Inference and Evidence 1

Inference and Evidence 1 1. The Two Bases of Rationality We have a variety of attitudes to the truth of propositions: believing that p is true, hoping that p be true, desiring that p become true, assuming, wanting, pretending, wishing, regretting and so on that p is true. Some of these attitudes are truth oriented in a way that other are not, that is, they are in one way or another responsive to truth, or at least they should be. This is the case with belief, knowledge, acceptance, presupposition. A belief is true when the proposition believed is true. It is impossible to know something that is false and one cannot rationally accept or presuppose something believed or known to be false. On the other hand, one can desire, wish, hope, assume, want or pretend that something be the case, even if it is neither true nor thought to be true. Let us call the attitudes of the first kind veritatic attitudes and the second kind, conative attitudes. 2 The latter attitudes, to the extent that they relate to the truth of the proposition within their scope, do so only in a subtle and roundabout way. For instance, finding out that my pockets are empty is typically incompatible with hoping that there is 1,000$ in my left pocket. Desiring that something be the case may be incompatible with desiring that it won t. Veritatic attitudes, however, relate to truth much more directly. Truth for them is a constitutive aim, a standard of success, or a norm in some intuitively plausible sense (more on this later). Specifically, the truth of the proposition in their scope must play some role in the rational formation of these attitudes. But rationality also introduces a distance from truth. If the notion of rationality is to have a genuine role in the assessment of attitudes, it cannot be the case that the rationality of attitudes depends on their truth. If being rational just is having a true veritatic attitude, then rationality is a wheel that turns nothing. Put differently, the veritatic attitudes of godly beings with direct access to the truth are not rational nor irrational, but simply correct. Since we do not have unlimited and unmediated access to the truth, what we actually rely on in the formation of veritatic attitudes is often evidence. In order to form true beliefs about the world or to know empirical facts about it, we rely on evidence indicating what is 1 This paper is based on and partially repeats previous (and continued) work with Peter Pagin and with Assaf Sharon. 2 Sometimes (following Anscombe (Intention**)) the distinction between these two types of attitude are described in terms of direction of fit. Conative attitudes are not attitudes reflecting independent features of the world, rather, for these attitudes to be fulfilled (e.g. wishes) the world needs to comply with the attitude. Veritatic attitudes are meant to fit (aside from abnormal cases) independent features of the world. 1

the case. We seek evidence for determining our veritatic attitudes, in order that they be properly and sufficiently grounded in the way things are. The available evidence will usually determine whether a veritatic attitude is rational or justified. Typically if the available evidence does not support a veritatic attitude enough 3, that attitude is not rational (or justified). But attitudes are not formed solely on the basis of evidence. We often form attitudes by reasoning from existing attitudes to new ones. Believing that p and believing that p implies q, we can infer and form the belief that q. As Harman (1986) 4 showed, we can also forego our belief in p or in the implication. This kind of reasoning is based on a guarantee provided by logic valid deductive inference patterns are truth-preserving. If reasoning is governed by methods guaranteed to preserve truth, inferring that either (some of) the premises are false or the conclusion is true, is solid. Forming veritatic attitudes by truth-preserving methods is therefore rational. We have, then, two epistemic resources by which rational attitudes are formed: evidence and inference. One provides us with indications about the truth and the other guarantees its preservation as we transition from attitudes to other attitudes. This picture can be labeled as the two bases of rationality view. It is the aim of this article to argue that this broad and appealing picture is incoherent. The basic reason for this incoherence is because evidence and truth behave very differently, especially when it comes to the ways in which they allow us to form new attitudes on the basis of existing ones. Evidence is truth directing. Deductive inference is reasoning from assumed truth. Modus Ponens, for example, tells us that if two premises of the forms φ and φ ψ are true, then ψ is also true ( denotes material implication and φ and ψ are propositional variables). Insofar as truth is concerned, this authorizes the transition from the premises to the conclusion, or the rejection of one or both premises. But this does not apply to evidence. Both φ and φ ψ may be supported by the available evidence, while ψ is not. Truth is transitive over valid inferences. Evidence is not. So, if the 3 At this stage I will leave the question of the degree to which an attitude must be supported to count as rational unspecified. Naturally, different veritatic attitudes will have different (perhaps vague) sufficiency conditions for being rational. 4 One of Harman s famous examples is the following (1986: 5): Mary believes that if she looks in the cupboard, she will see a box of Cheerios. She comes to believe that she is looking in the cupboard and that she does not see a box of Cheerios. At this point, Mary's beliefs are jointly inconsistent and therefore imply any proposition whatsoever. This does not authorize Mary to infer any proposition whatsoever. Nor does Mary infer whatever she might wish to infer. Instead she abandons her first belief, concluding that it is false after all. I ll come back to this and related examples later on (p. **). 2

evidence so prescribes, one should hold fast to both premises while rejecting the conclusion, if one s attitudes are to be rational. We can either follow truth-preserving MP inferences, or follow the evidence. We cannot do both. The Two Bases view is wrong. I will argue that we should side with evidence and not with inference. 2. Veritatic Attitudes Unlike conative attitudes, veritatic attitudes are about getting things right, maintaining to some degree of assurance (different attitudes may involve different standards) that things are one way rather than another. Such assurance or indication is, at least with respect to most empirical matters, provided by what we call evidence. What determines whether having a certain attitude with respect to a certain propositions is rational, is the available evidence. Similar things can be said about deductive inferences. They indicate which propositions are true by establishing their truth-preserving relation to other, rationally established, propositions. Logically valid inference rules (or patterns) are truth-preserving in the sense that their premises cannot be true without truth of their conclusions. So the natural thought, as just stated, is that if veritatic attitudes concern truth, having an attitude that is rational with respect to the premises of a valid inference pattern and noticing that the valid logical pattern guarantees the truth of the conclusion, one should either have the same attitude with regard to the conclusion or forgo this attitude with respect to one or more of the premises. Believing, say, that it will rain tomorrow and that if it will rain, the golf tournament is off, one should infer that the golf tournament will be canceled (or forgo one of the beliefs the inference is based on). Moreover, suppose it is known with certainty that one is rational in accepting the two modus ponens premises. Surely, it would seem, one would be rational in accepting its valid conclusion. The same is true for other inference rules - in virtue of preserving truth, they may guide us in forming rational veritatic attitudes on the basis of other rational veritatic attitudes. Following this line of thought, valid inferences place constraints on rationality. Let us say that an inference rule is rationally valid, if and only if, having a rational veritatic attitude towards its premises and drawing the inference, entails that it is rational to have this attitude toward the conclusion or no longer having it toward one or more of the premises. Rationally Valid Inference (RVI): A deductive inference rule or pattern is rationally valid for an attitude (A) if and only if for every proposition p and every subject S, if S 3

properly infers p from a set of propositions to which S rationally has attitude A, then it is rational for S to have attitude A with respect to p (A R (p)). This is a criterion for the rational validity of any given inference rule. It leaves open the possibility that some deductive inference rules are rationally valid while others are not. Even more modestly, some inference rules can be rational for some veritatic attitude and not for others. I shall focus, for now, on the very basic deductive inference rule of modus ponens (MP). MP is rationally valid with respect to a veritatic attitude A if the following holds: Modus Ponens Rational Validity (MPRV): For all propositions p and q and rational attitudes A of a subject S, if S has made the proper modus ponens inference, A(p), A(if p then q), A(q) cannot all be rational for S. Similar claims can be formulated with respect to modus tollens, disjunctive syllogism (A(p or q), A( q), and A(p) are not all rational for S) and so forth. According to the Two Bases view, due to its relation to truth, i.e its preservation from premises to conclusion, MP is rationally valid for veritatic attitudes. This is one way of capturing the idea that the rationality of veritatic attitudes is (at least partially) dependent on the inferences by which they are formed. 5 But the Two Bases view is also committed to another idea. The rationality or irrationality of veritatic attitudes depends not just on inferences, but also on evidence. This dependence can be formulated as follows: 6 Rational Evidence Sensitivity (RES): For all subjects S, total evidential states e, rational veritatic attitudes A, and propositions p, if S is in evidential state e, and e does not support p, then A(p) is not rational for S. 5 MPRV seem highly intuitive, especially since it is weaker than other formulations in several ways: first, its formulated synchronically and makes no commitment to dynamics - the inference has been made and evidence is assumed to be fixed (unlike Harman s example above (note **). Second, MPRV assumes the premises are held rationally, i.e. rationally believed, presupposed, accepted, etc. Third, it is a wide scope not a more demanding narrow scope principle. I will later comment on the wide and narrow scoping issue (p. **). Finally, froth, modus ponens itself is weaker than inferences, or so it seems, like conjunction introduction. One who rejects deductive closure generally, may still insist on modus ponens closure. See p. (**). 6 There are weaker versions of this idea. For instance, one might claim that e cannot support p more than it supports p if it is rational for S to, e.g. Accept or believe that p. Moreover, the principle does not specify what it means to be supported by evidence. Nevertheless, as I hope will become evident, these ways of refining RES are not important for the arguments below. One can set any standard one wants, high or low, and one might understand evidential support in many ways, and still the major argument will go through. It is sufficient for my purposes here that rationality requires only some sensitivity to evidence. 4

Our question is whether these two principles, RES and MPRV, are compatible for rational veritatic attitudes. I will first claim that at least for attitudes that need not be conclusively supported (or entailed) by the evidence in order to be rational, they are not compatible. The general reason for the incompatibility is because logical operations on attitudes that are not conclusively supported allow doubt to accumulate. Consequently, as captured in different ways in Kyburg s lottery paradox (1961) and Makinson s preface paradox (1965), non-conclusive evidence for a set of propositions taken individually, need not support every proposition that can be deductively inferred by conjunction introduction (p,q p q). Later, after considering the rational validity of MP inferences, I will come back to conclusively supported veritatic attitudes and argue that they too do not lend support to the Two Bases of rationality view. 3. The Incompatibility Argument The argument is meant to expose how deep the tension is between evidence and inference: Let us use the index R (in subscript) to denote that a veritatic attitude A is rational, S to denote a subject, and e to refer to a complete evidential state (it is assumed that e incorporates relevant a priori knowledge). Now assume that relative to e, veritatic attitude A is rational for S with respect to propositions p 1,p 2,p 3 p n. So relative to the state e at a particular time t for S: A R (p 1 ), A R (p 2 ), A R (p 3 ). A R (p n ). Since e does not entail any of the propositions p 1,p 2,p 3 p n, it is possible that while the support e lends to each of them is sufficient for attitude A to count as rational with respect to it, this evidence does not support their conjunction (p 1 p 2 p 3 p n ). Hence, if the rationality of attitude A is evidence sensitive, A(p 1 p 2 p 3 p n ) is not rational at t for S according to RES. In other words, S can be rational if she lacks the attitude: A(p 1 p 2 p 3 p n ). This possible evidential relation, is exemplified in different ways by the lottery and preface paradoxes, it suffices to put significant pressure on the conjunction introduction rule: for all S at t, if A R (p 1 ), A R (p 2 ), then by the logical operation of conjunction introduction, A R (p 1 p 2 ) for S at t. As these and other examples had clarified, contrary to conjunction introduction, it is not rational for S to have a veritatic attitude toward p 1 p 2 p 3 p n although this conjunction can be derived by conjunction introduction from conjuncts that can rationally be in the attitudes scope individually. 5

What about a weaker rule of inference such as modus ponens (permitting only two premises)? The following argument shows that, insofar as doubt can accumulate, modus ponens is on a par with conjunction introduction 7 : 1. MPRV: It cannot be true that A R (p), A R (p q), and A R (q) at t for S, if S has properly inferred q by modus ponens from p and from p q. 8 2. RES: For all veritatic attitudes A, subjects S, and propositions q, if S s body of evidence e doesn t support q at t, then it is not the case that A R (q) for S at t. 3. A R (p 1 ), A R (p 2 ), A R (p 3 ),. A R (p n ) for S at t. 9 [Assumption] 4. S s evidence e at t, does not support p 1 p 2 p 3 p n. [Assumption] 5. A R (p 1 p 2 p 3 p n ) for S at t. [4, 2] 6. A R (p 1 (p 2 (p 1 p 2 ))) A R (p 1 ) [PL, 3] 10 7. A R (p 2 (p 1 p 2 )) A R (p 2 ) [MPEV, 6, 3] 8. A R (p 1 p 2 ) [MPEV, 7] 9. A R (p 1 p 2 (p 3 (p 1 p 2 p 3 ))) [PL] 10. Repeating Modus Ponens inferences from tautologies as in the sequence 6-8, S will infer and come to have the rational veritatic attitude A R (p 1 p 2 p 3. p n ) at t. Yet 10 contradicts 5 - the attitude A(p 1 p 2 p 3. p n ) can t be both rational and not-rational for S at t. One of the assumptions or steps in the argument must be denied. Before proceeding to examine ways of avoiding the reductio while retaining 1 and 2 let me mention two general points about the argument. First, if cogent, the argument shows that whatever one wants to say about the rational validity of conjunction introduction, one should say the same about the rational validity of MP. Second, overcoming the reductio argument requires radical steps. We must either reevaluate the role or nature of evidential support and its relation to veritatic attitudes, or understand the role of inference rules with respect to rationality very differently from the way it is normally conceived. 7 For a similar argument based on Pagin (1990) see Pagin and Spectre (MS). 8 A stronger result than the one I will focus on here is possible, if we assume a tighter connection between evidence and veritatic attitudes, namely, that one is rational, say, in believing that q is true, even though one has inferred q by modus ponens inferences that are believed rationally. 9 The number of rational epistemic attitudes is finite but otherwise unrestricted. More on this on p. **. 10 I m assuming that for S it is rational to have an veritatic attitude to tautologies. But this can be argued for separately by inferring p 2 (p 1 p 2) from p 1 directly (p 1 (p 2 (p 1 p 2))) or by direct support from e, since e either strictly implies the tautology vacuously, or it is already part of the evidence as assumed above (p. **). 6

4. Resisting the argument The contradiction between 10 and 5 implies the incompatibility of 1 (the rational validity of modus ponens) with 2 (rational evidence sensitivity). I will argue, first, that they are indeed incompatible and then that 1 should be rejected and not 2. In this section proposals for avoiding the argument and its conclusion are inspected. Presumably, no one will seriously question that 10 follows from 1-5. If 3 and 4 are true, denying 5 amounts to denying 2, so does not avoid the argument s conclusion. Dodging the argument, therefore, requires a rejection of either 3 or 4. Now, 3 on its own seems non-negotiable since rejecting it entails an unreasonable skepticism about the very possibility of rational veritatic attitudes. We are down then to assumption 4. The question, specifically, is: can assumption 4 be rejected when 3 is true? In other words, if modus ponens is rationally valid (MPRV), the following claim must be true: Validity Thesis: If given e it is rational for S to have attitude A to each of the propositions p 1, p 2, p 3, p n, it must be the case that evidence e supports their conjunction sufficiently for S to rationally have attitude A toward the conjunction (A R p 1, A R p 2,... A prn A R (p 1 p 2 p 3 p n )). 11 To defend VT, it is necessary and sufficient to claim that rational doubt cannot accumulate when forming new rational attitudes on the basis of modus ponens. Or, if you want, that the conclusion of every modus ponens inference will be at least as evidentially supported as its least supported premise. On any standard conception of evidential support, evidence for p 1, p 2, p 3, p n need not support p 1 p 2 p 3 p n. If doubt is allowed to enter at the premises, there s no avoiding its accumulation at the conclusion. So to defend VT it is necessary to either eliminate doubt from the premises or to abandon these standard conceptions of evidence. I will examine proposals of both kinds in turn. 4.1. Rational Uncertainty? One way of rejecting the compatibility of 3 and 4 is by denying that rational attitudes are compatible with any measure of rational doubt. This means that having a rational veritatic attitude requires having evidence that strictly entails the proposition in its scope. This is the case, let us suppose, with rational certainty. If the evidence I have strictly entails p, it is 11 denotes something like conceptual or a priori entailment. 7

rational for me to be be certain that p, and irrational to also have some doubt about its truth. Timothy Williamson has defended a similar claim about knowledge. For the case of knowledge (i.e. A( )=K( )), 4 is incompatible with 3. Perhaps this is the place to note that although Williamson himself does not motivate his view in this way, it is a significant virtue of his account of knowledge that it can preserve the epistemic validity of MP along with the less intuitive conjunction introduction rule. On his view, when I know that p and know that if p then q, I know that q, full stop. Few knowledge accounts can deliver this result without falling pray to skeptical consequences. 12 Williamson s theory avoids the reductio, then, at least for knowledge. On his theory all knowledge is evidence (this follows from the stronger claim that evidence is knowledge, the K=E thesis). Hence, for all p and subjects S, if S knows that p, S s evidence conclusively supports p (p entails p). The reductio argument, then, is blocked: if S knows p 1...p n, then S s evidence conclusively supports each of them as well as their conjunction. Any deductive inference rule is epistemically valid for known premises since the evidence (viewed as knowledge), does not merely support any logically valid conclusion, the evidence entails it. But whatever merit the evidence-knowledge equation has, it cannot be a complete response to the argument. This is mainly because it applies only to knowledge and not to other veritatic attitudes. To generalize Williamson s view of knowledge to other attitudes it is necessary to treat not only any known proposition as evidence, but also to treat as evidence any rational (or justified) belief, presupposition, rationally accepted proposition, and so forth. Aside from the implausibility of this type of account, it would entail either a revision of the notion of evidence or a revision of how we think about these attitudes. Neither option seems workable. The first entails the non-factivity for evidence and as a consequence it would in turn violate the evidence=knowledge thesis. The second, though not incoherent, would entail that all rational veritatic attitudes are factive, from which other significant implausibilities follow. Not only would it be impossible to believe a falsehood rationally, all veritatic attitudes would not be rational unless they were absolutely and conclusively supported by the available evidence. This type of view places virtually impossible requirements on the rationality of veritatic attitudes. Even for knowledge it is not clear that Williamson s theory can offer a proper response to the argument. Following a paper by John Hawthorne and Maria Lasonen-Aarnio (2009), 12 Worse, sometimes epistemologists mistakenly formulate the closure principle in a way that they cannot defend in face of an argument of the 1-10 form that is phrased in terms of knowledge. Unless all knowledge is entailed by the subjects evidence, knowing that p and knowing that if p then q, will not necessarily suffice for knowing that q. 8

Assaf Sharon and I have argued (Sharon and Spectre (forthcoming)) that Williamson s view entails significant costs precisely in virtue of its adherence to the conjunction introduction principle for knowledge. Without considering other related arguments we present there in detail, one problem directly relates to this view s implausible commitment regarding rationality: Regardless of the evidence one has for (many) future contingent propositions, the chance that they will not occur is rarely 0 (or alternatively Ch(p)<1 for many contingent propositions p). Assuming conjunction introduction is epistemically valid for knowledge and that one can know propositions relating to future events 13, one can infer conjunctions (as in 10) from many such items of knowledge, thereby coming to know their conjunction. This means, as Hawthorne and Lasonen-Aarnio elaborate, that though the chance that the conjunction may be low, the epistemic probability (roughly the conditional probability of a given proposition on the evidence/knowledge) is 1. This result imposes a sharp separation (as Williamson (2009) admits in his response) between chance and epistemic probability. But simply separating the two, however sharply, is not enough. Knowing future events will take place is not the only possible knowledge one might have. One can calculate and come to know what the chance is that this conjunction (of future events) is true. Assuming the chance of each conjunct is independent of the chance of the others, the calculated chance can be arbitrarily close to 0 and known to be so by calculation. Suppose φ-ing will allow me to avoid a highly undesirable outcome!, if and only if p 1 p 2 p 3 p n (the conjunction of known future events) is true. If p 1 p 2 p 3 p n is false, however, I will avoid outcome! if and only if I do not φ. Is it rational for me to φ or is it rational for me to refrain from φ-ing? It is hard to see how to settle this question within the Williamsonian framework. Both possible actions would be based on knowledge and so they would both be rational, it seems, on the suggested account (taking practical rationality to be action properly based on one s evidence). Should I go with the known chances or should I go with the epistemic probability? It seems that it can t be both - either φ-ing is rational or not-φ-ing is rational. 14 From the inside, so to speak, I have a rational belief that the chance I will avoid! by φ-ing is 13 That the propositions are about the future is not a necessary assumption of the argument. Since it it less controversial that future contingents are chancy, this simplification is dialectically convenient. 14 Perhaps one would say that it is impossible to both rationally believe that p and believe that it s chance of being true is false so the purported knowledge of both propositions is impossible. Though this seems like the right thing to say, it s hard to see how one could say it and still accept a separation between chance and epistemic probability the way Williamson (2009) advises. Moreover, it isn t clear to me how to reconcile this suggestion with the validity of conjunction introduction. Isn t this just another way of saying that knowledge does not collect over conjunction if the subject is (rationally) calculating chances? This question is another way of presenting some of the deliberation in the main text. 9

negligible but I also believe rationally that I I will avoid! if I φ. Conversely, if it is rational for me to refrain from φ-ing it is hard to see how believing, accepting, presupposing that the conjunction is false, are not rational for me as well. But if- it is also rational for me to φ and to believe that the conjunction is true, then by conjunction introduction it is rational for me to believe that the conjunction is true and false. Something must give here. The problem is not just that I would not know what to do in this and similar cases, the problem is that there is very good reason to suppose that both mutually exclusive actions are rational and are based on reasons that relate to different incommensurable dimensions of regarding the same events (on the suggested theory). In other words, both are based on the same overall evidence. This is because if it is irrational for me to believe the conjunction is false because I believe that the chance it is true is extremely low, then chance and epistemic probability are not independent after all. And assuming that I can t know something I believe irrationally, it turns out that conjunction introduction is not epistemically valid for knowledge. But if I can believe rationally that it s true, despite the seemingly opposing belief, then we are back to the problem of rationally believing on one dimension that all the events will take place, and believing that at least one of them won t on a different dimension. But even if this is the right thing to say, since I must do something about!, and rationality can t just be up to me, the incommensurability idea just can t work, at least not in this case 15. This view, it seems, commits its advocates to cases where an overall evidential state supports both φ-ing and notφ-ing, both believing p, and believing p. If both are rational conjunction introduction (and MP) must be rationally invalid. 16 I take it that even if Williamsonians have a reply to this latter challenge, the first worry should persuade us that 3 and 4 are compatible for some veritatic attitudes, i.e. that some veritatic attitudes are rational despite not being conclusively supported by the evidence at the subject s disposal. (Unless a rational belief cannot be false, belief is one such attitude). 4.2. Hedged propositions There is another way to deny the compatibility of 4 with 3. A possible 17 (yet, to my knowledge, underdeveloped) option in the face of worries of the kind presented here, 15 Other cases when I get the same prize whatever I do are not the issue here. 16 Perhaps it deserves to be emphasized that a view that accepts the possibility that it is rational for a subject on the same overall evidence to rationally believe that p and also believe that p is in trouble regardless deductive inferences. 17 John Broome, in conversation, suggested such a view to me. I m not sure whether he advocates it. David Enoch mentioned that others are working on an idea of this type. 10

without denying the possibility of partial doubt, is to imbed the uncertainty within the propositions. According to this view, one does not rationally believe that one s lottery ticket is a loser, which would lead to epistemic invalidity of deductions. Instead, one believes that it is highly probable that one s ticket did not win the lottery or that it is very likely that the ticket did not win. One does not believe that one will go to the movies tonight, but that it is very likely that he will do so. These hedged propositions, unlike their unhedged standard counterparts, can be conclusively supported by the evidence. When they are conclusively supported, certain attitudes with respect to them, such as belief, are rational. As things stand, it is not clear how the details of this view are to be spelled out, and I will not try to speculate about them. 18 Nevertheless, let me say what it would mean to try and defend the epistemic validity of modus ponens (or conjunction introduction) on such a view. Likelihood can be either objective or epistemic. Since objective likelihoods do not strictly follow from ones evidence, adding conjuncts by conjunction introduction will result in a high probability that one of the objective likelihoods is not true 19. To block the argument, then, this view must be about epistemic likelihoods, that is likelihood that something is true given one s evidence. So the suggestion will have the following general form: a rational belief that p is to be understood as the belief that it is likely (given one s evidence) that p is true. In symbols B R (p)= df B(lp) (where l is a variable that ranges over the epistemic probabilities and is high enough to satisfy the sufficiency conditions for B). A second necessary feature of such an account is that Pr(lp e)=1 (Pr(p e)=l), i.e., the evidence for every rational belief is conclusive. Now given these features, both necessary for claiming that 3 and 4 are incompatible, inference rules, instead of being shown to be rationally valid are deemed useless. To see this, suppose that B R (lp 1 ), B R (lp 2 ), B R (lp n ). Since the evidence for each parenthetical item is conclusive, we may safely use conjunction introduction to arrive at a conclusively supported conjunction. It will be rational for the subject who rationally believes these conjuncts to also believe the conjunction (B R ((lp 1 ) (lp 2 ) (lp n )). Note that the conjunction is within the scope of the belief operator, and so we have now something that is logically stronger than every one of the premises.) Yet this is a superficial victory. To see this observe that the conjunction is compatible with both of the following: (l(p 1 p 2 p n )) and (l(p 1 p 2 p n )). 18 One major issue concerns comparisons between rationally believed hedged propositions. We might sometimes face situations where it is important whether one proposition is more likely than the other. But then hedging is not the right phrase, since it becomes important in those contexts what (epistemic) probability is entailed by the evidence. Few if any of such propositions would be rational. 19 If one does not think that objective likelihoods are true or false one will not be entitled to use truth preserving inferences anyhow. 11

In other words, the conjunction (lp 1 ) (lp 2 ) (lp n ) gives us no information over and above the information we already had. If on my evidence it is rational to believe that tomorrow it is likely to rain and rational to believe that it is likely that there is a golf tournament, I have no idea if it is rational or not to believe that it is likely that there will be a golf tournament in the rain. To determine that, I need to check my evidence to see, for instance, if it all came from the same source, if some of it preceded the rest and if so in which order did it appear and so forth. 20 The gist of the claim here is that rather than resolving the problem with inferences, a suggestion along the lines of hedged propositions displays the very same problem, only this time in the content of rational attitudes. The inference may go through formally, but still the only guide we have for what to think about the world, so to speak, is the evidence. One is no better off for making proper inferences. (Note that this is in conflict with the Two Bases view.) There are other issues here, but let me just briefly mention one. If knowledge entails rational belief, then it will be rare to find unhedged knowledge (knowledge is almost always qualified and partially about one s epistemic state). A logician telling me, for instance, that q is a logical truth will result in knowledge that for me q is most likely a logical truth (until I learn to prove it). 21 This is a somewhat surprising and to my mind unwanted feature of this view. 4.3. Evidence and Probability We have so far looked at potential responses to the argument by requiring certainty of rationality in one way (conclusive evidence) or another (hedged propositions). I now move to responses that do not require certainty by the evidence but rather prevent the accumulation of doubt in other ways. To do this, that is to deny the possibility of 4 while 1,2,and 3 are true, is not an easy task. To appreciate the difficulty, consider standard probabilistic interpretations of the evidence-for relation (evidential support). If p and q are probabilistically independent, it may be the case that both Pr(p e) and Pr(q e) are greater than a rational number r in the interval [0,1] when 0<r<1 and yet Pr(p q 20 This example is a bit misleading, since presumably all these issues about the evidence need to be incorporated prior to the formation of the likely conjunctive belief. Nevertheless we can get extreme examples by adding more conjuncts and the example gives a simple model for what will be true with the longer conjunctions. If you are not persuaded, think of lottery vs. Independent propositions. 21 Analogous to the present type of suggestion is the semantic invalidity of modus ponens under some versions of fuzzy logic. Machina (1976: 70) defines an inference to be valid just in case the conclusion is at least as true as its falsest premise. He also (1976: 68) defines the value of a conditional as: v(p q) = 1-v(p)+v(q), if v(q)<v(p), 1, otherwise. So, with v(p)= 0.9 and v(q)=0.8, v(p q)=0.9. Hence, by Machina's definitions, the inference p, p q q, has a conclusion more false than its falsest premise. On Machina's view, this gives the proper diagnose of the sorites paradox: it relies on invalid inference steps. Thanks here to Peter Pagin. 12

e)<r. The probability of p1 p2 p3 pn on the evidence, then, may be arbitrarily close to 0 while the conditional probability of each conjunct might be arbitrarily close to 1 (greater than any r). Independence is not necessary to demonstrate the accumulation of rational doubt over conjunctions. Assuming the evidence-for relation is consistent with the standard calculus, if p entails the evidence e, the probability Pr(p e) will be greater than the prior probability of p. The same will be the case for Pr(p q e), it too will have a greater probability than the prior Pr(p q) (Pr(p q e)=pr(p q e)/pr(e) = Pr(p q)/pr(e) which will be greater than Pr(p q) (assuming Pr(e)<1)). So the evidence e supports p, supports q, and supports their conjunction. But e will not normally support p q to the same degree as it does p and does q independently. Naturally, the priors will be Pr(q)>Pr(p q)<pr(p) unless we assume a special case where p entails q, q entails p, or their probability is 1 or 0. But more to the point Pr(p q e) is lower than Pr(p e) and Pr(q e): Pr(p q e)=pr(p q)/pr(e) (since e is entailed by each of the conjuncts) and Pr(p e)=pr(p)/pr(e) and likewise for q. Hence, unless Pr(p q)=pr(p),it must be that: Pr(p q e)<pr(p e). But we already saw that unless p entails q or the other way around, the probability of P(p q) is lower than Pr(p) (it cannot be higher, of course, since p q entails p). So unless the evidence is conclusive (contrary to the general assumption), or the propositions are equivalent, fending off the argument requires the rejection of any account of evidential support that is consistent with the standard probability calculus (and perhaps some non-standard accounts as well). 4.4. Non-Probabilistic Support MPRV is very intuitive, so perhaps a rejection of the standard probabilistic interpretation of evidential support is a cost worth paying. Note that it is not enough merely to dispense with the probabilistic analysis of evidence. First, the evidence-for relation must be such that inconclusive evidential support does not decrease over deductive inference rules (independently of probability). Second, veritatic attitudes themselves must not be gradable. Finally, third, independently of the attitudes and the underlying evidential support structure, rational doubt cannot accumulate on any other independent dimension. For instance, in the above argument by Hawthorne and Lasonen-Aarnio, risk or rational doubt must be shown not to accumulate while the evidential support remains constant 22. If it can accumulate, it may become rational independently of the evidential relation to withhold judgement or 22 I will ignore the kind of risk accumulation associated with the deductive process itself. For arguments against deductive closure along these lines see Lasonen-Aarnio (2008) and Prawitz (2008). 13

disbelieve long inferred conjunctions. Let me say something about these challenges for responding to the argument by denying probabilistic interpretations of evidential support. Suppose evidential support is essentially non-probabilistic, call it qualitative support. To respond to the argument by denying 4 while accepting 3, one would have to claim that qualitative support does not allow rational doubt accumulation over deductive logical inferences. This means, straightforwardly, that propositions having to do with loss of fair lotteries cannot be rationally believed, they can t be rationally accepted, or presupposed. But even weaker attitudes can t be rational with respect to such propositions. Here one might be tempted by the following thought. Suppose we take the most undemanding rational attitude that is still veritatic: there is something (possibly minimal) to be said for the truth of p. Now since modus ponens is a truth-preserving inference, if there is something to be said for the truth of p, and something to be said for the truth of if p then q, then there is something to be said for q. Namely, that q follows by modus ponens from things for which something for their truth can be said. So at least that can be said for the truth of q. But this can t be right, for suppose we know for certain that either the butler is home (p) or he s at the movies (q), but there is no movie theater at his home ( (p q)). So there is something to be said (let us suppose) for the truth of p and something to be said for the truth of q. But surely there is something to be said for the tautology if p, then if q, then p-and-q, namely, that it is a tautology. But then by modus ponens and the idea above about how it transmits the attitude of having something to be said for its truth, there is something (perhaps very minimal) to be said for the truth of if q, then p and q. And since there is something to be said for the truth of q, there is something to be said for p-and-q, which there isn t. So any idea about evidential support, qualitative or probabilistic, will have to give up the MPRV or place sufficiency conditions to stave off absurdity. 23 Suppose a qualitative sufficiency condition is imposed. Still, qualitative or not, evidential support will have to make a distinction (a non-probabilistic one) between stronger and weaker evidence. Let us symbolize a sufficient evidential support relation for a veritatic attitude A, as E A (e,p) (where e is the total evidence a subject has, and p is a proposition). Qualitative support that is not necessarily sufficient, will be E(e,p) (without the superscript). On this view (as well as most others) it will be the case, presumably, that if p is logically equivalent to q (q p p q), then if E(e,p), then E(e,q), but if p is logically weaker than q (q p p q), e will support p at least as much as it supports q. In particular if E(e,p) and E(e,q), then on this view (given it s transitive nature) (pvq) is supported (at least as much and) 23 Note that giving up epistemic validity allows having weak rational veritatic attitudes. 14

possibly more than p and more than just q are by e. For instance, assuming E(e,p) is not equal to E(e,q), then either E(e,pvq)>E(e,p) or E(e,pvq)>E(e,q) or both. Now taking a case where the latter holds (i.e. E(e,q)<E(e,pvq)>E(e,q)) and given equivalence, means we have E(e, q)<e(e, ( p q))>e(e, p). This double in-equation means that e supports the claim that p-is-false-and-q-is-false more than it supports the claim that p is false. The same can be shown about the relation between e and q. Evidence e counts against the conjunction more strongly than it does against each of the conjuncts. If evidential support can accumulate, evidential doubt can also accumulate over conjunction introduction (assuming that equivalence holds). In other words, unless it is shown how rational doubt can be prevented from accumulating while rational support can accumulate, there is no reason to suppose that doubt won t accumulate enough to push a long conjunction below the sufficiency line: E A (e,p 1 ) E A (e,p 2 )... E A (e,p n ) and E A (e,p 1 p 2...p n ). If qualitative evidential support can increase over disjunction, it can decrease over conjunctions. So either these assumptions are refuted, or qualitative support as such is no refuge from rational doubt accumulation. This argument can be turned into a general worry about qualitative support: if qualitative support can accumulate, so can doubt. I think that the qualitative support view must accept this much. It would be strange (to say the least) if deductive rules are epistemically valid for attitudes, but not for the logic of evidential support on which the rationality of the attitudes is founded 24, it seems that a qualitative account of evidence will have to give up on very basic logical relations which it refuses to give up in general, i.e. when rational attitudes are in play. If the view does not give up the logical relations regarding evidential support, once the evidence suffices to rationalize a certain attitude, it must behave differently than it presumably did when the attitude was insufficiently supported. When evidence is collected doubt remains and presumably can accumulate (if support can), but in order to preserve conjunction introduction, the quantitative evidence view must hold that once it surpasses the sufficiency condition, doubt no longer accumulates over conjunction introduction. This does not seem like a promising avenue to further develop. It seems that the only remaining option is to find a distinction between cases of accumulation and cases of no accumulation (of rational doubt or support) that is categorical - it separates types of propositions (or types of support) from others. This means that with regard to the nonaccumulative category, we can have rational veritatic attitudes and in that case there is no problem with conjunction introduction and MP. This also means that in the other category no 24 For those who think that in any case evidential support is not transitive, this is no cost at all. But for those who believe that it is transitive when it comes to the attitudes, forgoing logical operations on the evidence should be a significant cost. 15

matter how much evidence one has for the accumulating cases, if they fall short of rational certainty one cannot have a rational veritatic attitude about them. This too does not seem promising. But, trouble does not stop here. Any view of this kind whatever line it goes on to develop must accept some questionable consequences over and above a bifurcation of the logic of evidence. Suppose I carefully copy phone numbers into my phone book. Knowing that I m careful and skilled in copying phone numbers, I m rational in having a variety of veritatic attitudes with respect to the copied phone numbers, such that a s number is X, b s number is Y, etc. Of course, my evidence does not entail that I ve copied any one of these numbers correctly, and yet, having these rational attitudes entails (according to the class of views under consideration) that it is also rational to have these attitudes towards the conjunction of all the phone number propositions. A familiar and powerful preface type intuition, must be, then, considered irrational all the way down (so to speak). In fact, it s irrational to suppose, believe, accept, etcetera that one of my veritatic attitudes is probably mistaken. Moreover, it is not even rational for me to withhold judgement on whether they are all correct. 25 Consider a case where having spots on one s forehead is a well documented symptom of illness X. Suppose further that identifying that a subject has these spots (by an expert in favorable conditions) is enough for a veritatic attitude about this subject having illness X to be rational. So long as the certainty required for the rationality of this attitude falls short of full certainty, more and more instances of subjects having the spots can be added to create a long conjunction of such propositions. The more subjects you add the harder it will be to say that one can rationally have the same veritatic attitude with regard to the conjunction as one is with regard to each of them. What do these examples show? How does the tension between rational premises and irrational (though deductively valid) conclusion arise? The answer is that the evidence in each case remains fixed, while possibilities of error accrue. Each new patient displaying the symptom and each new number copied add a new possibility of error. There is nothing forcing us to view these possibilities as corresponding to the probability calculus, and yet, the added possibilities of error are accumulative. In other words, while the evidence remains fixed, doubt accumulates and places added pressure on the rationality of the conjunction of the veritatic non-conclusively supported attitudes. This is a general reason why one would be 25 For a very good presentation of this and other preface type arguments, see Christensen (2003). 16

well advised not to lay too much hope on finding a sufficiency condition that relies on qualitative accumulated evidence that will be closed under logical operations. 26 4.5. Conclusive and Inconclusive Evidence The only way, it seems, to avoid the incompatibility is to assume that veritatic attitudes are rational only when conclusively supported by the evidence. For conclusively supported attitudes, clearly, the problem doesn t arise. If there is no uncertainty or rational doubt to begin with, there is nothing to accumulate over inference. But this idea, I argued, is not plausibly applicable to all rational veritatic attitudes (and significant problems arise even for those for which it is). But even if it could apply to all rational veritatic attitudes, it would not be free from the essential upshot of the argument the rational futility of inference. Although requiring conclusive evidence for the rationality of veritatic attitudes would resolve the incompatibility, it would only do so at the price of making inferences rationally redundant. If evidence is conclusive, a conclusion of a logically valid deductive argument will alway be derivable directly from the evidence (or from whatever propositions we take to represent the evidence). Though I do not deny that deductive inferences are useful when operating on the evidence, when applied to propositions that are in the scope of veritatic attitudes they have no rational role. Their function is purely heuristic. Thus, for the two presumed bases of rational veritatic attitudes, one is indispensable (we need the evidence since that is what needs to be conclusive), the other is redundant. The inferences from rationally certain attitudes are useful in the context of discovery, but have no genuine role in the context of justification. A similar issue applies to inconclusively supported attitudes where doubt is not allowed to accumulate. Say two agents S 1 and S 2 have the same evidence and the same attitudes. S 1 formed his attitudes by inference while S 2 formed them on the shared evidence. The only important question regarding the rational status of the attitudes seems to trace back to the evidence. That is, what will determine which of the veritatic attitudes is rational will depend 26 Several offers of this kind have been proposed, non of them have been able, as far as I can see, to answer this challenge generally. This does not mean that these offers do not provide a good account for the cases for which they were proposed. Stewart Cohen s (**) account of context change due to salience of error deals nicely with lottery paradox for knowledge and justified beliefs. On his account the conjunction introduction rule for knowledge and justification is not valid and so no claim is made that the general challenge can be met. Other accounts, like Martin Smith s (20**), that do attempt to meet the challenge generally, are indeed vulnerable. Smith s skillful account deals very well with justified beliefs when separating them from unjustified lottery type beliefs. It does this by an appeal to normic conditions that are more or less like ceteris paribus laws. Though Smith may be right in saying that lottery beliefs are not justified (I m not yet convinced), it is hard to see how the account can defend the conjunction rule in all cases. Conditions can be normal for independently formed beliefs while it is also (other things being equal) normal that at least one of the justified beliefs is false. The phonebook and symptom cases can be worked out in this way but even harder are unrelated beliefs. 17