Synthese (2008) 164:19–44DOI 10.1007/s11229-007-9214-5

The evidential support theory of conditionals

Igor Douven

Received: 4 July 2006 / Accepted: 14 June 2007 / Published online: 15 August 2007© Springer Science+Business Media B.V. 2007

Abstract According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically sig-nificant relation between the conditional’s antecedent and its consequent. This paperpoints to some linguistic data that our current best theories of the foregoing type ap-pear unable to explain. Further, it presents a new theory of the same type that does nothave that shortcoming. The theory is then defended against some seemingly obviousobjections.

Keywords Conditionals · Probability · Semantics · Bayesian epistemology

According to so-called epistemic theories of conditionals, the assertability/accept-ability/acceptance of a conditional requires the existence of an epistemically relevantrelation between the conditional’s antecedent and its consequent. It is the workinghypothesis of this paper that some such theory is correct at least for indicative con-ditionals. However, below I will point to some linguistic data that our current bestepistemic theories of conditionals appear unable to explain. I further present a newtheory of the same type that does not have that shortcoming and that also seems towithstand further scrutiny.

As a preliminary point, I should note that the present paper’s main focus will beon simple indicative conditionals, that is, indicative conditionals the antecedents andconsequents of which do not contain conditionals, and then chiefly on the assertability/acceptability conditions of those conditionals, and not so much on their truth condi-tions (if they have any). Also, I will leave compounds of conditionals undiscussed.This is not much of an omission if those are right who have argued that a substantialpart of the embedded conditionals we encounter in daily speech can be reduced to

I. Douven (B)Institute of Philosophy, University of Leuven, Leuven, Belgiume-mail: igor.douven@hiw.kuleuven.be

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unembedded ones and that of the remaining ones we are unable to make senseanyway.1 Unless otherwise indicated, by “conditional” I shall mean “simple indic-ative conditional.”

1 Epistemic theories of conditionals

I begin by briefly reviewing what are arguably the most prominent epistemic theoriesof conditionals, to wit, those of Adams, Jackson and Lewis, and Mellor.2

Of these, Adams’s theory is the simplest. It says that both the assertability and theacceptability of a conditional for a given person are measured by the person’s degreeof belief in the consequent conditional on the antecedent. That is, where Pr(·) is theperson’s degrees of belief function at a certain time, Pr(Q | P) measures both theassertability and acceptability of “If P, Q” for her at that time.3 So, for instance, tosay that a given conditional is highly assertable for the person is to say that her degreeof belief in the consequent given the antecedent is close or equal to 1. For Adams, thisis all there is to conditionals. In particular he does not think we need some story abouttruth conditions for conditionals, for on his view conditionals lack truth conditions.

On the latter point he is in disagreement with Jackson and Lewis (among others).In their view, the conditional “If P, Q” has the truth conditions of the material impli-cation P → Q. This does not mean that according to them a conditional is highlyassertable or acceptable if the corresponding material implication is true. Rather theyhold that “If P, Q” is highly assertable for one/acceptable to one iff P → Q is highlyprobable on one’s degrees of belief function and is “robust” with respect to P, mean-ing that Pr(P → Q | P) is high and close to Pr(P → Q). As Jackson (1987, p. 31)notes, the latter conditions boil down to requiring that Pr(Q | P) be high. He furtherargues that the natural way to generalize these assertability/acceptability conditionsto degrees of assertability/acceptability other than “high,” yields Adams’s view that aconditional’s assertability/acceptability for a given person at a given time is measuredby the person’s degree of belief at that time in the consequent given the antecedent(Jackson 1987, p. 32). How do conditionals possess these assertability/acceptabilityconditions if it is not in virtue of their truth conditions? According to Jackson andLewis the answer lies in the conventional meaning of the word “if,” just as it is due tothe conventional meaning of “but” that “P but Q” is unacceptable/unassertable unlessthere is some sort of contrast between P and Q.4,5

1 See, among others, Jackson (1987, pp. 127–137) and Edgington (1995, p. 382 ff).2 See Adams (1966, 1975), Lewis (1976), Jackson (1979, 1987), and Mellor (1993).3 Adams assumes, as will I throughout the paper, that degrees of belief always are rational in that theyconform to the probability calculus. I should also note that a person’s degrees of belief at a certain timeare always assumed to be degrees of belief relative to her background knowledge at that time. So, to bemaximally precise I should say that for Adams Pr(Q | P, K ) measures the acceptability/assertability of “IfP, Q” for a given person at a given time, where K is the background knowledge the person possesses atthat time. I will mostly suppress reference to the background knowledge, however.4 Lewis first thought that the assertability/acceptability conditions of conditionals were governed by aconversational implicature, but in the postscript to his 1976 paper he adopted Jackson’s proposal that theimplicature is a conventional one.5 Jackson and Lewis put their theories in terms of assertability only (later Jackson 1984, p. 72, declaredthat it had been better put in terms of “assentability”). As Edgington (1986) pointed out, however, their

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Mellor’s theory needs a bit more explaining. On his disposition theory ofconditionals, the conditional “If P, Q” “expresses a disposition to infer Q from P”(Mellor 1993, p. 236) or, as he puts it in slightly different terms, “fully to accept asimple ‘If P, Q’ is to be disposed fully to believe Q if I fully believe P …” (ibid.),where “fully to believe a proposition P” is to believe P to a degree close, but notnecessarily equal, to 1 (p. 233). Given a realist understanding of dispositions, whichMellor endorses, the foregoing translates into: “to be disposed to infer Q from P isto have an intrinsic property G such that believing P while I am G will cause me tobelieve P” (p. 241; italics omitted). The theory generalizes to degrees of acceptance ofa conditional in the following way: to accept a conditional to degree n (0 � n � 1) isto have an intrinsic property such that coming to believe the antecedent will cause oneto believe the consequent to degree n, provided one retains the said intrinsic property(p. 236). Mellor (p. 234 n6) is explicit that he is not concerned with assertability. Itis worth noting, though, that, although less explicitly, he also is not really concernedwith acceptability; the disposition theory purports to state what it is to accept a condi-tional, not when a conditional is acceptable. (If it makes sense to say that it is alrightto have a certain disposition,6 then Mellor’s theory naturally suggests that “If P, Q”is acceptable exactly if it is alright to have a disposition to infer Q from P. But I canmake my point against Mellor purely in terms of acceptance.)

2 The linguistic data

This section starts by presenting some linguistic data concerning conditionals andthen shows how they cast aspersion on each of the theories just depicted. Consider thesentences (1α) and (1β):

(1) There will be at least one heads in the first 1,000,000 tosses of this fair coin

if

{there is a heads in the first ten tosses. (α)

Chelsea wins the Champions League. (β)

Assuming that we are considering some fair coin to be tossed at least 1,000,000 times,and making no special assumptions about the context of utterance, it appears that(1α), while not a deep truth perhaps, is perfectly assertable/acceptable and that (1β)is assertable/acceptable, if at all, to a vastly lesser extent.

Footnote 5 continuedconcern should at least also have been with the acceptability of conditionals. But while the notion ofconventional meaning (or conventional implicature) first surfaced in discussions about the rules of goodconversation, and thus about what can and cannot be properly asserted, I do not see any impediment toholding that it is a factor in what we can accept, too. For instance, while I believe that Sue and Jim fellin love and then got married, none of my beliefs would be rightly expressed by means of the sentence“Sue and Jim fell in love and yet they got married.” Given that I do not believe there to exist any contrastbetween their falling in love and their getting married, it is not just the case that I cannot properly assert theaforementioned sentence, I cannot—and, as I will point out in the text below, should not—believe it either(even though I can, and do, believe the nearly identical “Sue and Jim fell in love and then got married”).6 This it may well do. It at least appears that we have no difficulty understanding such phrases as that it isalright to be disposed to infer “2 + 2 = 4” from the axioms of Peano Arithmetic or that it is not alright tobe disposed to infer “2 + 2 = 5” from those axioms.

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First off, let me note that this example stands for a whole type of data that areproblematic for the theories of Sect. 1, but a general description of the type must bepostponed to the next section. Second, we cannot without further ado assume that theexample is of more than superficial relevance to theories of conditionals. It might be,after all, that the asymmetry in the assertability of (1α) and (1β) has an explanationpurely in Gricean terms. If so, the example could hardly be brought to bear against anyof the theories of the previous section—unless the theory or theories were somehowincompatible with Gricean pragmatics, which none of them seems to be—nor could itbe used in support of the new theory to be offered in Sect. 3, both of which I intend todo. And prima facie, (1β) does suggest that a Gricean explanation of the asymmetryshould work. For the sentence seems unassertable on the grounds that it suggests aconnection between Chelsea’s winning the Champions League and there being a headsin the first 1,000,000 tosses, a connection that, we may reasonably assume, does notexist, so that the suggestion is misleading. And it is one of the core insights of Griceanpragmatics that even a true sentence may be unassertable because by asserting it aspeaker would mislead her audience by suggesting something false. Still, assumingthe Gricean machinery, how exactly is the suggestion of there existing a connectionbetween the truth of (1) and that of (β) supposed to come about?

It may be tempting to invoke Grice’s (1989a, p. 26) first Maxim of Quantity here,according to which we should make our contribution to a conversation as informativeas is required for the purposes of the conversation, or, in Jackson’s (1979, p. 112) morespecific formulation, we should “[a]ssert the [logically] stronger instead of the weaker(when probabilities are close).” First note, though, that whether (1) on its own reallyis logically stronger than (1β) will depend on what the truth conditions of these sen-tences are, if the latter has any at all (which, as was said, at least Adams would deny).If we grant Jackson and Lewis, and also Mellor,7 that, as they think, the conditional“If P, Q” has the truth conditions of P → Q, then (1) is indeed logically strongerthan (1β), and the probabilities of (1) and of (1β) are very close to each other (seebelow for the exact values). But even so, the strategy seems hopeless, for given thepresent assumption, (1) is also logically stronger than (1α), and the probability of (1)is very close to that of (1α) as well. Accordingly, the proposed Gricean explanationwould seem to explain both too much and too little: it would “explain” why (1α) isunassertable—but (1α) is, as we said, highly assertable—and it would thereby fail toexplain the asymmetry in the assertability of (1α) and (1β).8,9

7 See Mellor (1993, Sect. 4).8 See also Jackson (1979, Sect. 2).9 In his discussion of conditionals, Grice (1989b, p. 61) literally says about the first Maxim of Quantity that“to make a less informative rather than a more informative statement is to offend against the first maximof Quantity, provided that the more informative statement, if made, would be of interest.” An anonymousreferee suggested that the contexts in which (1α) is assertable are ones in which that sentence is of interestbut (1) is not, namely, theoretical contexts in which we are suspending judgement about the truth of (1),or are ignoring its truth value. Since—the referee further argued—in those contexts (1) is not assertable(because of the suspension of judgement), (1β) is not assertable either. That, according to him/her, explainsthe asymmetry between (1α) and (1β). While Grice does not say much about the notion of interestingnesshe has in mind, it would seem that, in a colloquial sense of the word, (1α) can be perfectly assertable bothif, as in most contexts will be the case, it is a dull and uninteresting claim and if (1) is of interest. More gen-erally, since, as seems undeniable, people assert all sorts of uninteresting things—including uninteresting

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Another maxim to consider in this connection is the second Maxim of Quality,according to which we should have adequate evidence for what we say (Grice 1989a,p. 27). Supposing that (1β) has the truth conditions of “Chelsea wins the ChampionsLeague → There will be at least one heads in the first 1,000,000 tosses,” and thusthose of “¬Chelsea wins the Champions League ∨ There will be at least one heads inthe first 1,000,000 tosses,” there seems to be an immediate problem with this sugges-tion. For I do have excellent evidence for the disjunction, and thus, given the currentsupposition, for (1β), namely, all the evidence I have for the second disjunct, that is,(1). But Grice (1989b, p. 61 f) is clearly aware of this problem and demands that theevidence for the conditional must be of a non-truth-functional kind, meaning that itmust not be just evidence for either the consequent or the negation of the antecedent.Why is that? Well, otherwise I am clearly violating the first Maxim of Quantity, forif I have excellent evidence for, say, the consequent, then I should assert that, and notthe logically weaker conditional. As we just saw, however, this cannot be right. For ifit were, (1α) should be unassertable, which patently it is not.10

Some might intuit that an appeal to the Maxim of Relevance (or Relation) might behelpful here.11 This maxim, about which Grice (1989a) is exceedingly brief, simplystates that we should make our contribution to a conversation a relevant one. Andwhile it may be that asserting (1β) could only be of relevance if its antecedent andconsequent were related in some intuitive way, and would hence create the—giventhe minimal suppositions we made about the context of utterance—misleading im-plicature that they are in fact thus related, that one will usually make an irrelevantcontribution to a conversation by asserting a conditional whose antecedent and con-sequent fail to be related (in the requisite sense) is not something that follows fromthe Maxim of Relevance, nor, it seems, from any other of Grice’s maxims. In fact, itrather seems to be the sort of thing to be explained by a theory of conditionals. At aminimum, a theory of conditionals that can explain why it is an irrelevant contributionto a conversation to assert a conditional whose antecedent is irrelevant to its conse-quent is, all else being equal, to be preferred to one that must accept the foregoing as aprimitive fact. (My own explanation for (1β)’s unassertability indeed will have to do

Footnote 9 continuedconditionals—that are not therefore unassertable in any pretheoretically plausible sense, I think that thecategories of assertability and interestingness should be kept separate. It is hard to conceive of a reasonwhy the Gricean would want to disagree with this. After all, the central idea of Gricean pragmatics is thatby asserting a truth one may implicate a falsehood and thereby mislead one’s audience. And although it iswell imaginable how in a conversation between well-educated people an assertion of (1α) may be devoid ofany interest (though it may then still serve some broadly social goal, like keeping the conversation going),it will not normally be misleading; for instance, among well-educated people it will be pretty obvious thatthe asserter knows that (1) is overwhelmingly likely and so the audience will not be inclined to infer froman assertion of (1α) that the asserter takes herself not to be in the position to assert the stronger (1).10 Barker (1997, p. 205 f), in defending a material implication semantics of indicative conditionals, arguesthat someone may want to assert such a conditional even if she is in the position to assert either the nega-tion of the antecedent or the consequent, because she may want to convey the information that a certainconnection holds between antecedent and consequent. I have nothing to object to this idea. The theory tobe presented in the next section could in effect be regarded as an attempt to chart the nature of the requisiteconnection; Barker’s paper is silent on that.11 Though Grice apparently did not. His own paper on conditionals (1989b) does not contain a singlereference to the Maxim of Relevance.

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with what one may regard as a lack of relevance of the antecedent to the consequent,but this lack of relevance is then itself explained in more basic terms.)

Elsewhere I have defended the view that the practice of assertion is governed bythe rule that one ought to assert only what is acceptable to one.12 If this is correct, as Iwill here assume, then that brings the assertability of a conditional and its acceptabil-ity close enough to one another to make it harmless, or at least excusable, to refrainfrom differentiating between the two in most of what follows. Nevertheless, becauseit occurred to me that people tend to have clearer intuitions about the unassertabilitythan about the unacceptability of (1β), let me briefly point to a reason independent ofany considerations related to assertability for believing that (1β) should come out asbeing unacceptable, or at least not highly acceptable, on any adequate theory of theacceptability of conditionals.

Consider again the sentence

(2) Sue and Jim fell in love and yet they got married.

I do not accept this, I said in note 5, even though I do accept the sentence

(3) Sue and Jim fell in love and got married.

This may seem strange, especially if we agree with common lore that (2) and (3) havethe same truth conditions.13 But even if there is no semantic distinction between thetwo sentences, that does not imply that there could not be an important epistemicdistinction between them. And it seems that from the point of view of what one mightcall epistemic hygienics there may be a good reason for not holding (2) acceptable,and for not wanting to accept it, even if one does accept (3). Suppose I accept (2). Inow know that there is no contrast between Sue and Jim’s falling in love with eachother and their later marriage. But we may forget things we know. Finding among mybeliefs one to the effect that Sue and Jim fell in love and yet got married, and havingforgotten that the contrast suggested by the phrasing of my belief does not exist, I maycome to think that there was such a contrast. Doubtless there must have been somereason for storing this belief in its specific wording! I might for instance speculatethat perhaps Sue and Jim had both been married before and that at some point eachof them had declared to me that if they were ever to be in love again, they would notmarry for a second time, and that I had forgotten their declarations. The simple wayto forestall this mistake is to accept (3) but refrain from accepting (2).

The same applies to (1β). Even if I find “Chelsea wins the Champions League →There will be at least one heads in the first 1,000,000 tosses” acceptable, it would seemimprudent to accept the conditional (1β). Accepting that sentence might later lead meto wonder whether perhaps there was some connection I have forgotten about betweenChelsea’s winning and there being a heads in the first 1,000,000 tosses. Right now, Iknow that such a connection does not exist. But there is no guarantee that I shall not

12 See Douven (2006a); this is in opposition to the more widely held view that one ought to assert onlywhat one knows, as defended in Williamson (2000), Adler (2002), and DeRose (2002), among many others.13 Which may not be mandatory. See for instance Récanati (1989, 2002) for some arguments to the effectthat common lore is wrong in this respect.

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forget this.14 (Note that I have still not explained how accepting (1β) might lead mewrongly to believe, or at least surmise, that there must be some connection betweenthe sentence’s antecedent and its consequent. That explanation will have to wait untilafter the statement of our theory in Sect. 3.)

To see, then, why the pair (1α) and (1β) constitutes a problem for the earlier-pre-sented theories, first note the following facts. That there will be at least one heads in thefirst 1,000,000 tosses has a very high a priori probability, namely, 1 − 1/2106 ≈ 1.15

Conditional on (α), the truth of (1) is of course certain. But Chelsea’s winning theChampions League is, we may assume, probabilistically independent of there beingat least one heads in the first 1,000,000 tosses. So, conditional on either (α) or (β)

the probability of (1)’s being true is at most marginally different from 1. Furthermore,we can bring the probability of (variants of) (1) conditional on (β) as close as welike to that of (1) conditional on (α) by supplanting the number 1,000,000 in (1) by alarger one.

It follows immediately that, on Adams’s theory, both (1α) and (1β) are highly asser-table/acceptable for any rational person, for we may assume that for any such personPr ((1) | (α)) ≈ Pr ((1) | (β)) = high. Of course, on that theory (1α) is perfectlyassertable/acceptable while (1β) is not quite. But the theory predicts that the differ-ences in assertability/acceptability are only minute,16 and that is manifestly wrong. Itfurther predicts—on account of the fact mentioned in the last sentence of the previousparagraph—that by making some suitable substitution the assertability/acceptabilityof (variants of) (1β) can be brought arbitrarily close to the perfect assertability/accept-ability of (1α), which appears wrong too, for it seems that the intuitive unassertability/unacceptability of (1β) is fairly independent of the exact number of times the coin ispresumed to be tossed.17

Jackson and Lewis, too, get a wrong result for (1β), and for the same reason asAdams does: (1)’s probability is high conditional on (β), so that (1β) comes out as beinghighly assertable/acceptable on their account. Moreover, the degree of assertability/acceptability of this sentence is not appreciably different from that of (1α) on Jackson’sand Lewis’s theory.

14 Supposing that sentence pairs such as (2) and (3) have the same truth conditions, the point made in thisparagraph assumes that our beliefs are not just individuated by their propositional contents. But that theyare thus individuated has never seemed a plausible position to me: believing that Sue and Jim fell in loveand nevertheless married is, introspectively, and to me at least, not at all the same as believing that theyfell in love and therefore married. Incidentally, the idea that broadly Gricean considerations traditionallythought to apply only to conversation have relevance to what we can accept, too, was, to my knowledge,first hinted at in Harman (1982, p. 255).15 I take the assumption that the coin is fair to imply that the tosses are probabilistically independent.16 For all Adams suggests, on his view Pr(Q | P) and the assertability/acceptability of “If P, Q” varylinearly with one another.17 Kaufmann (2004) argues along different lines that Adams’s proposal sometimes deviates starkly fromintuition. However, as I argue in my (2007a), our intuitive verdicts about the cases Kaufmann depicts arehighly dependent on how the probabilistic information relevant to them is presented to us, and the kind ofverdict that appears to militate against Adams’s proposal may only follow because in Kaufmann’s examplesthat information is presented in a way that psychologists have long known to confuse people’s intuitions inpredictable ways. I thus conclude that the intuitions to which Kaufmann appeals are to be explained awayrather than explained.

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The story is somewhat different for Mellor’s theory. He gets (1α) rather than (1β)wrong. To see why, imagine that we are considering a fair coin that is about to betossed 1,000,000 times. Supposing “this fair coin” in (1) refers to this coin, I accept(1α) unhesitatingly. Note, though, that it seems reasonable to believe that there willbe at least one heads in the first 1,000,000 tosses regardless of the truth-value of (α);as we said, the probability that there will be a heads is a priori very close to 1, so closethat—one can reasonably suppose—anyone adopting that probability as her degree ofbelief in the proposition will qualify as believing it on Mellor’s account (or if she doesnot so qualify, we can readily adapt the example so as to make her qualify as believingit; let the coin be tossed 10100 times, for instance). Thus, although (1) follows from(α), I am not disposed to infer the former from the latter, simply because I believe(1) already, and I need not infer from anything that which I already believe. Put inMellor’s realist terms, should I come to believe (α) fully, that will not cause me tobelieve (1). How could it, given that, again, I already believe (1)? Accordingly, onMellor’s theory I do not count as accepting (1α). And yet by reasonable suppositionI do accept it.

One might respond that, on the contrary, someone who already believes (1) will bedisposed to infer it from anything. If that is so, the disposition theory does explain whyI accept (1α). Such a use of the term “disposition” appears strained to me, especiallygiven a realist conception of dispositions.18 But suppose we understand it. Then stillthe response fails. For, first, given the supposed notion of disposition, I should countas accepting (1β) as much or almost as much as I count as accepting (1α)—but asI said earlier, I do not find the former acceptable, or at least not nearly to the sameextent as I find (1α) acceptable, and accordingly I do not accept it, or at least not to thesame extent as I accept (1α). Secondly, and more importantly, it is simply false that Iam willing to infer (1) from anything. In fact, there are many things from which I amreluctant to infer (1) and that would even cause me to abandon my belief in (1)—forinstance, “There is no heads in the first 999,999 tosses” would do so.

There still might seem to be an easy escape route for Mellor. For why not have histheory say (or interpret it as saying) that to accept an indicative conditional “If P , Q”is to have the disposition to infer Q from P if one does not, or were not to, believe Qalready?19 But, first, it would at a minimum seem odd to analyze the acceptance ofindicative conditionals in terms of counterfactual conditionals. More importantly, onemay wonder whether the modification really helps. For would I not directly believe (1)under all circumstances were I told that this (pointing to the coin referred to in thesentence) is a fair coin that will be tossed at least 1,000,000 times? And of course wereI not told that, I would certainly not infer (1) from (α); from “there is one heads in thefirst ten tosses with this coin” it cannot be inferred that there will be a first 1,000,000tosses—the coin might be destroyed right after the tenth toss. So it may well be that, assoon as I am told that this fair coin is to be tossed 1,000,000 times, I will immediatelybelieve that there will be at least one heads among those tosses and so will not infer

18 Surely Mellor himself would not want to respond in this way. For he says on p. 437 of his (1993) thathe does not accept “If P, Q” for all P and Q that he fully believes, “since most of my beliefs are causallyindependent of each other.”19 Thanks to Hugh Mellor for suggesting this possibility (in personal correspondence).

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that from (α). To put this differently, note that while the disposition theory, modifiedin the way suggested above, may work in the following case:

(4) If there is one heads in the first ten of a total of 1,000,000 tosses with this faircoin, then there is at least one heads in the total of 1,000,000 tosses,

where the antecedent may cause us to believe the consequent—provided we do notalready believe that, say because we have already been informed that the coin is to betossed 1,000,000 times—it still gets matters wrong, given that in

(5) If there is no heads in the first ten of a total of 1,000,000 tosses with this fair coin,then there is at least one heads in the total of 1,000,000 tosses,

we are only negligibly less disposed to infer the consequent from the antecedent thanwe are in (4): as soon as we hear that there will be 1,000,000 tosses with a fair coin, thebelief that there will be at least one heads among those tosses will virtually enforceitself upon us. But while this disposition is not noticeably weaker than the corre-sponding disposition in the case of (4), one cannot plausibly maintain that the intuitiveassertability/acceptability of (5) is not noticeably less than that of the former (but notethat (5) would become as assertable/acceptable as (4) is if “if” in (5) were replaced by“even if”).

I will not consider possible escape routes for Adams’s theory or for the Jackson/Lewis theory here, because the theory to be presented in the next section seems suf-ficiently close to those theories—or, in the case of the latter, can easily enough bebrought close to it (see Sect. 4)—that it can be regarded as offering an escape route tothem.

3 The evidential support theory of conditionals

Sentences (1α) and (1β) may serve as a further illustration of the familiar point thatfor a conditional, at least a “typical” one (see p. 31 below), to be assertable/acceptable,or at least to be highly assertable/acceptable, its antecedent and consequent must, insome specific sense, be salient to one another. It was seen that the requisite connectionbetween antecedent and consequent is at best only incompletely captured in termsof the latter having a high probability conditional on the former or of there being adisposition to infer the consequent from the antecedent. What more, or perhaps alto-gether else, is needed? Part of my proposal is that the more—in the case of Adams’s,Jackson’s, and Lewis’s theories—or the else—in the case of Mellor’s—that is neededis some kind of evidential support, where the relevant notion of evidence is the Bayes-ian one, according to which A is evidence for B iff B’s probability conditional on Aexceeds B’s unconditional probability.

The proposal is not going to be that a conditional is assertable/acceptable iff its ante-cedent is evidence—in the Bayesian sense—for its consequent. My colleague Henry’squitting his job is evidence that I shall teach next year’s introductory course in socialphilosophy, because conditional on the former the latter is a bit more probable than itis unconditionally. But even the conditional probability is still exceedingly low, giventhat I simply lack the requisite background for teaching such a course. It is in effect

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much more likely that Tom, my other colleague with a specialization in social philos-ophy, will teach the course if Henry quits his job. Thus, “If Henry quits his job, I shallteach next year’s introductory course in social philosophy” is not assertable for me/acceptable to me, notwithstanding that its antecedent is evidence for its consequent.

Nor will the proposal be that the antecedent must make the consequent certain.20

“If Henry quits his job, then Tom will teach next year’s introductory course in socialphilosophy” is perfectly assertable for me/acceptable to me. But although the anteced-ent is evidence, and indeed very strong evidence, for the consequent, it is not true thatconditional on the antecedent the consequent has perfect probability. Even if Henryquits his job, a welter of things could happen to Tom, or to the department (or both),that would prevent him from teaching the course. Just as it is too strong a requirementthat a sentence (or proposition) have probability 1 for it to be assertable/acceptable,21 itis too strong a requirement that a conditional’s antecedent make its consequent certainfor the conditional to be assertable/acceptable.

Assume, mainly for convenience of presentation, that we have some thresholdvalue t close but unequal to 1 such that a proposition P is highly probable on adegrees of belief function Pr(·) exactly if Pr(P) > t. Then, to a first approximation,the proposal is this:

(6) “If P, Q” is assertable for one/acceptable to one iff, on one’s degrees of belieffunction Pr(·), P is t-evidence for Q,

where P is t-evidence for Q on one’s degrees of belief function iff, on that function, Pis evidence for Q—that is, as explained, Pr(Q | P) > Pr(Q)—and, moreover, one’sdegree of belief in Q conditional on P exceeds t—that is, Pr(Q | P) > t.

Note how already (6) is enough to get the right answers with respect to (1α) and (1β):Even though, quite plausibly, the probability of (1) conditional on (β) exceeds t,22 (β)

is not evidence for (1). By contrast, not only is the probability of (1) conditional on(α) above t, that probability is also greater than (1)’s marginal probability, meaningthat (α) is evidence for (1). As a result, (1α) comes out as being assertable/acceptableon our proposal, (1β) as being unassertable/unacceptable, or at least—if there is adifference—not as being assertable/acceptable, as required.

But while I believe the above proposal to be nearly correct, it cannot stand unadorned.The reason is that in its current form it faces some immediate counterexamples itself.For suppose you own a ticket in a lottery. It is common knowledge that the lotteryconsists of 100 tickets and that each of these has the same chance of winning. It isunknown, however, how many winners there will be. What is known, is that eitherthere will be a unique winner or half of the tickets will end up being winners, and thatthe chances of either of these possibilities obtaining are equal. Then it can be easilycalculated that the probability that your ticket will lose equals 149/200 = .745. Nowconsider the conditional

20 See Dudman (1992) for a proposal in this vein.21 See, among others, Foley (1992), Edgington (1995, p. 287, n50), DeRose (1996), and Douven (2006a);but see also Williamson (2000, Ch. 11).22 Or if you think it does not, let the number of tosses be 1010, or 10100, or …. In this connection it isnoteworthy that according to Jackson (1979, p. 118) “ ‘At least one of the five tosses will be a head’ [saidof a fair coin to be tossed five times] is probable enough … to warrant assertion.”

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(7) If there is only one winner, then your ticket is a loser,

and assume for the time being that t = .9.23 Then the antecedent of (7) is t-evidencefor its consequent: Pr(Your ticket is a loser | There is only one winner)= .99 > t >

.745 = Pr(Your ticket is a loser). As a result, the conditional comes out as beingassertable/acceptable on the above proposal. I submit, though, that just as we find thesentence “Your ticket is a loser” unassertable/unacceptable previous to the drawing ofthe lottery and on the supposition that there is exactly one winner, we find (7) unasser-table/unacceptable on the supposition that either there will be one winner or half of thetickets will win.24 After all, were we to accept (7) and then come to learn, still previousto the drawing, that the lottery has a unique winner, it would be reasonable and alsonormal to “detach” the consequent of (7) (i.e., to apply modus ponens), whereby wewould accept “Your ticket is a loser” previous to the drawing.25

Here it is helpful to consult the philosophical literature on acceptance rules, that is,rules purportedly stating under what condition or conditions it is rational to accept asentence for (at least) arbitrary non-conditional sentences.26 The following has longbeen thought to be a plausible candidate for such a rule:

(8) A sentence P is acceptable, given background knowledge K , if Pr(P | K ) > t.

As was famously pointed out in Kyburg (1961), though, (8) is untenable if we wantto maintain the principle that acceptability is closed under conjunction—which mostphilosophers nowadays think we should maintain.27 For given the foregoing rule, itis rational to accept of each ticket in an n-ticket lottery that is known to be fair and tohave exactly one winner, and with 1 − 1/n > t, that it will lose. But since it is known,and thereby acceptable, that the lottery will have a winner, it follows from the closureprinciple that the inconsistent conjunction “Ticket #1 is a loser & · · · & Ticket #nis a loser & One of tickets ##1–n is the winner” is acceptable—which of course is aconclusion one wants to avoid.

In response to this result, various authors have come up with proposals of thefollowing schematic form:

23 The assumption that t = .9 is a rather common one in the literature on so-called acceptance rules(cf. below in the text). See, for instance, Kaplan (1981, p. 308), Moser and Tlumac (1985, p. 128), andKyburg (1990, p. 64); Foley (1992, p. 113) assumes that t = .99. It will be obvious, however, that nothingof essence hangs on the precise value of t.24 See in the same vein DeRose and Grandy (1999, p. 416, n7). But see also the discussion of the urnexample in Appiah (1985, p. 171 f), which involves a lottery by another name.25 Although it must be noted that, while “detachment” is in general a reasonable response to learning theantecedent of a conditional, it is not always; see Sect. 4 under (iv).26 Not all authors who write about such rules seem to have realized that conditionals may require specialtreatment in this respect. This may seem surprising. Part of the explanation may be that much of the work onthese rules antedates Lewis (1976) proof that the probability of a conditional is not generally the probabilityof its consequent conditional on its antecedent. If the two could be equated, then, given that virtually allacceptance rules that have been proposed are of a probabilistic nature, it would seem at least prima facienot implausible to think that the acceptability conditions of conditionals coincide with those of other typesof sentences.27 Kyburg is one of the rare dissenters on this point; see in particular his (1997). See in the same vein Foley(1992). For reasons given in my (2002, Sect. 2) I am with the majority.

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(9) A sentence P is acceptable, given background knowledge K , if Pr(P | K ) > t,unless defeater D applies to P given K .

It is supposed that the defeater D applies to “lottery sentences” such as those of theform “Ticket #i is a loser,” at least given that the background knowledge containsno more relevant information about the lottery than we assumed. Moreover, D is toapply as selectively as possible to lottery sentences: most high probability sentencesare supposed still to qualify as acceptable on account of their high probability. Theproposals differ on what D should be, and there is currently none that enjoys morethan modest popularity. For the time being, however, I will suppose that there is a ruleof the form (9) that we can agree on.28

Given this supposition, we can introduce the notion of t*-evidence, defining P to bet*-evidence for Q given background knowledge K iff (i) P is t-evidence for Q given Kand (ii) the defeater D does not apply to Q given K ∪ {P}, that is, the backgroundknowledge with P (hypothetically) added to it.29 Then the proposal in full—whichwe may dub the evidential support theory of conditionals—is this:

(10) “If P, Q” is assertable for one/acceptable to one, given background knowledge K ,iff, on one’s degrees of belief function Pr( · | K ), P is t*-evidence for Q.

On this proposal we can be assured that (7) comes out as being not assertable/acceptable: relative to the assumptions we made about the lottery plus the conditional’santecedent, D—whatever its exact content—simply must apply to the consequent lestwe have the lottery paradox back. But what about our sentences (1α) and (1β)? I saidthat the defeater was meant to apply as selectively as possible to lottery sentences,but “lottery sentence” is a vague expression, and it is not immediately evident that itdoes not apply to (1). In effect, there seems to be a salient commonality between thatsentence and, for instance, the sentence “Your ticket is a loser”: whatever reasons wemight have for accepting these sentences must be of a purely statistical nature. And ifon the supposition that (α) is known D applies to (1), then (1α) no longer comes outas being assertable/acceptable. Moreover, even if D does not apply to (1) given (α),but if D still does apply to (1) given (β), then it would appear that we do not need thefirst clause of (10) in order to explain the asymmetry between (1α) and (1β). In thatcase, our theory would at a minimum be needlessly complicated.

First, however, it should be clear that given (α) the defeater does not apply to (1);the former, after all, implies the latter. Indeed, no one thinks that the defeater applies to“Your ticket is a loser” on the supposition that it is known, or even only highly likely,that (say) Kate’s ticket is the winner. Second, it seems no less clear that we would notwant the defeater to apply to (1) given (β). The important thing to notice here is thatgiven (β) as a supposition, there still is no analogue to the lottery paradox involving (1)or similar sentences. Even though, on that supposition, there is a negligible chance

28 Although it is somewhat discouraging in this respect that, as Douven and Williamson (2006) show,what prima facie seemed to be the most plausible such rules—namely those stateable in terms that areprobabilistic or broadly logical—are all untenable.29 There is no need to consider cases where P is inconsistent with the background knowledge, for ifK ∪ {P} � ⊥, then Pr(P & K ) = 0, so that when K constitutes the background knowledge, P cannotpossibly be t-evidence, or evidence at all, for Q—i.e., in that case already clause (i) of the definition oft*-evidence fails to be satisfied.

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that there is no heads in the sequence of 1,000,000 tosses, there is no inconsistencyin assuming that in a given collection of sequences of 1,000,000 tosses with a faircoin—a collection of any cardinality—there is no sequence with zero heads. So, forany n ∈ N, given (1) and n similar sentences about other sequences of 1,000,000 tosseswith a fair coin, and supposing (β), it does not follow that the conjunction of all thesesentences is inconsistent.30 Hence, given that the clause concerning the defeater wasadded to (8) with the purpose of blocking the lottery paradox, and given that (1) doesnot appear to give rise to that paradox (or some close cousin of it) on the suppositionthat (β) is part of the background knowledge, it seems reasonable to assume that Ddoes not apply to (1) on that supposition.31

Now follow some further comments on the theory, beginning with an importantdisclaimer. As has been frequently observed in the literature, the word “if” is put toa great many uses, uses that at least on an intuitive level may be very different andsome of which we may even be inclined to think of as being “misuses” of the word(Dudman 1984, p. 148); cf. also Bennett (2003, Ch. 8). Some might still hope for aunified theory of conditionals that correctly predicts all possible judgements about anyconditional any competent user of the language might ever make. Note, though, thatit is by no means a priori that such a theory can be had. Indeed, in view of the manyfailed attempts to come up with a satisfactory unified account of conditionals, or evena satisfactory account of conditionals at all, it seems more realistic to suppose that theright approach to conditionals is of a more modest and piecemeal variety. In this spirit,I offer the evidential support theory of conditionals as a theory of ordinary or normaluses of conditionals. “Normality” is here to be taken in the entirely unproblematicstatistical sense of the word: most conditionals we encounter in quotidian speech haveassertability/acceptability conditions as are specified by (10).

One important class of conditionals that are not normal in the previous sense, andwhich I briefly want to highlight here because they are of some relevance to thelinguistic data presented in Sect. 2, consists of what Bennett (2003, p. 122 ff) callsnon-interference conditionals.32 The purpose of the use of such a conditional at least

30 Given a frequentist conception of chance, it follows that if we keep producing sequences of 1,000,000tosses with a fair coin in the (idealized) long run, then there are bound to be ones with zero heads. After all,to say that there is a positive chance that in such a sequence there will be no heads is, on that conception, tosay that in the infinite long run there will be sequences with no heads. But given that, as Keynes famouslynoted, in the long run we are all dead, there is no inconsistency in supposing that we will never encountera sequence of 1,000,000 tosses with a fair coin that does not contain at least a single heads.31 This is not to say that (1) does not give rise to an analogue of the lottery paradox whatever we suppose.Take the following sentence: “The sequence referred to is one of a collection of a hundred actual suchsequences in one of which there occurs no heads.” If we add this (hypothetically) to our backgroundknowledge, then it is fairly easy to arrive at a contradiction from the supposition that, given that backgroundknowledge, (1) is still acceptable. This corresponds to the fact that, as far as I can see, the followingconditional is unassertable/unacceptable:

If sequence S of 1,000,000 tosses with a fair coin is one of a collection of a hundred actual suchsequences in one of which there occurs no heads, then there will be at least one heads in S.

Note, though, that in this case already the first clause of the evidential support theory (10) gives us thecorrect result, for in the present case the antecedent is evidence against the consequent. After all, here we

have Pr(consequent) = 1 − 1/2106> Pr(consequent | antecedent) = .99.

32 Burgess (2004, p. 567) also suggests classifying these conditionals as non-standard ones.

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in part is to indicate that the truth of its antecedent is irrelevant to that of its conse-quent. A common characteristic of these sentences, which suggests an easy litmustest for them, is that their assertability seems either not at all or positively affected bysubstituting “even if” or “whether or not” for “if” in them, or by inserting the word“still” before their consequent clauses.33 It is worth noting that whether or not a con-ditional can be taken to indicate non-interference may be a context-dependent matter.For instance, given the minimal assumptions we made about the context of utteranceof (1α) and (1β), (1β) is, as we said, at a minimum not highly assertable. But nowlet us make some further assumptions about that context. Suppose, for instance, thatwe had been discussing the question what would be an acceptable price for a certainbet that pays big if there is at least one heads in the first 1,000,000 tosses of the coin,and that someone had suddenly changed the topic of the conversation by starting totalk about Chelsea’s chances of winning the Champions League. We might then quiteappropriately assert (1β) to bring the conversation back to our previous discussion.Note, however, that with these assumptions added there seems to be no difference inassertability whether we leave in or out the bracketed words in

(11) (Even) if Chelsea wins the Champions League, there will (still) be at least oneheads in the first 1,000,000 tosses with this fair coin,

in line with what we just said about non-interference conditionals.34

The second comment has to do with the fact that (10) is purely qualitative. If onethinks that conditionals can be assertable/acceptable to a greater or lesser degree, thenone obvious way to turn (10) into a quantitative theory is to add the following phrase toit: “… and ‘If P, Q’ ’s degree of assertability/acceptability, respectively unassertabili-ty/unacceptability, is measured by Pr(Q | P).” Given that in the remainder I will onlyassume the qualitative theory, I will leave aside the question whether the foregoingsuggestion is entirely correct.35

By way of a third comment let me point to the fact that the theory explains why eachof the extant accounts of conditionals has at least some initial plausibility. According

33 Jackson (1987, p. 44) comes close to considering the evidential support theory of conditionals by con-sidering the view that for “If P, Q” to be highly assertable Pr(Q | P) must not only be high but alsosignificantly higher than Pr(Q). He is very brief about it, however, because he thinks it can be dismissedout of hand, on the basis of one alleged counterexample, to wit, the conditional “If Reagan is bald, no onein the press knows it.” He seems right to note that the conditional is highly assertable even though “theabsolute probability of ‘No one in the press knows Reagan is bald’ is higher than its conditional probabilitygiven he is bald” (ibid.). It would thus seem that on our theory the conditional is not assertable either. Note,though, that here we have a clear instance of a conditional whose intuitive assertability would, if anything,go up were we to replace “if” in it by “even if,” or by “whether or not.” I therefore take the conditional tobe an abnormal one and hence to be outside the intended scope of our theory.34 Another class of abnormal conditionals that has attracted some special attention from philosophers is thatof the so-called biscuit conditionals. DeRose and Grandy (1999) claim to give an account of conditionalsthat is able to treat biscuit conditionals on a par with normal conditionals, but see Bennett (2003, p. 125 f)for a critique of this claim.35 The theory suggests further quantitative questions that it seems worth investigating and that I can onlyhint at here. For instance, that the degree of assertability/acceptability of a conditional “If P, Q,” provided itis assertable/acceptable, is in part a function of Pr(Q | P) seems more than plausible to me. But an interest-ing issue is whether assertability and acceptability are solely determined by conditional probability. A notunreasonable conjecture would seem to be that they also have something to do with the measure to whichthe antecedent supports the consequent, which many take to be given by the function Pr(Q | P) − Pr(Q)

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to Adams, Jackson, and Lewis, high probability of the consequent given the antecedentis both necessary and sufficient for a conditional to be (highly) assertable/acceptable.My proposal agrees with the necessity part of the claim but denies that high conditionalprobability is sufficient. And if we assume that, modulo the complications arising fromlottery sentences pointed to above, a non-conditional sentence is acceptable to one iffit has a probability above t on one’s degrees of belief function, then the proposal mayalso help to explain Mellor’s inferential intuitions. For in that case, coming to knowthe antecedent of an acceptable conditional will often lead one to accept the conse-quent upon learning the antecedent, namely, if one does not already unconditionallybelieve the consequent to a degree exceeding the threshold, and if coming to know theantecedent will not alter one’s conditional degree of belief in the consequent given theantecedent (see Sect. 4), which seem to be fairly common conditions.36

Related to the foregoing, it may be noticed that it is now easy to construct exampleslike the one involving (1α) and (1β), which create difficulties for the accounts of Sect. 1,and, more generally, to describe the type of data that militate against those accounts.For those of Adams, Jackson, and Lewis, the recipe is to take any conditional whoseconsequent is highly probable conditional on its antecedent, but no more probable thanit is unconditionally. According to their accounts, such conditionals should be highlyassertable/acceptable, but following the cited recipe I have had no difficulty findingones that are intuitively not highly assertable/acceptable. For Mellor’s theory, on theother hand, the recipe is to take a conditional whose consequent is already uncondi-tionally high enough to warrant assertion/acceptance but still evidentially supportedby the conditional’s antecedent. I do not accept such conditionals on Mellor’s theory,but here too the recipe makes it easy to multiply examples like (1α) that intuitivelyare highly assertable/acceptable, and that I do accept, their having a consequent witha high unconditional probability notwithstanding.

The next comment points to the fact that (10) makes it perfectly explainable howasserting (1β) will normally create a misleading implicature, and why it is epistemi-cally imprudent to accept it. The latter is straightforward: if the acceptability conditionsof a conditional include that the antecedent is evidence for the consequent, then find-ing (1β) among my beliefs, I will assume that when I accepted it the acceptabilityconditions were met and thus that Chelsea’s winning the Champions League mustsomehow be or at least have been evidence for there being at least one heads in the

Footnote 35 continued(though there are more options here; see, e.g., Fitelson 1999). Another aspect of the use of conditionals thatcan be made quantitative is the extent to which one is warranted in putting emphasis on the word “if” in aconditional. For instance, if I think it rather unlikely that Jim will come to the party, I may say:

If Jim will come to the party, Sue will come too,

or, similarly,

If Jim comes to the party—and this is a big “if”—then Sue will come too.

How appropriate, or felicitous, is it to emphasize “if” in uttering the former sentence, or to add the remarkbetween dashes in the latter? Here, too, there is an answer which almost suggests itself, namely, that theappropriateness varies linearly with 1 − Pr(Jim comes to the party), and similarly for other conditionals.This hypothesis too seems open to empirical testing.36 I am here taking for granted that learning proceeds by conditionalization.

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first 1,000,000 tosses. I may then wonder how that can be, and speculate about possiblemechanisms that could explain the probabilistic connection—all of which is spurious,of course, for there is no such connection. As to asserting (1β), a hearer who assumesthat the speaker is cooperative will conclude that the assertability conditions are metand thus that (1β)’s antecedent is t*-evidence for its consequent, or at least that that isso according to the speaker’s degrees of belief function—which is misleading becausein reality the antecedent and consequent are probabilistically independent accordingto the speaker (or so we may assume).

It is worth noting that similar explanations of why (1β) creates a misleading impli-cature are not available to Adams, Jackson, or Lewis. On their theories, asserting (1β)basically would conversationally implicate only that the conditional probability of itsconsequent given its antecedent is high—which is not misleading at all: it is simplytrue.

Finally, nothing said so far compels me to take a stand on the issue of whetherconditionals have truth conditions and, if they do, what these are. But for those con-vinced by Lewis’s (1976, p. 84 f) argument that any adequate account of conditionalsmust attribute truth conditions to them,37 let me notice that a conservative approach,and also a prima facie plausible one,38 would be to modify just slightly Jackson’s andLewis’s proposal and let “If P, Q” have the truth conditions of P → Q and associatewith “if” the conventional implicature that P is t*-evidence for Q instead of the im-plicature that Q is robust with respect to P (which in the previous section was seento fail). It will be clear by now that I would then take the implicature to govern notonly what we can assert but also what we can accept (as Jackson and Lewis may havemeant to take it all along; see note 5). There is no need to worry that if this option istaken it might happen that a conditional is assertable/acceptable on my account andyet does not have a high, or even has a low, probability. For it is a simple theorem ofprobability theory that Pr(¬P ∨ Q) � Pr(Q | P). So if on my theory “If P, Q” isassertable/acceptable, and thus Pr(Q | P) > t, then also Pr(¬P ∨ Q) > t and hence,on the current proposal, Pr(If P, Q) > t.39

37 For a more recent argument see Kölbel (2000). There are relatively many who are not convinced, how-ever. Apart from Adams, whose opposition to the view that conditionals have truth conditions was alreadynoted, these include Gibbard (1981), Appiah (1985), Edgington (1986, 1995, 2000), Bradley (2000), andBennett (2003, Ch. 7); see Douven (2007b) for a reply to Bradley.38 Which is of course not the same as saying that it will solve each of the problems that authors have shownto beset the Jackson/Lewis theory.39 We need not be conservative on the point of truth conditions. As far as I can see, the evidential sup-port theory is perfectly compatible with, for example, the possible-worlds theory of indicative conditionalsrecently presented in Nolan (2003). But then again, he may claim to have no need for a separate theory ofthe assertability conditions of conditionals, for he suggests that his theory allows us to say simply that forconditionals assertion goes by probability of truth. The correctness of this is hard to assess, however, for hedoes not tell how we are to determine the probability of truth of a conditional, given the truth conditions ithas on his account (he hints, in note 41, that this probability is given by an imaging function as defined inLewis (1976), but relegates elaboration of that hint to future work). At any rate, the example he gives onp. 261, in which he assesses the probability of a given conditional, is utterly unconvincing to my eyes, andmakes me suspect the claim about the relationship between assertion and probabilities of conditionals is atleast in its generality false.

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4 Anticipated objections

In this section I present, and try to defuse, some objections that might be raised againstthe evidential support theory of conditionals.

(i) A whole batch of related objections may involve what initially appear to belogical principles that any theory of conditionals should validate. Other authors havealready exposed many of these principles as being not “logical” at all, and even asdefective. For instance, it may at first seem obvious that the acceptability of condition-als should be transitive in the sense that if “If P, Q” and “If Q, R” are both acceptableto a person, then “If P, R” should be acceptable to her as well. And if this is elevatedto a logical principle, then it is one my account fails to respect, for, as can be easilychecked, P may be t*-evidence for Q, Q may be t*-evidence for R, and yet P mayfail to be t*-evidence for R. But as an example of Adams (1966) shows, this is howit should be. For it may well be acceptable that if Jones is elected, then Brown willresign, and also that if Brown dies before the election date, Jones will be elected, butit can hardly be acceptable that if Brown dies before the election date, he will resign.Adams (1998, Ch. 6) is a real treasure trove of similar counterexamples against otherallegedly logical principles; see also Bennett (2003, Ch. 9).

But here is one that Adams does not discuss: It would seem reasonable to think thatif one accepts “If P, then Q and R,” then one should be willing to accept both “If P,Q” and “If P, R.” I am not aware of any counterexample in ordinary language againstthis principle, so I cannot exclude that it is deserving of the epithet “logical.” That mayseem to be a problem for my account, for P may be t*-evidence for the conjunctionof Q and R on one’s degrees of belief function without being t*-evidence for eitherQ or R on that function. Of course, if P is t*-evidence for the conjunction of Q and R,so that the conjunction is highly likely conditional on P, then both Q and R must behighly likely conditional on P, too. But the point is that P need not be evidence foreither conjunct; it may even be evidence against both.40 In other words, the inferentialpattern is “high probability–preserving,” but not “evidential support–preserving.” Itthus creates no difficulty for Adams (nor for Jackson and Lewis), which explains whyAdams does not discuss it.

40 To see this, assume for concreteness that t = .9 and consider the following probability model: Pr(P&Q&R) = .87407407; Pr(P & Q & ¬R) = .06666667; Pr(P & ¬Q & R) = .00648148; Pr(P & ¬Q & ¬R) =.02037037; Pr(¬P & Q & R) = .02893407; Pr(¬P & Q & ¬R) = .0006435; Pr(¬P & ¬Q & R) =.00274455; Pr(¬P & ¬Q & ¬R) = .00008528. As is easy to verify, on this model all of the followingconditions hold simultaneously: (i) Pr(Q & R | P) > t > Pr(Q & R); (ii) Pr(Q | P) < Pr(Q); (iii)Pr(R | P) < Pr(R).

At this juncture one could object that, for all we are assuming about it, the defeater D to which theacceptance rule (9) refers might, for all P, Q, and R, apply to the conjunction of Q and R given P wheneverthe latter is t-evidence for that conjunction but not for both Q and R separately. In that case P would bet*-evidence for none of “If P, then Q and R,” “If P, Q,” and “If P, R.” But there is no reason whatsoeverto believe that is so. In fact, the second clause of (10) is meant to block a very particular type of counterex-ample to (6). Without loss of real generality, we may therefore as well suppose that none of the examples ofconditionals considered in this or the next section is beset by the kind of problem that besets (7). Otherwiseput, we may for the remainder as well make the simplifying assumption that if P is t-evidence for Q, thenit also is t*-evidence for Q.

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And there appears to be a converse problem, namely, that it would seem reasonableto think that if one accepts “If P, Q” and accepts “If P, R,” then one should also bewilling to accept “If P, then Q and R.” However, P may be t*-evidence for both Qand R separately and yet not for their conjunction.41

Here is a third seeming problem: Suppose “If P, R” and “If Q, R” are both accept-able. One might think then that “If P and Q, then R” should be acceptable too. Yeton my account there is no guarantee that it is, for it is possible that both P and Q aret*-evidence for R while their conjunction is no t*-evidence for R.42

As to the first two problems, suppose that for all or virtually all instantiations ofthe inferential patterns

(i) If P, then Q and R (i) If P, then Q and R(∴) If P, then Q (∴) If P, then R

as well as of the pattern

(i) If P, then Q(ii) If P, then R(∴) If P, then Q and R

that we encounter in real life and that we deem intuitively valid, it holds that the ante-cedent P is t*-evidence for each of the consequents, that is, for the conjunction of Qand R as well as for Q and R individually. That supposition is quite compatible withthe existence of probability models such as the ones given in notes 40 and 41. Andif true, it explains our intuition that “If P, Q” and “If P, R” are both acceptable iff“If P, then Q and R” is acceptable. For although not strictly a logical principle, forall, or at least most, practical purposes it may be regarded as one. Some independentreason to believe the supposition to be true arises from the fact that it would explainthe intuition that led Hempel (1945) to adopt as an adequacy condition for theories ofconfirmation his Special Consequence Condition (according to which evidence for aproposition is ipso facto evidence for every logical consequence of that proposition).It would further explain the non-negligible surprise of many students when they aretaught that on a Bayesian theory of evidence, evidence for a conjunction need not beevidence for any of the conjuncts. (While some may still take the intuition causingthe surprise to speak against the Bayesian theory of evidence, and perhaps in favor ofsome Hempelian account of evidence, I suspect that nowadays most prefer to explainthe intuition along the lines suggested in the present paragraph.)

41 As is for instance the case in the following probability model (still assuming that t = .9): Pr(P &Q & R) = .0491016; Pr(P & Q&¬R) = .00421875; Pr(P &¬Q& R) = .00421875; Pr(P &¬Q&¬R) =.00105469; Pr(¬P & Q & R) = .170625; Pr(¬P & Q & ¬R) = .231055; Pr(¬P & ¬Q & R) = .231055;Pr(¬P & ¬Q & ¬R) = .308672. Here we have that P is t*-evidence for both Q and R, but while it isevidence for the conjunction of Q and R, it is not t*-evidence for that conjunction: Pr(Q & R | P) = .838.42 Consider, for instance, this probability model: Pr(P & Q & R) = .0912252; Pr(P & Q & ¬R) =.0202723; Pr(P & ¬Q & R) = .147819; Pr(P & ¬Q & ¬R) = .00336943; Pr(¬P & Q & R) = .147819;Pr(¬P & Q & ¬R) = .00336943; Pr(¬P & ¬Q & R) = .455; Pr(¬P & ¬Q & ¬R) = .131126. Then wehave Pr(R | P) = Pr(R | Q) = .91 > Pr(R) = .841863 > Pr(R | P & Q) = .818182. Still assuming thatt = .9, this verifies the claim.

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As to the third problem, while it may often be the case that an instance of “If Pand Q, then R” is acceptable if the corresponding instances of “If P, R” and “If Q,R” are both acceptable, there exist ordinary-language examples showing that we arenot onto a logical principle here, nor even onto one that may be assumed to hold forall practical purposes. Consider, for instance, the following sentences:

(12) If Kate’s ticket wins the lottery, then Jim’s is a loser.

(13) If Sue’s ticket wins the lottery, then Jim’s is a loser.

(14) If both Kate’s ticket and Sue’s ticket win the lottery, then Jim’s is a loser.

My background beliefs about the lottery include that it is a large and fair lottery. How-ever, I do not know, but only deem it highly likely, that there will be a unique winner.Then it would still seem reasonable if I were to find the conditionals (12) and (13)acceptable.43 However, the antecedent of (14) challenges what I deem highly likely,to wit, that the lottery has a unique winner. Of course if that is wrong, then that Kateand Sue win does not exclude, nor necessarily make it unlikely, that John will win too.Accordingly, I do not find (14) acceptable.

(ii) Another batch of objections might come from the fact that, on our understandingof evidence, P can only be evidence for Q if both 0 < Pr(P) < 1 and 0 < Pr(Q) < 1.This means that on the evidential support theory, conditionals whose antecedent and/orconsequent have either probability 0 or probability 1 for a given person are not asser-table for/acceptable to that person. Does this accord with intuition? Let us considerthe possible cases.

There is general agreement that conditionals are what Bennett (2003, pp. 54–57)calls zero-intolerant, meaning that “nobody has any use for ‘If P, Q’ when for himPr(P) = 0” (p. 55; italics omitted and notation adapted for uniformity of reading).44

So what the theory says about conditionals with probability 0 antecedents is clearlyright.

43 For reasons given in the previous section, we need not worry that the defeater of (9) applies to “Jim’sticket is a loser” on the supposition that Kate’s, respectively Sue’s, ticket wins. Of course, on the suppositionthat Kate has a winning ticket, I will accept not only that Jim’s ticket is a loser, but also that Sue’s ticket is aloser, that Harry’s ticket is a loser, and so on, for all other tickets in the lottery. However, in contrast to whatwe have in the Lottery Paradox, I do not know that the conjunction “Ticket #1 is a loser and ticket #2 is aloser and … ticket #n is a loser,” where tickets ##1 − n are supposed to be all the tickets in the lottery exceptKate’s, must be false, supposing that Kate’s wins. Quite the contrary; that conjunction is highly likely tome on the said supposition.44 This is generally, but not universally, agreed upon. Some think that one may believe that Oswald killedKennedy and at the same time reasonably believe that if Oswald did not kill Kennedy then someone elsedid (see, e.g., Gillies 2004, p. 585 f). That seems right to me only if we are using “believe” in a rather looseway, and in a sense in which it does not quite mean the same as “accept” (as we have been using the latter,namely, as believing to a degree close to 1). It seems perfectly alright, for instance, to say that I believe thatOswald killed Kennedy if that is to mean that I am relatively confident (but not necessarily near to sure)that he did, and that I believe also, or even accept, that if he did not do it, someone else did. By way oftest, one may compare “Oswald killed Kennedy, at least I’m relatively confident of that; but if he didn’t,then someone else did” with “Oswald killed Kennedy, but if he didn’t, then someone else did.” The formersounds more or less okay to my ears; the latter sounds positively odd.

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Conditionals with probability 0 consequents seem to pose more of a problem. Beingconfident that John did not pass the exam, I may reasonably say: “If John passed theexam, I’m Peter Pan.” I am very sure that I am not Peter Pan, so the antecedent cannotpossibly be t*-evidence for the consequent. How, then, do I explain the assertability ofthe conditional? Recall what we said earlier, appealing to (10) together with Griceaninsights, about how asserting a conditional conversationally implicates that the ante-cedent evidentially supports the consequent. It will be obvious to all, and obvious toall that it is obvious to all (and so on), that it is false that I am Peter Pan, and thusalso that there can be no evidence for my being Peter Pan. As a result, a hearer willunderstand that by implicating that, per impossibile, John’s passing the exam wouldbe evidence for my being Peter Pan, I really intend to convey the thought that I amconfident that John did not pass the exam. The same point can be made with respectto acceptability. At present, I strongly believe that John failed the exam, but supposeI lose that belief simply by forgetting it. If, then, I find among my remaining beliefsone represented as “If John passed the exam, I’m Peter Pan,” I no doubt will correctlyinfer from it that it is highly unlikely that John passed the exam.

Conditionals whose antecedents and/or consequents have probability 1 can begrouped together. In general, we find it misleading if someone asserts such a con-ditional. If the antecedent has probability 1, then mostly we think that it would bebetter were “if” replaced by “given that,” “since,” or “as.” And someone who assertsa conditional of whose consequent she is certain will generally be convicted of mis-leading us by making it seem as if the truth of the consequent depends on somethingthe truth of which is still to be settled. For reasons of epistemic hygienics, basicallythe same point applies to accepting such conditionals: by doing so we risk mislead-ing our future self. All this is not to say that such conditionals are never assertable.Suppose, for instance, that I try to explain to someone that Chelsea cannot fail to winthe championship, now that it has a seven-point lead on Arsenal, which is in secondposition, and there are only two more matches to go, but the person thinks I am wrong,although she knows Chelsea has a seven-point lead and there are only two matchesto be played. Then I may well say: “But if Chelsea is seven points ahead of Arsenal,and in each match there are at most three points to be gained, then the championshipcannot escape Chelsea. Can it?” Here both the antecedent and the consequent haveprobability 1 for me. So how can the conditional be assertable for me, as it surely is?The right answer seems to be that, in such cases, one does not make a conditionalclaim but gives a kind of demonstration to the addressee of how she ought to reasonto arrive at a conclusion we are fully convinced of but that for her is still a matter ofdoubt.45

I am not in a position to claim that the foregoing remarks cover all conditionals withprobability 0 consequents nor all conditionals with probability 1 antecedents and/or

45 See in this vein also Edgington (1997, p. 112). The same may apply with respect to conditionals suchas “If 7 + 5 = 12, then 7 + 6 = 13,” which seem perfectly assertable (see Atlas 2005, p. 89). This need notbe our general response to conditionals whose antecedents and consequents are mathematical truths/false-hoods. While according to standard probability theory all logical and mathematical truths are to receiveprobability 1 and all logical and mathematical falsehoods probability 0, there is a pretheoretically clearsense of probability in which it can be reasonable to assign a probability other than 1 to, for instance,a mathematical truth. For instance, prior to the enunciation of Andrew Wiles’s proof of Fermat’s Last

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consequents. It is worth re-emphasizing, though, that the theory of conditionals pre-sented in the previous section is meant as a theory of normal conditionals. And itseems that conditionals of the sorts just discussed are much less common than thosewhose antecedents and consequents have non-extreme probabilities (for instance, youmay want to ask yourself how often you have heard, or asserted, a “Peter Pan condi-tional”46). So even if there should exist examples of conditionals of the aforementionedkinds whose intuitive assertability or acceptability conditions are not specified by (10),they would not necessarily be counterexamples to that theory.

(iii) A further potential problem for my theory is that if Mellor (1993, p. 235 f,247 f) is right about future-referring conditionals, such as

(15) If Oswald doesn’t kill Kennedy, someone else will,

then my proposal might fail. For according to Mellor, Oswald may, prior to his deci-sion to make the antecedent true, have had no degree of belief in the antecedent andhence—Mellor’s suggestion is—he can have had no conditional degree of belief inthe consequent given the antecedent. Nevertheless, it is at least imaginable that (15)was acceptable to Oswald prior to his decision to kill Kennedy. If Mellor’s suggestionis correct, however, then according to my proposal (15) could not possibly have beenacceptable to Oswald.

In response, let me first note that it is not quite so evident that Oswald may have hadno degree of belief whatsoever in the antecedent. Even if he was still to decide whatto do, we may suppose that he had some inkling of the outcome of his deliberations,however vague perhaps. So he may well have been able to assign at least a vague(interval-valued) probability to the antecedent.47

More importantly, what matters for my proposal to work in this case is that, paceMellor, Oswald can plausibly be supposed to have had a degree of belief (if per-haps only a vague one) in the consequent as well as a conditional degree of belief(which may also have been vague) in the consequent given the antecedent. As to theformer, the question whether the consequent of (15) is true will have been highlyrelevant to Oswald while he was still pondering whether to kill Kennedy. We maythus assume that Oswald had a (possibly vague) unconditional degree of belief inthe consequent of (15). As to the latter, Mellor’s supposition that if Oswald had nodegree of belief in the antecedent, he can have had no conditional degree of belief inthe consequent given the antecedent, is of course true if we assume that that condi-tional degree of belief is the ratio of Oswald’s degree of belief in the conjunction

Footnote 45 continuedTheorem, it was certainly reasonable (in the pretheoretical sense) to assign that theorem some probabilitylower than 1. Given this notion of probability, there is of course no difficulty accommodating conditionalsof the designated type within our general approach. Alternatively, we might treat the notion of mathematicalevidence as being sui generis, and requiring an account of its own (see Martin 1998 for some first stepstoward such an account). That might allow us to say that, for instance, “If ZFC is consistent, then so isZFC + V = L” is assertable/acceptable because the antecedent is very strong mathematical evidence (thisbeing the analogue of t*-evidence) for the consequent.46 Tellingly, Jackson (1987, p. 35 f) classifies Peter Pan conditionals as non-standard ones; so do Heylenand Horsten (2006).47 Cf. Levi (1974).

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of the antecedent and the consequent and his degree of belief in the antecedent.But even though the ratio definition of conditional degrees of belief seems to sug-gest something about the order of determination of the degrees of belief involved—first one must know the values of the terms in the definiens, then one can calcu-late the value of the conditional degree of belief on the basis of them—variousauthors have convincingly argued that that suggestion is wrong.48 I may be com-pletely unable to say something definite about my degree of belief in A as wellas about my degree of belief in A & B and yet I may be very clear about theirratio and thus about my conditional degree of belief in B given A. Suppose, forinstance, you are told that from a large but unspecified number n of lottery tick-ets, numbered 1 through n, all of which have the same chance of being drawn,precisely one is to be drawn. Since you have not been informed about the exactnumber of tickets in the lottery there is no way to determine a credence in “Ticket#1 will be drawn”; all you will be able to say is that it must be small. Still, youwill have no difficulty determining your conditional degree of belief in “Ticket #1will be drawn given that one of tickets #1 and #2 will be drawn.” Even more tothe point, suppose Oswald indeed had no degree of belief in the antecedent of (15),as per Mellor’s suggestion. Would he have had difficulty determining his degree ofbelief in the proposition that Oswald does not kill Kennedy conditional on the prop-osition that Oswald does not kill Kennedy? Surely Oswald must have thought thatit is 1. I suspect that, even if Mellor is right about Oswald’s possibly not havinghad a degree of belief in the antecedent of (15), Oswald would still have been ableto answer the question whether he thinks it is more likely that someone will killKennedy, given the supposition that he does not kill Kennedy, than that someonewill kill Kennedy, period, and also the question whether he thinks it is very likely ornot quite so likely that someone will kill Kennedy given that he (Oswald) will notdo it.

So, even if it is true that we may sometimes be entirely in the dark about our degreeof belief in a proposition, it still does not follow that there may be conditionals forwhich we fail to have a corresponding degree of belief in the consequent conditionalupon the antecedent, nor that, if we do have such a conditional degree of belief, wemay be unable to say that it is greater than our unconditional degree of belief in theconsequent and/or that it is greater than a given threshold value.49

48 See, among others, Edgington (1986, p. 188 f, 1995, p. 266 f, 1997, p. 109), and Hájek (2003).49 According to Mellor (1993, p. 235 f), (15) refutes what he calls the belief-in-Adams theory, that is,the theory that states that to accept a conditional is to believe that one has a high conditional credencein the consequent given the antecedent. It might seem that those wanting to defend the theory can saveit from Mellor’s objection by appealing to my defense, too. That still does not save the belief-in-Adamstheory, however, for as a moment’s reflection suffices to make plain, the type of counterexample againstthe epistemic theories of conditionals presented in section 1 is a counterexample against the former aswell.

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(iv) Naturally, we must also consider conditionals of the well-known type exem-plified by, for instance,

(16) If Reagan worked for the KGB, there will be no evidence to be found for that.50

We can well understand how these could be perfectly assertable/acceptable to some-one, but on my proposal their assertability/acceptability might at first seem somethingof a mystery, for the proposal may seem to imply both that there could be evidence for aproposition and that if there really is evidence for that proposition, then it must be con-cluded that there is no evidence for the proposition. But that is not really implied, forit is perfectly compatible to suppose both that the antecedent of (16) is t*-evidence forthe consequent and that if someone came to know—and thus to have excellent evidenceand so, we may suppose, t*-evidence—that Reagan worked for the KGB, she would not“detach” the consequent of (16). As several authors have noted, “conditionalization”(as Bayesians say) is not the only possible rational response to the receipt of newinformation; another is to revise one’s conditional degree(s) of belief given the newinformation.51 Sentence (16) illustrates this point: if we were to find out that Reaganworked for the KGB, and thus were to get evidence for that, we would revise—in factlower to 0—our degree of belief in the proposition that there is no evidence that Rea-gan worked for the KGB conditional on the proposition that he did work for the KGB.Accordingly, the conditional in that case would no longer be assertable/acceptable.There is no inconsistency with our now finding it assertable/acceptable or even assert-ing/accepting it (supposing we do now find it assertable/acceptable or assert/acceptit).

(v) The final problem I want to discuss has to do with the fact—as it appears tobe—that we may be unsure about the assertability/acceptability of a conditional. Theproblem is to explain how this can be on my account, for it seems that in order to findout whether a conditional is assertable for me/acceptable to me I just have to check (soto speak) my degrees of belief function at the relevant “locations.” So, how could I beunsure whether or not my degree of belief in the consequent given the antecedent of agiven conditional is both above my unconditional degree of belief in the consequentand above t?

This problem is only compelling as long as we assume the standard Bayesian pic-ture of what it is to have degrees of belief. On that picture, we have at any momentof our conscious (mature) lives determinate degrees of belief in all propositions thatare expressible in our language, and hence, given the ratio definition of conditionalprobability, we also have determinate conditional degrees of belief in all propositionsgiven any other proposition or propositions. However, it is widely recognized, and

50 See van Fraassen (1980); also the postscript to Lewis (1976). Note that it would be wrong to dismiss (16)as a non-interference conditional. Although we can certainly imagine contexts in which it would be mostnaturally interpreted as one, there are others in which it could not be thus interpreted. Imagine, for instance,that it has just been asserted that if Reagan worked for the CIA, then, sooner or later, we will come to haveevidence for that. “On the other hand,” it might then be added, “if he worked for the KGB, we will neverfind any evidence for that.” It would sound quite wrong if “if” in that sentence were replaced by “even if”or by “whether or not.”51 See, for instance, Jackson (1979, p. 123 f) and Howson (2000, p. 139).

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we already implicitly granted it in our discussion of future-referring conditionals, thatthis picture is starkly idealized. In reality, we often have only vague—sometimes onlyvery vague—degrees of belief in many propositions. And if I have only a vague degreeof belief in some proposition and/or a vague degree of belief in that proposition con-ditional on a second proposition, then I may well be unable to determine whether thelatter is t*-evidence for the former.52

Even leaving vagueness apart, there is the problem that the Bayesian picture assumesour degrees of belief to be directly accessible to us at any moment. As is explained inDouven and Uffink (2003), this may have been a reasonable assumption in the opera-tionalist/behaviorist climate in which (modern) Bayesianism emerged,53 and in whichdegrees of belief were equated with dispositions to engage in acts of certain kinds,but it ceases to be reasonable once we grant, as many nowadays think we should, thata person’s degrees of belief correspond to certain states of her mind. Given such arealist attitude toward degrees of belief, it is no less plausible to hold that some of ourdegrees of belief may at least sometimes be inaccessible to us, or at least not fullyaccessible to us, than that for instance our desires are not always fully accessible tous—which has the status of a near-truism in contemporary psychology.

A different but related response is to interpret the function Pr(·) not as a person’sdegrees of belief function but as measuring something more objective, like for instancea reasonable degrees of belief function, where reasonableness may have to do withbeing informed “well enough” of the available relevant evidence. One might then beunsure about the assertability/acceptability of a conditional because, even though onone’s degrees of belief function the antecedent is t*-evidence for the consequent, oneis unsure whether one’s degrees of belief function is a reasonable one.54

Acknowledgements I am profoundly grateful to two anonymous referees for providing very valuablecomments on an earlier version of this paper. Comments from Leon Horsten, Hugh Mellor, and Christophervon Bülow have also been of great help. A version of the paper was presented at the University of Groningen.I am indebted to the audience for stimulating questions and remarks.

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