no instance of the conditional or if

every instance of the conditional occurs within the scope of a negation.

Definition 4 A Conditional Inference Relation is a relation | from (finite) sets of

sentences of L to sentences of L , satisfying, in addition to &Acc, I, E, I, E, I,

E and the structural requirement Reflexivity:Acc | A B if and only if, A | B.

As well as the following restricted forms of the structural constraints:

R-Cut If | A and | B, then , A | B, provided that B is categorical.

R-Cautious Monotony If | A and , A | B, then | B, provided

that B is categorical.

Here are some sample properties (the proofs are left as an exercise, for the proof of

(2) it is helpful to use Theorem 2 below):

Observation 1 For any conditional inference relation |:
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1.3 Discussion

From the perspective of classical logic, the distinctive feature of conditional inference

relations is that they fail to satisfy:

Monotony If | B then , A | B.

Cut If | A and , A | B, then | B.

Failure of Monotony is of course the defining feature for all non-monotonic log-

ics. Given the intended interpretation of conditional inference relations the failure of

monotony corresponds to the fact that what you accept on a given set of assumptions ,

need not be accepted upon further suppositions . In particular, a belief (which is

an unconditional acceptance) may be bracketed when reasoning under a supposition,

so that we may have | A without having B | A.

For instance, say that Jane, noticing that her cat comes inside drenched in water,draws the reasonable conclusion that it is raining outside. That is, after Jane has made

her observation we have

|j It is raining outside

Her belief is no firmer, however, than that she can coherently consider the possibility

that she is wrong; on the supposition that it isnt raining, she no longer accepts that

it is raining. Instead, on supposing (for the sake of the argument) that it isnt raining,

she brackets her belief that it is raining and accepts that someone poured water on thecat, that is, we have

It isnt raining outside |j Someone poured water on the cat,

but not

It isnt raining outside |j It is raining outside.

Or say that Bill saw Jane leaving home this morning, apparently without an umbrella.Thus, on the supposition that it rained during the day, Bill would conclude that Jane

got wet:

It rained |b Jane got wet.

On the other hand, while unlikely, Bill is willing to consider the possibility that Jane

carried a hidden umbrella, so he also accepts:

It rained, Jane carried a hidden umbrella |b Jane didnt get wet,


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The distinguishing property of arguments that have the structure of an reductio ad

absurdum is that there are suppositions that one cannot coherently accommodate (that

is, they lead one to accept a contradiction). So for instance, Jane accepts that 1 + 1 = 2

and she cannot coherently accommodate the supposition that 1 + 1= 2. So we have:

|j 1+1 = 2,

and

1+1 = 2 |j 1+1 = 2.

but we also have:

1+1 = 2 |j 1+1 = 2.

Accepting both a claim and its negation is the defining feature of a reductio argument

and in classical logic, as well as here, it leads to a logical explosion: one accepts

everything.

While non-monotonicity is widely accepted as a feature of everyday reasoning, the

failure of Cut is more contentious (for instance, Gabbay 1985, holds it as one of the

defining properties of an inference relation). But, I will argue, it is a distinguishing

feature of conditionals in the kind of inference relations considered here that they

systematically violate Cut.It should already be clear that there is a difference in the logical status of accepting

that A as opposed to supposing that A. This is obvious already in allowing for non-

monotonicity. As we have seen, in allowing for non-monotonicity one can coherently

have both:

| A

and

, B | A.

Note, however, that one is committed to accepting anything one supposes to hold, so

we always have:

A, | A,

and we never have

A, , B | A.
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fact that A occurs on the right-hand side of the relation | Athe fact that A is

(hypothetically) acceptedis no guarantee that it will remain on the right-hand side

no matter what we add to the left-hand side; but as soon as A occurs on the left-hand

side, this is also a guarantee that it will occur on the right-hand side: no matter what

, we always have , A | A.Suppositional reasoning allows us to explore the possibility that we are wrong in

some of our beliefs (of course, they also allow us to explore possibilities that are

consistentwith what we believe); in supposing that A one explores a possibility (A)

that may go contrary to what one believes. So some of the conditionals that we accept

explore the possibility that we are wrong in what we believe to be the case. But the

conditionals that we accept never explore the possibility that we are wrong in what

we assume or suppose to be the case. For instance, we have:

A, A | B,

for an arbitrary B (one cannot coherently explore the possibility that both A and A

are true). So we have:

A | A B,

for an arbitrary B; under the assumption that A the conditional A B explores a

possibility (A) that cannot be coherently explored while one is assuming that A.

On the other hand it is not in general the case that just because we have:

| A,

we also have:

A | A.

One need not accept a contradiction when making an assumption that goes contrary

to what one believes. So it is not in general the case that when we have | A, we also

have, for an arbitrary B:

A | B.

Thus it is not in general the case that whenever we have | A, we also have:

| A B.

So here we begin to see the problem with Cut (and unrestricted Cautious Monot-

ony). For Cut says that | A together with A | A B, entails | A B.

It saysimplausibly when conditionals are involvedthat defeasible beliefs can be


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undermine what one otherwise believes (but not what one otherwise assumes), and so

when reasoning with conditionals one cannot treat what one otherwise believes as on

par with assumptions.

We see this feature over and over again:

Yes: A | A B. No: If| A, then | A B.

Yes: A B | (A&C) B. No: If| A B, then | (A&C) B.

Yes: A, A B | B. No: If| A and | A B, then | B.

Yes: A B | B A. No: If| A B, then | B A.

Yes: A B, B C | A C. No: If| A B and | B C,

then | A C.

Yes: A B | A B. No: If| A B, then | A B.

All the properties of the left-hand column (the yes column) hold for conditional

inference relations (they are easy to prove). On the other hand, none of the propertiesof the right-hand column (the no column) in general hold for conditional infer-

ence relations (this can easily be shown given the representation results below). If

we allow unrestricted Cut, all of the No-properties would follow from the Yes-

properties.

Why is this important? Because there is good evidence from considered judgements

(e.g. Adams 1975) that none of the No-properties are valid in general. It is important

because one of the cardinal mistakes that is endemic in the literature on conditionals is

to take the evidence against the No-properties as suggesting that the Yes-properties

are not valid, or that they pertain to some other conditional. It is important becausethe opposite mistake is just as common; the mistake of dismissing the evidence against

the No-properties on the grounds that the Yes-properties belong to the core of clas-

sical logic, a deeply entrenched theory of reasoning. It is important because the two

opposing views typically do not realize that they are discussing two distinct sets of

logical properties, and that the systematic disagreement between the two parties is due

to a systematic confusion about the properties they are talking about. Give up Cut and

this false conflict is exposed.

The first pair (with the right-hand side being a well-known paradox of implica-

tion) has already been discussed. Consider the second pair. The failure of antecedent

strengthening is generally accepted for natural language conditionals: one can accept

If Jim struck the match (M), it lit (L) while denying If Jim struck the match and

the room was empty of oxygen (O), the match lit. In the present framework this is

respected as we can have:

| M L ,

and

| (M&O) L .
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M L | (M&O) L .

The third Yes/No-pair (two different forms of modus ponens) has already been

discussed and similar stories can be told for the remaining pairs. I will not belabour

the point any further.

2 Semantics

2.1 Classical semantics

Definition 5 A classical valuation is an assignment V of truth values {t, f} to the

sentences of L satisfying:

A A

t f

f t

A B t f

t t f

f t t

A&B t f

t t f

f f f

Note that the conditional is here interpreted as the material implication.

Definition 6 A is a classical consequence of, in symbols | A, if and only if for

every classical valuation V: if V(B) = t for all B , then V(A) = t.

Theorem 1 (Supra-classicality) For any Conditional Inference relation |, if | A,

then | A.

Theorem 2 For any Conditional Inference relation |:

1. If A | B and B | A, then , A | C if and only if, B | C.

2. , A, B | C if and only if, A&B | C .

Any conditional inference relation contains the classical consequence relation. Fur-

thermore, it is easily seen that the classical consequence relation is itself a conditional

inference relation. Thus the classical consequence relation is the smallest conditional

inference relation: it captures the requirements of rationality that hold regardless of

what is otherwise accepted. For this reason I think we can think of classical logic as

a theory ofpurely suppositional reasoning.

An important property (Theorem 2) is that classically equivalent sentences are sub-

stitutable when they appear as assumptions. Thus, in particular, A B and A B

have the same role in reasoning when they appear as assumptions: to assume that

A B is to exclude the possibility that A is true and B is false, i.e. to assume that

A B. This does not mean that A B and A B are substitutable under accep-

tance. It is consistent with the requirements of conditional inference relations that one

can accept A B without accepting A B.
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and the consequent to be true. This is mirrored in the material truth conditions: it

is wrong to accept a conditional with a true antecedent and false consequent as the

conditional is false under these conditions, and it is wrong to reject a conditional with

a true antecedent and true consequent as the conditional is true under these conditions.

Note, however, that there is no general tendency for people to accept a conditionalon the basis that they hold the antecedent of the conditional to be false, yet according

to the material truth conditions a conditional is true if its antecedent is false. Still

worse, people actually and reasonably reject conditionals that they according to the

material truth conditions believe are true: I believe that Oswald murdered Kennedy,

I do not believe that if Oswald didnt murder Kennedy, then my grand-mother did,

indeed I would reject this conditional as preposterous. So this part of the material

truth conditions cannot be grounded in usage, but then what reason can be offered for

holding that a conditional is true as soon as its antecedent is false?

In my view, the strongest justification for the material truth conditions for the con-ditional stems from two theoretical commitments: (i) classical logic is the correct

theory of purely suppositional reasoning, (ii) an inference in purely suppositional

reasoning is correct only if it preserves truth. Note that (i), according to the strictures

of classical logic, on assuming A B one is committed to accepting A B. So

(ii) A B is true whenever A B is true; the latter is true when A is false, so

A B is true when A is false. If one holds, as I do, that the conclusion of this

argument is false, one must reject one of the premises (i) or (ii). I think we should

reject (ii) and with it the presupposition that every conditional sentence is either true

or false.

2.2 Non-bivalent semantics

If we agree that one can correctly infer A B whenever one has assumed that

A B, and agree that in purely suppositional reasoning a valid inference cannot

take one from a true premise to a false conclusion, we should conclude that A B

cannot be false when A is false. In a bivalent setting this implies that A B is true

when A is false, but if we accept that conditionals express partial propositions, and

so can lack truth value, we can settle for a different conclusion: A B lacks truth

value when A is false.

The basic idea that actual usage of a particular kind of construction is important in

fixing the meaning ofand hence truth conditions forthe construction can hardly

be controversial. Empirical studies on how speakers in various situations evaluate

conditionals support the contention that conditionals lack truth value when the ante-

cedent is false, for instance Politzer (2007), in a review of the empirical psychological

investigations of reasoning with conditionals concludes In sum, reasoners seem to

consider the not-A case as irrelevant to the truth value of conditional statements [of

the form If A, then B] (p. 80). Admittedly it is difficult to interpret data on how

speakers evaluate claims as true or false, but when the data points in one direction
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Definition 7 A non-bivalent-valuation is an assignment W of truth values {t, f, } to

the sentences of L such that:

A A

t ff t

f

A B t f

t t f f

t f

A&B t f

t t f f f f f

f

Theorem 3 If a classical valuation V and a non-bivalent valuation W coincide on

the atomic formulas, then for any A:

V(A) = t if and only if W(A) = f

Proof Just check the truth tables.

So B is a classical consequence of A if and only if A is false in every non-bivalent

valuation where B is false. That is, on the non-bivalent interpretation, classical logic is

the strongest logic that guarantees that if the conclusion of an argument is false, then

some premise is false. On this interpretation the distinctive feature of classical logic

is not that it preserves truth (from the premises to the conclusion), but that it does not

introduce falsity (in the conclusion from non-false premises).

2.2.1 Conditional negation

A non-bivalent semantics allows for semantical distinctions that cannot be made in the

classical bivalent semantics. In particular, it allows for a different notion of negation,

inneror conditional negation, here represented by :

A A

t f

f t

This form of negation is of interest as it makes (A B) and A B logically

equivalent and provides a plausible path to the property:

| (A B) if and only if, A | B.

This negative Ramsey Test is the plausible requirement that one should reject a con-

ditional like If Jane applied, she got the position if and only if one, on the assumption

that Jane applied, rejects the claim that Jane got the position. Conditional negation thusseems more appropriate in the present setting than the outer negation (A B)


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This is a very strong form of negated conditional. It seems clear that we do occasionally

negate conditionals with a form of negation that is closer to than to .

Inner and outer negation can (of course) coexist within the same language, but they

can also be defined in terms of one another. A unary connective is definable in a

language if for any A, there is a sentence B in the language such that A is logicallyequivalent to B (has the same truth value in every valuation).

Theorem 4 is definable in L.

Proof Actually, we define both and the operator T(A) (A is true) with the truth

conditions:

A T(A)

t t

f f

f

Define T and as a function from sentences of L to sentences of L as follows:

1. T(p) = p.

2. T(A&B) = T(A)&T(B).

3. T(A B) = A&T(B).

4. T(A) = A.

Now define A as (T(A) A) A. It is straightforward to check that T(A) and

A, as defined, satisfy the requisite truth conditions. Note that | (A B) if and only if (by definition of) | A B if and

only if, A | B.

I have chosen to characterize conditional inference relations using outer rather than

inner negation to emphasize the connection to classical logic (if conditional negation

had been taken as the primitive form of negation, the inference from A B to

(A B) would come out valid, and while this seems plausible enough, it is not

classically valid). See Cantwell (2008b) for a treatment of the logic of conditional

negation.

3 Preferential models

So far the notion of accepting a proposition on the basis of an assumption has been

treated as a primitive. But it can be useful to study a model of rational acceptance. Here

such a model is developed and it is shown that the model completely characterizes the

class of conditional inference relations.

Let U be a set of points (possible worlds) and I be a function that to each prop-

ositional letter p assigns a subset of U. Each point u in U generates a non-bivalent

valuation Wu where Wu (p) = t if and only ifu I(p), and Wu (p) = f if and only if

u I(p). Let |A| denote the set of points in U where Wu (A) = f. A proposition P
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Definition 8 A selection function is a function from propositions to subsets of U,

with the restrictions:

1. (|A|) |A|.

2. If (|A|) |B|, then (|A| |B|) = (|A|).3. (|A| |B|) (|A|) (|B|).

The intended interpretation of (|A|) is that it is an agent relative function that selects

the A-worlds (the worlds where A is not false) that the agent considers most plausi-

ble or most (epistemically) preferred.7 Due to the structural similarity between the

present model and the model of KLM (Klaus et al. 1990) I will call a triple (U, I, )

a preferential model.

Preferential models are not intended to provide an agent-relative semantics for

conditionals (the semantics of conditionals was dealt with in the last section), but are

instead intended to provide a structure rich enough to model the epistemic state of arational agent and how the agents epistemic state changes upon making an assumption.

Let represent the initial epistemic state of the agent and A represent the

epistemic state of the agent upon supposing that A. Here A is defined:

A(|B|) = (|A| |B|).

That is, by assuming that A, the agent screens off all possible worlds where A is false

from further consideration. Note that if is a selection function, then so is A.

The task of the remaining sections is to give an account of how an epistemic stateis related to the propositions accepted under different assumptions. That is, when

represents the epistemic state of some rational agent j , we want to show how

characterizes the relation |j .

3.1 The categorical case

Definition 9 An acceptance relation on the categorical language L0 and a preferential

model M is a relation such that:

B if and only if (||) |B|.

So B is accepted on the assumptions if and only if B is true in the most preferred

-worlds. It follows that:

B if and only if || B if and only if ||(U) |B|,

That is, B is accepted on the assumptions if and only if B is true in the worlds thatare most preferred after the agent has made the assumptions .
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We have the following representation result:

Theorem 5 For any preferential model (U, I, ), is a categorical inference

relation.

Theorem 6 For any categorical inference relation |, there is a preferential model

(U, I, ) that characterizes |, i.e. there is a preferential model (U, I, ) such that:

| A if and only if A.

3.2 The conditional case

The acceptance recipe for categorical propositions does not carry over to conditionals.

Say that you believe that A, then (according to the semantics proposed above) youbelieve that A B lacks truth value, and this in itself does not give grounds for

either accepting or rejecting the conditional. An acceptance recipe that works better

with partial propositions is the following (here True(A) is the set of worlds where A

is true and False(A) is the set of worlds where A is false):

A if and only if ||(True(A) False(A)) True(A).

Or, in the special case when is empty:

A if and only if (True(A) False(A)) True(A).

That is, you accept that A if and only if A is true in the most preferred worlds where

A has a truth value.

This acceptance recipe allows us to derive the Ramsey Test:

A B if and only if A B.

Proof Note first:

(E) (False(B) True(B)) |A| = True(A B) False(A B).

Now:

1. ( A)(False(B) True(B)) True(B)

if and only if (by definition of)

2. ((False(B) True(B)) |A|) True(B)

if and only if (by (E))

3. (False( A B) True( A B)) True(B).


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As |A| True(B) = True(A B), (4) is equivalent to:

5. (False(A B) True(A B)) True(A B).

So: A B if and only if, by the acceptance recipe, (1) if and only if (5) if and only

if, by the acceptance recipe, A B.

Informally: you accept a conditional like If Jane applied she got the position if

and only if the conditional is true in all the most preferred worlds where it has a truth

value; it has a truth value only in those worlds where Jane applied and is true only in

those worlds where she applied and got the position. So you accept the conditional if

and only if the most preferred worlds where Jane applied are worlds where Jane got

the position.

Unfortunately, the above acceptance recipe works badly when we consider con-

ditionals embedded in conjunctions. What does it take for a conjunction to be true?Presumably that both conjuncts are true. What does it take for a conjunction to be

accepted? Presumably that both conjuncts are accepted. In a bivalent setting the seman-

tical and pragmatic theses go hand in hand, but when we turn to conditionals we face a

problem. If a conditional lacks truth value when the antecedent is false then it cannot

happen that both of the following are true (and so their conjunction cannot be true):

1. If the coin landed heads, Jim won the bet.

2. If the coin didnt land heads, Jim lost the bet.

Yet clearly, one can acceptboth conditionals at the same time (if one believes that Jimhas bet on heads, and if one does not know how the coin landed) and, hence, accept

their conjunction. Thus, it is consistent with the requirements of rationality that one

can accept a claim (such as the conjunction of the above conditionals) that cannot be

true (but that might be false, maybe Jim didnt bet on heads).

The acceptance recipe needs to be revised. For the moment I suggest the following:

Definition 10 The acceptance relation characterized by a preferential model M is a

relation defined recursively:

p if and only if (||) I(p).

A if and only if (||) U |A|.

A&B if and only if A and B.

A B if and only if, A B.

We can now prove two representation theorems:

Theorem 7 For any preferential model (U, I, ), is a conditional inference

relation.

Theorem 8 For any Conditional inference relation |, there is a preferential model


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3.3 Comments

Note the weight that the proposed semantics of conditionals pulls in the above account.

The idea is that an agent j satisfies A B p if p is true in the most plausible

worlds where A B is not false. That is, when an agent assumes that A B shethereby excludes from consideration any world where A B is false. This presup-

poses that A B can be false, i.e. that it can have a truth value (although it does not

presuppose that A B always has a truth value). Thus it is an essential aspect of the

above model and the resulting representation theorem that conditionals can have truth

values.

The semantics also places constraints that are satisfied by the model. It is a require-

ment of rationality that if one accepts that a conditional is true, then one accepts

the conditional. Furthermore, if one has consistent beliefs one should not accept a

conditional that one believes is false. Both these constraints are satisfied when a non-bivalent semantics is employed (the first property is not satisfied when the conditional

is interpreted as a material implication):

Obervation 2

1. If (|A|) True(B), then A B.

2. If (|A|) = and (|A|) False(B), then A B.

Proof (1) Assume that (|A|) True(B). Proof by induction over the length of B.

B = p. Trivial.

B = C&D. Note that (|A|) True(C&D) if and only if (|A|) True(C) and

(|A|) True(D) if and only if (by the induction hypothesis) A C and A D

if and only if A C&D.

B = C. Trivial.

B = C D. Note that if (|A|) True(C D), then (|A|) |C| and so

(|A|) = (|A&C|). So (|A&C|) True(C D) True(D) and so, by the

induction hypothesis, A&C D and so A C D.

(2) By Corollary 2 (in Sect. 5.5), if A B, then (|A|) |B|. So assume

that (|A|) = and that (|A|) False(B). It follows that (|A|) |B| and so

A B.

Nevertheless, as conditionals express partial propositions the fit to semantics is not

perfect: an agent can believe a conditional even though she does not believe that it is

true, indeed even though she believes that it lacks a truth value (the conditional must

then be true in the, according to the agent, most plausible worlds where it has a truth

value).

Note that with the representation theorems at hand, one can prove a number of

negative results. For instance, one can show that the following is nota requirement of

rationality:

If | A and | A B, then | B.
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example in the introduction, r is a world where Reagan won, a is a world where

Anderson won and c is a world where Carter won. Of these r is the most plausi-

ble, followed by c, followed by r. Assuming that |Reagan won| = {r} (similarly for

Anderson and Carter) and |A Republican won| = {r, a} we have:

Reagan won.

A Republican won.

For the most plausible world is r where Reagan, who was a Republican, won. We also

have:

A Republican won (Reagan won Anderson won).

For the most plausible world where a Republican won and it wasnt Reagan, is theworld a where Anderson won.

But we also have:

Reagan won Carter won.

For the most plausible world where Reagan didnt win is c where Carter won. For this

reason we do not have:

Reagan won Anderson won.

Thus we see that one of the forms ofmodus ponens is not satisfied in all preferential

models, hence we know (from the representation theorems) that it is not entailed by

the requirements on conditional inference relations.

Of course, a restricted version of this form of modus ponens still holds:

Observation 2 If| A and | A B, then | B, provided that B is categorical.

The proof is left as an exercise.

4 Implausible properties

As already noted conditional inference relations contain classical logic (where is

interpreted as the material implication), thus we have all the paradoxes of implica-

tion, such as A | A B, that have fueled part of the literature on conditionals.

I have already argued that the property A | A B (together with a number of

related properties) is in fact a plausible requirement of rationality when the premise

A is viewed as an assumption, and that it should not be confused with its implausible

sibling If| A, then | A B.

Still, there remain some implausible properties. The problem with the strong clas-


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(A B) (A B) for any A and B. But clearly it is not a requirement of

rationality that I accept:

Either it is the case that if the coin landed heads then Napoleon is still alive, or it is

the case that if the coin didnt land heads then Napoleon is still alive.Plausibly, a necessary condition for accepting a disjunction is that one does not reject

both disjuncts, but in the above disjunction I can properly reject both disjuncts, so

I cannot be required to accept their disjunction. Classical logic thus entails require-

ments, even on purely suppositional reasoning, that seem too strong. In its defense

one can note that it is not possible for both disjuncts (A B) (A B) to be

false at the same time (if one disjunct has a truth value, the other disjunct lacks truth

value), thus we are not required to accept a claim that might be false. But this just

shows that avoidance of falsity cannot be sufficient grounds on which to characterize

acceptability conditions when we are dealing with partial propositions. Semantics isnot enough.

In this paper I have cheated as I have defined A B as (A&B), so A B

will be true if neither A nor B are false but this means that A B cannot be interpreted

as A or B. Given the way that A B has been defined, (A B) (A B) is

not an obviously implausible property (both conditionals cannot be false at the same

time). So this just leaves open the important problem of characterizing the logic that

governs the English or when it combines with conditionals, a problem that has not

been properly addressed here.

What has been shown here is how a classical treatment of the connectivesincluding the conditionalcan be combined with principles of non-monotonic logic.

This yields all the theorems of classical logic (the same class of valid formulas as the

classical consequence relation), but invalidates the classical reasoning patterns that

are inappropriate when we turn to reasoning that is not purely suppositional. Classical

logic is simple in many ways and very powerful but too strong to account for all our

intuitions, so clearly there is more work to be done.

5 Proof of theorems

5.1 Proof of Theorem 1

The proof does not invoke Restricted Cut. Proof strategy: show that for every condi-

tional preferential inference relation |: if | A, then there is a classical evaluation

of the connectives (where is interpreted as the material implication) where every

element of is true and A is false.

The construction is standard. is a |-consistentset if for no finite subset 0 of

: 0 | . is a |-inconsistentset if there is some finite subset 0 of such that

0 | . is a maximal |-consistent set if no superset of is |-consistent.

Lemma 1 Any |-consistent set can be extended to a maximal |-consistent set
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1 0 = .

2.a i +1 = i {Bi }, ifi {Bi } is a |-consistent set.

2.b i +1 = i {Bi }, otherwise.

Let =

i

i .

First show that is a |-consistent set. Assume that it is not, then there is some

finite set such that | . As is finite there is some i such that i .

Take any B i . Due to the inconsistency of, | B. Thus by R-Cautious Mono-

tonicity, , B | , that is {B} is a |-inconsistent set. Reiterating this procedure

for each B i we find that i | . (In effect, we have shown that even if | is

not in general monotonic, a |-inconsistent set remains inconsistent no matter what

we add to it). As 0 is a |-consistent set there is some 0 j < i such that j is a

|-consistent set but j {Bj } and j {Bj } are both |-inconsistent sets. That

is, j , Bj | . But then j | Bj by I; similarly, j | Bi by E. So j is

|-inconsistent contradicting the assumption that j is |-consistent.Next show that is a maximal |-consistent set. By the construction we know that

for every B, either B or B . In either case we would get a |-inconsistent

set by extending by any sentence B.

Lemma 2 Take any maximal |-consistent set and define VC:

VC(A) = t if and only if A .

VC(A) = f if and only ifA .

Claim: VC is a classical valuation of L (with interpreted as the material

conditional).

Proof Note that VC is well-defined as for no A do we have both A and A .

Furthermore, for every A, either VC(A) = t or VC(A) = f (thus VC is bivalent).

(). First: VC(A) = t if and only ifA if and only if VC(A) = f. Second:

Assume that VC(A) = f. It follows that A . Thus A and so A ,

so VC(A) = t.

(&). Assume that VC(A&B) = t, then A&B . We have (due to Reflexiv-

ity) A&B, A | A&B and so (due to &Acc) A&B, A | A. We also have (due to

Reflexivity) A&B, A | A. So (due to &Acc) A&B, A | A&A. As A&B

and as is consistent, A not . Similarly B . But then A, B and so

VC(A) = VC(B) = t.

Assume that VC(A&B) = f, then (A&B) . We have (A&B), A, B |

A&B (due to Reflexivity and &-IE). We also have (due to Reflexivity) (A&B),

A, B | (A&B). So (due to &-IE) (A&B), A, B | (A&B)&(A&B). Thus, as

(A&B) , we cannot have both A and B in and so either A or B .

But then either VC(A) = f or VC(B) = f.

(). Assume that VC(A B) = t, then A B . Assume for reductio that

VC(A) = t and VC(B) = f. Then A, B . By Reflexivity B, A B | A

B andso,by(Acc), B, A B, A | B. But we also have B, A B, A | B.

So by (&Acc) B, A B, A | B&B. But then is not a |-consistent set. Thus


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(A B), B | (A B)&(A B). Thus B and so B and VC(B) =

f. Similarly, as (A B), A, A | B, we have (A B), A | A B. But

then (A B), A | (A B)&(A B). So A and VC(A) = t.

So assume that | A. Note that {A} is |-consistent (as otherwise, by -E, | A). So {A} can be extended to a maximal |-consistent set . As

and A , it follows that for each B , VC(B) = t while VC(A) = f. As VC is a

classical valuation we have shown that A is not a classical consequence of.


(1) Assume that A is classically equivalent to B (it follows by Supra-classicality that

, A | B and , B | A). It is enough to show one direction of the Lemma: If, A | C, then , B | C. Proof by induction over the length of C.

C = p. Assume that , A | p. As , A | B we have , A, B | p by Restricted

Cautious Monotony. As , B | A we have , B | p by Restricted Cut.

C = D. Assume that the claim of the Lemma holds for D. Assume that ,

A | D. As , A | B we have , A, B | D by Restricted Cautious Monotony

(note that every instance of in D occurs within the scope of ). As , B | A

we have , B | D by Restricted Cut.

C = D E. Assume that the claim of the Lemma holds for D and E. Assume

that , A | D E. Then , A, D | E, but then (by the induction hypothesis), B, D | E and so , B | D E.

C = D&E. Assume that the claim of the Lemma holds for D and E. Assume

that , A | D&E. Then , A | D and , A | E, but then (by the induction

hypothesis) , B | D and , B | E and so , B | D&E.

(2) Show that , A, B | C if and only if, A&B | C. Proof by induction over

the length ofC.

C = p. Assume that , A, B | p. As , A, B | A&B we have , A, B,

A&B | p by Restricted Cautious Monotony. As , A&B | A and , A&B | B we

have , A&B | p by two applications of Restricted Cut. Assume that , A&B | p.

As , A&B | A and , A&B | B we have , A, B, A&B | p by two applica-

tions of Restricted Cautious Monotony. As , A, B | A&B we have , A, B | p

by Restricted Cut.

C = D. Assume that , A, B | D. As , A, B | A&B we have , A, B,

A&B | D by Restricted Cautious Monotony. As , A&B | A and , A&B |

B we have , A&B | D by two applications of Restricted Cut. Assume that

, A&B | D. As , A&B | A and , A&B | B we have , A, B, A&B |

D by two applications of Restricted Cautious Monotony. As , A, B | A&B we

have , A, B | D by Restricted Cut.

C = D E. , A, B | D E if and only if , A, B, D | E if and only if

(by the induction hypothesis) , A&B, D | E if and only if, A&B | D E.
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We need to show that a given satisfies all the constraints imposed on Categorical

inference relations. This is fairly trivial for &Acc, I, I, E, I, E and Reflexivity,

so here it is shown only that satisfies E, Cut and Cautious Monotony.E. Assume that , A C and , B C, i.e. that (|| |A|) |C| and

(|||B|) |C|. By requirement 3 on , ((|||A|)(|||B|)) (|||A|)

(|| |B|) and so ((|| |A|) (|| |B|)) |C|. But then (|| (|A| |B|))

|C| and so (|| |A B|) |C|, i.e. , A B C.

Cut and Cautious Monotony. Assume that A, i.e. (||) |A|. By require-

ment 2 on , (||) = (|| |A|). So , A B if and only if B.


The proof is omitted as the proof of Theorem 8 below can easily be adapted to the

categorical case.


We need to show that a given satisfies all the constraints imposed on Conditional

Inference relations. This is trivial for Acc and &Acc. So we need to show that the

remaining conditions are satisfied.First a number of auxiliary properties are presented.

Lemma 3 If B is categorical then A B if and only if (|A|) |B|.

The proof, which proceeds by induction over the length of B is quite trivial and is

omitted.

Lemma 4 If|A| = |B|, then A C if and only if B C.

Proof Assume that |A| = |B|. Proof by induction over C. The cases when C = pand C = D are trivial.

C = D E. As |A| = |B|, we have |A| |D| = |B| |D| and so |A&D| =

|B&D|. So A D E if and only if A, D E if and only if (by the induction

hypothesis) B, D E if and only if B D E.

C = D&E. A D&E if and only if A D and A E if and only if (by the

induction hypothesis) B D and B E if and only if B D&E.

Lemma 5 If A, B C, then (|A|) = (|A&(B C)|).

Proof Assume that A, B C. Proof by induction over the length of C.

C = q. We have (|A| |B|) |q|. Thus (|A| |B|) |B q|. Furthermore


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C = E F. It is assumed that A, B E F, i.e. A, B, E F, i.e.

A, B&E F. Now: (|A|) = (|A&((B&E) F)|) = (|A&(B (E

F))|) and we are done.

C = E&F. A, B E&F. Thus A, B E and A, B F. By the induc-

tion hypothesis (|A&(B E)|) = (|A&(B F)|). But then (|A&(B E)|) |B F| and so (|A&(B E)|) = (|A&(B E)&(B F)|) =

(|A&(B (E&F))|).

C = E. A, B E. So (|A&B|) |E|. But then (using the same argument

as the base case when C = p), (|A|) = (|A&(B E)|).

Corollary 1 If A B, then (|A|) = (|A&B|).

Proof Assume that A B. Then A, B.ByLemma 5, (|A|) = (|A&(

B)|) = (|A&B|).

Corollary 2 If A B, then (|A|) |B|.

Lemma 6 If|A| |B|, then A B.

Proof Proof by induction over the length of B. The cases when B = p and B = C

are trivial.

B = C D. Assume that |A| |C D|. Thus |A&C| |D|. By the induction

hypothesis A, C D and so A C D.

B = C&E. Assume that |A| |C&D|. So |A| |C| and |A| |D|. By theinduction hypothesis A C and A D, so A C&D.

Lemma 7 I If A, then A B and B A.

E If, A C and, B C, then , A B C.

I If, A , then A.

E If, A , then A.

I If A&A, then .

E If , then B, for any B.

Proof (I.) Assume that A. We need to show that A B, i.e. that

(A&B). By Corollary 2, (||) |A|. So (||) |(A&B)|. But

then (A&B).

(E.) Assume that , A C and , B C. Proof by induction over C.

C = p. It is assumed that , A p and , B p. So (|| |A|) |p| and

(|| |B|) |p|. Thus (by Requirement 3), (|| (|A| |B|)) |p| but then

, A B p.

C = D E. As , A D E and , B D E, we have As , A, D E and , B, D E. By the induction hypothesis , A B, D E and so As

, A B D E.

C = D&E. As , A D&E and , B D&E, we have As , A D,
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Synthese (2009) 171:4775 73

(I.) Assume that , A . By construction (|| |A|) = . Furthermore

|| = (|| |A|) (|| | A|). By Requirement 3, ((|| |A|) (|| | A|)) =

(||) (|| |A|) (|| |A|). So (||) (|| |A|). But then

(||) |A|. So A.

(E.) Proof by induction over A.A = p. Assume that , p . Then (|||p|) = . But then (via Require-

ment 3) (||) |p| and so p.

A = B&C. Assume that , (B&C) . Then (|| |(B&C)|) = .

(|| (|B| | C|)) = . But then (|| (|B| | C|)) = (|| | B|) =

(|| |C|). So , B and , C . By the induction hypothesis

B and C and so B&C.

A = B C. Assume that , (B C) . Then (|||(B C)|) = .

As |(B C)| = |B| | C|, (|| |B| | C|)) = . But then , B, C

and so, by the induction hypothesis, , B C. So B C.A = D. Assume that , D . Then (|| |D|) = . So (||)

|D|. But then D.

(I.) Assume that A&A. By Corollary 2, (||) |A&A|. But

|A&A|

= so (||) = and so .

(E.) Assume that . Then (||) = . Prove by induction over A that

A. The cases when A = p and A = B are trivial.

A = B&C. By the induction hypothesis B and C so B&C.

A = B C. As

, , B

. By the induction hypothesis , B

C.So B C.

Corollary 3

Reflexivity , A A.

R-Cut If A and B, then , A B, provided that B is categorical.

R-Cautious Monotony If A and , A B, then B, provided that B

is categorical.

Proof Reflexivity. An immediate consequence of Lemma 6.

R-Cut and R-Cautious Monotony. Assume that A. By Corollary 2, (||) |A|. Thus (||) = (|| |A|). By Lemma 3, , A B if and only B as

long as B contains no conditional.


Given a categorical inference relation | we need to construct a preferential model

(U, I, ) such that |=.

Let | be a categorical inference relation. Let U be the set of all maximal classically

consistentsets of sentences of L.Define:
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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74 Synthese (2009) 171:4775

Note (i) that for any u U and sentence A: u |A| if and only if A u. Note (ii)

that is well-defined. For if|A| = |C|, then A and C are classically equivalent. By

Theorem 2, A | B if and only ifC | B.

Let M = (U, I, ).

Lemma 8 M is a preferential model.

Proof We need to show that has the requisite properties.

(1) Show (|A|) |A|. Note that for any A, |A| = |A|. By Reflexivity A |

A. From Theorem 2, A | A. Thus for every u |(|A|), A u, i.e.

u |A| and so u |A|.

(2) Show that if |(|A|) |B|, then |(|A&B|) = |(|A|). Assume that

|(|A|) |B|.

First show that A | B. Assume for reductio that A | B. Let

= {C|A | C and C is categorical}.

Assume that |C B. Then there are C1, . . . , Cn such that C1& &Cn |CB. Then C1& &Cn |C B. By Supra-classicality, C1& &Cn | B as

A | Ci , 1 i n, A | C1& &Cn . By R-Cut A | B contrary to assumption.

Thus |C B. But then {B} can be extended to a maximal classically consistent

set u UL . As u, u (|A|) and as B u, u |B| contrary to assumption.So A | B.

By R-Cut and R-Restricted monotony A | C if and only if A, B | C, for any

categorical C and as, by Theorem 2, A, B | C if and only if A, B | C, we have

thus (|A|) = |(|A&B|).

(3) Show that (|A B|) (|A|) (|B|). Assume that u (|A|) (|B|).

Then there are categorical C and D such that A | C and B | D such that C, D u.

By Acc, A | C D and B | C D. By E, A B | C D. As C, D u,

C D u. so u (|A B|).

Lemma 9 Where is the acceptance relation defined by the model M , for all A and

B: A B if and only if A | B.

Proof By induction over the length of B.

B = p. p is categorical so A | p if and only if (|A|) |p| = I(p) if and only

if A p.

B = C. C is categorical so A | C if and only if (|A|) |C| = U |C|

if and only if A C.

B = C&D. A C&D if and only if A C and A D if and only if (by theinduction hypothesis) A | C and A | D if and only if A | C&D.

B C D A C D if d l if A C D if d l if (b h


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References

Adams, E. W. (1975). The logic of conditionals. Dordrecht: Reidel.

Alchourrn, C.,Grdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet functions

for contraction and revision. Journal of Symbolic Logic, 50, 510530.

Arl-Costa, H. (1995). Epistemic conditionals, snakes and stars. In L. Farinas del Cerro, G. Crocco, &

A. Herzig (Eds.), Conditionals, from philosophy to computer science (pp. 193239). Oxford University

Press. Vol. 5 of Studies in Logic and Computation, Series editor: D.M. Gabbay.

Arl-Costa, H. (1999). Belief revision conditionals: Basic iterated systems. Annals of Pure and Applied

Logic, 96, 328.

Arl-Costa, H. (2007). The logic of conditionals. Stanford Encyclopedia of Philosophy.

Bennett, J. F. (2003). A philosophical guide to conditionals. Oxford: Oxford University Press.

Brandom, R. (1994). Making it explicit. Cambridge: Harvard University Press.

Broome, J. (2001). Normative practical reasoning. Proceeding of the Aristotelian Society (pp. 175193).

Supplementary Volume 75.

Broome, J. (2002). Practical reasoning. In J. Bermdez & A. Millar (Eds.), Reason and nature: Essays in

the theory of rationality (pp. 85111). Oxford: Oxford University Press.Cantwell, J. (2008a). Indicative conditionals: Factual or epistemic? Studia Logica, 88, 157194.

Cantwell, J. (2008b). The logic of conditional negation. Notre Dame Journal of Formal Logic, 49(3),

245260.

Dummett, M. (1997). Elements of intuitionism. Oxford: Clarendon Press.

Edgington, D. (1995). On conditionals. Mind, 104, 235329.

Gabbay, D. (1985). Theoretical foundation for non-monotonic reasoning in expert systems. In Proceeding

NATO Advanced Study Institute on Logics and Models of Concurrent Systems (pp. 439457). Berlin:

Springer.

Grdenfors, P., & Makinson, D. (1991). Relations between the logic of theory change and nonmonotonic

logic. In A. Fuhrmann & M. Morreau (Eds.), The logic of theory change, Berlin: Springer Verlag.

Gentzen, G. Investigations into logical deduction. In M. E. Szabo (Ed.), In the collected works of Gerhard

Gentzen. Amsterdam: North Holland.Hansson, S. O. (1992). In defense of the Ramsey test. Journal of Philosophy, 89, 522540.

Klaus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumu-

lative logics. Artificial Intelligence, 44, 167207.

Levi, I. (1988). Iteration of conditionals and the Ramsey test. Synthese, 76, 4981.

Levi, I. (1996). For the sake of the argument. Cambridge: Cambridge University Press.

Lewis, D. (1976). Probabilities of conditionals and conditional probabilities. Philosophical Review, 85,

297315.

McGee, V. (1985). A counterexample to modus ponens. Journal of Philosophy, 82, 462471.

Politzer, G. (2007). Reasoning with conditionals. Topoi, 26, 7995.

Prawitz, D. (1965). Natural deduction: A proof-theoretical study. Almqvist & Wiksell.

Quine, W. V. O. (1950). Methods of logic. New York: Holt, Rheinhart and Winston.Ramsey, F. P. (1929/1990). Law and causality. In D. H. Mellor (Ed.), F. P. Ramsey: Philosophical papers.

Cambridge: Cambridge University Press.

Stalnaker, R. (1975). Indicative conditionals. Philosophia, 5, 269286. Reprinted in Ifs, ed. W. L. Harper,

R. Stalnaker and G. Pearce 1976 by Reidel.

Weatherson, B. (2001). Indicative and subjunctive conditionals. The Philosophical Quarterly, 51(203),

200216.


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