Consistency and Epistemic Probability Doris Olin

Is consistency always epistemically virtuous? Is it possible for a set of rational beliefs to be inconsistent? The traditional view has been that logical consistency is a minimum requirement for rational belief. Recently, this traditional view has been challenged, and is now in some disrepute. The issue is not only of interest in its own right, but also has bearing on several other aspects of our conception of justified belief. In particular, it is a critical issue for the coherence theories of justification which have been so prominent of late, for coherence is normally understood in such a way as to presuppose logical consistency. (1) Three distinct lines of argument against consistency can be discerned in the current discussion (although not always clearly distinguished there): (i) the paradox of the lottery, (ii) the fallibility argument, the core of which is an inference from my fallibility in the past to the claim that my present justified beliefs also contain error, and (iii) what I call `the epistemic probability argument', which is based on the rules of the classical probability calculus. I shall focus, in this paper, just on what I take to be the most significant threat, the third argument. (2) An examination of this argument will prove of interest both for its implications concerning consistency and, as well, for what it reveals about epistemic probability. The strength of one's evidence is surely a critical factor in assessing warranted belief. So the epistemic probability of a proposition, understood as the degree of confirmation or support provided by the total available evidence, would seem to be a central notion in epistemology. Epistemic probability, of course, is to be sharply distinguished from other types of probability, in particular from statistical probability; the latter has to do with the relative frequency with which a given property or event occurs within a certain reference class. It is the concept of epistemic probability which bears directly on issues of warranted belief. The degree of epistemic probability adequate for warranted belief is commonly thought to be high, but less than certainty. The argument for inconsistency I want to examine may be put informally as follows. Consider a large set of statements which I am warranted in believing. My evidence for each, let us suppose, is very strong, but not perfect; that is, these statements are not certain on the basis of my evidence. So for each of my beliefs there is some risk of error. If a number of these statements are conjoined, the risk of error associated with the conjunction is still greater than that associated with any individual belief. Given a sufficiently lengthy conjunction, the likelihood of error will increase to the point that I am justified in believing that the conjunction is false. Hence, I am justified in believing each member of the original set, as well as the negation of their conjunction. Since this is an inconsistent set, it follows that I am justified in believing inconsistent propositions. (3) This is the core of the argument for the rationality of inconsistency. The argument can be stated more formally, and more precisely, in terms of the rules of the probability calculus. The multiplication rule for the epistemic probability of a conjunction is: Cr(p & q) = Cr(p) x Cr(q / p) (where `Cr' designates degree of confirmation). It follows from this rule that if a statement q is added as a conjunct to some statement p, the probability of the conjunction will be less than that of p provided that Cr(p) _ 0 and Cr(q / p) < 1. Thus, if p is a contingent statement and p does not logically imply q, then Cr (p & q) must be less than Cr(p). Suppose now that A_{1}, ..., A_{n} are logically independent contingent statements which I am justified in believing; (4) they might be about what I ate for breakfast, what I read in the newspaper, and so on. Their degree of confirmation is high (but less than 1). It follows from the above that the probability of the conjunction A_{1} & ... & A_{n} will decline with the addition of each conjunct; with enough conjuncts, the degree of confirmation of the conjunction will be very low. But the probability of a statement and its negation are complementary: Cr(~p) = 1  Cr(p). So the degree of support for ~(A_{1} & ... & A_{n}) will be very high; given sufficient components, it will be high enough to warrant belief. Consequently, I will be justified in believing each of A_{1}, ..., A_{n}, as well as ~(A_{1} & ... & A_{n}). That is, I will be justified in holding inconsistent beliefs. The standard presentation of this argument in the literature makes use of the conjunction rule. It is worth noting, however, that another version of the argument can be constructed focusing on the rule for the probability of a disjunction. The general rule for disjunctions is: Cr(p v q) = Cr(p) + Cr(q)  Cr (p & q) We can thus see that if a statement q is added as a disjunct to some statement p, the probability of the disjunction will be greater than that of p provided that Cr(q)  Cr(p & q) > 0, which will be the case if Cr(q) > Cr(p & q). So if p is a contingent statement and q does not logically imply p, the probability of p v q must be greater than that of p. Now consider a large set of logically independent contingent statements B_{1}, ..., B_{n}, where each B_{i} is the negation of a statement I am justified in believing. Given sufficient components, the degree of support for the disjunction of B_{i}'s will be high, and I may be justified in believing it. It is thus possible that I be warranted in believing each of ~B_{1}, ..., ~B_{n}, as well as B_{1} v ... v B_{n}. But this is an inconsistent set of beliefs. I want to show that these arguments for the rationality of inconsistent belief are unsound. For purposes of illustration, I shall focus on the version which draws on the multiplication rule for conjunctions. The plausibility of this argument rests, in part, on the fact that the conclusion, ~(A_{1} & ... & A_{n}), does not strike us as particularly informative or significant. My strategy will be to apply analogous probabilistic reasoning to carefully tailored examples. The conclusions which emerge are substantive, informative and, I shall argue, utterly implausible. To accept the epistemic probability argument against consistency is thus to be forced to grant absurdities. Consider: A medical research laboratory is working on different aspects of AIDS research. A list is prepared, extremely lengthy as it happens, of natural medications and therapies which have at some time been claimed to have some impact on the disease. Each item on the list is investigated by the research team as a potential cure. A_{1} is the claim that the first item on the list is not a natural cure for AIDS; and so on. The total evidence amassed by the research team provides strong support for each A_{i}. It follows from the now familiar reasoning that given sufficient A_{i}'s, the degree of confirmation of the negation, ~(A_{1} & ... & A_{n}), will reach the level associated with warranted belief. Consequently, we will be warranted in believing each of A_{1}, ..., A_{n}, as well as the negation of their conjunction. This much is analogous to the original argument. Note, however, that ~(A_{1} & ...&A_{n}) logically implies: A: There is a natural cure for AIDS. It follows that the degree of confirmation of A is at least as high as that of the negation of the conjunction; and if we are warranted in accepting ~(A_{1} & ... & A_{n}), then clearly we are also warranted in accepting A. But these are absurd results. Surely we are not prepared to grant that belief in A, belief that there is a natural therapy which cures AIDS, could ever be warranted on this basis. And it seems absurd to suggest that the results achieved by the research team could, in principle, provide strong confirmation for A. Repeated failures to establish that a particular natural regimen cures AIDS cannot provide strong confirmation or adequate evidence for the positive thesis that there is a natural cure for AIDS. The degree of support for A, it appears, is no higher than that of the negation of the least probable A_{i}. The consequences of this analogous argument, then, are simply unacceptable; and it is difficult to see how one might nevertheless justify accepting the epistemic probability argument. (Note that the same absurd consequences will flow from the disjunction rule. Let B_{1} be the claim that the first item listed is a natural cure for AIDS. Given sufficient components, the degree of support for the disjunction of B_{i}'s may reach the level associated with warranted belief. But the statement B_{1} v ... v B_{n} implies that there is a natural regimen which cures AIDS.) The AIDS case provides a particularly striking example with which to illustrate the absurdity of this form of reasoning; but other effective examples are readily available. One might take each A_{i} to be the denial of a report that a particular person has witnessed an event which constitutes a miracle (in which case A would be the assertion that someone has witnessed a miracle); or each A_{i} could be interpreted as the denial of a claim that a particular individual has exhibited psychic powers (in which case A would be the claim that some individual has exhibited psychic powers); or, finally, each A_{i} might be taken as asserting that a particular claimant is not Anastasia, daughter of the Czar of Russia (in which case A would be the assertion that a particular claimant is Anastasia). Each A_{i}, let us suppose, will be strongly confirmed. But surely no one would be tempted to justify belief in A by indicating as evidence a large set of A_{i}'s. What general conclusions can be drawn here? First, what has emerged is that if the epistemic probability argument is successful in establishing the possibility of rational inconsistent belief, then it will also be the case that strong confirmation for the thesis that there is a natural regimen which cures AIDS (or that a miracle has occurred) is possible, in principle, on the basis of a series of entirely negative findings, or discredited positive claims. The consequent of this conditional is, in my view, quite incredible. I conclude, therefore, that the antecedent must be denied, that is, that the epistemic probability argument for inconsistency fails. Second, the diagnosis: the argument for inconsistency fails because epistemic probability, degree of support, does not conform to the axioms of the classical probability calculus. In the examples cited, the degree of confirmation of each B_{i} can not be combined, or added up, to produce a substantially higher degree of support for the disjunction. Nor does the degree of confirmation of the conjunction of the A_{i}'s appear to follow the multiplication rule. (I leave open the possibility that these rules may be adequate for some cases.) This is not, of course, an entirely novel conclusion. Others have challenged the fit between the principles of the calculus and degree of support or confirmation. (5) Still, the issue has been controversial and far from clear cut. Relatively few philosophers have been prepared to endorse the view that epistemic probability diverges from the calculus. The argument advanced here , I believe, is more compelling than previous discussions; the implications of the conjunction and disjunction rules which have been revealed here are simply unacceptable. It is one thing to grant, as the epistemic probability argument purports to show, that there is error in my present justified beliefs. But it is quite another to suppose that a series of entirely negative findings may provide adequate evidence for a substantive thesis — for instance, that some individual has exhibited psychic powers. Is there any possibility of salvaging the disjunction and conjunction rules of the calculus? Two objections to the analogous arguments would have this effect, and should be considered here. First, it might be suggested that we erred in the probability values we ascribed to the A_{i}'s in the examples above. That is, each A_{i} should be assigned a probability of 1 on the grounds that we are confident that each A_{i} is true. (Similarly, each B_{i} should be assigned a value of 0.) Such an assignment would ensure that the probability of the appropriate conjunctions does not decrease. This result would be achieved, of course, only at the cost of abandoning a standard convention of probability theory, namely, that 0 and 1 be reserved for logically impossible and necessary statements, respectively. One unfortunate consequence of giving up this convention is that we will no longer have any way to distinguish the degree of confirmation of contingent statements such as the A_{i}'s from that of logical truths. More significantly, it is difficult to see how one could use this option to rule out every analogous argument without assigning a value of 1 to any statement we are clearly justified in believing (and thus a value of 0 to its negation). But the effect of such an assignment would be the complete collapse of the argument for the rationality of inconsistent beliefs. Second: Another suggestion is that the rule of negation must be rejected. More specifically, it might be proposed that the probability of a statement and its negation need not equal 1; (6) and that the negation of any statement we are justified in believing will have a probability of 0. This proposal would also clearly stop the analogous arguments. The probability of the disjunction of the B_{i}'s would not increase given that each B_{i} has a probability of 0. In the version which makes use of the conjunction rule, we could not infer a high probability for ~(A_{1} & ... & A_{n}) from the low probability of the conjunction. Of course, for the very same reasons, both versions of the argument for the rationality of inconsistency would also fail. Each of these proposals, it appears, blocks the analogous arguments only to the extent that it also defeats the argument for inconsistency. So there is no way to accept the latter without also granting the former. Further, each objection involves rejecting some aspect of the traditional calculus. What seems clear, then, is that epistemic probability cannot conform to the calculus as a whole. It remains a plausible diagnosis, although not the only one possible, that the rules for conjunction and disjunction are, at least in part, the source of the problem. (7) Let me, in closing, briefly pursue some consequences of this diagnosis. If the epistemic probability of conjunctions and disjunctions does not conform to the calculus, then a natural alternate model would be that the probability of a disjunction is equal to that of its most probable disjunct, and the probability of a conjunction is equal to that of its least probable conjunct. (8) Certainly, this model seems plausible in the cases discussed above. Unfortunately, it does not appear to hold in every instance. Let me illustrate just with regard to the disjunction rule (analogous points can be made concerning conjunction). The most obvious exception is found in the category of logical truths. Clearly, the probability of P v ~P, for example, will be higher than that of either of its components if neither P nor ~P has a probability of 1. But the problem also occurs with contingently true disjunctions. Suppose there are exactly three candidates in an election, Adams, Brown and Cowley. No new names can now be added to the list of candidates, and no candidate has a clear lead. Let A be the statement that Adams will win; and so on. The disjunction A v B v C will have a very high degree of confirmation, since one of the three must win; and it will be higher than any of its disjuncts, since no candidate is highly likely to be the winner. So again the alternate model fails, in this case for a contingent statement. It is interesting to note that what unites these two examples is that the probability of the disjunction, in either case, is not affected by the assumption of the negation of any disjunct. In the second case, for instance, Cr(A v B v C) = Cr(A v B v C / ~A). These are two clearcut counterexamples to the new model; there may well be others. However, it would be rash to suppose that epistemic probability is never correctly calculated by the alternate rules. Nor, for that matter, should we infer that the traditional rules are never accurate. What seems clear is that neither model holds universally. To conclude: I take the analogous arguments to show that degree of confirmation, or epistemic probability, does not conform to the classical probability calculus. This conclusion should have significant implications for epistemology generally. One immediate consequence is that the epistemic probability argument for inconsistency must fail. Thus, one threat to the traditional view of consistency is now removed. 
Notes (1) In The Theory of Epistemic Rationality (Cambridge: Harvard University Press, 1987), 96102, Richard Foley argues that the coherentist position cannot accommodate the rationality of inconsistent beliefs. (2) In another paper, `The Fallibility Argument for Inconsistency', Philosophical Studies 56 (1989), 95102, I show that the reasoning in the fallibility argument is defective at several points. The lottery paradox, I believe, can ultimately be handled by distinguishing between statistical and epistemic probability, and denying a perfect correlation between the two. This general approach is taken by J.L. Pollock, `Epistemology and Probability', Synthese 55 (1983), 231252, although I do not agree with the details of Pollock's argument. (3) Some examples of this type of argument can be found in: Richmond Campbell, `Can Inconsistency be Reasonable?', Canadian Journal of Philosophy 11 (1981), 254; Richard Foley, op. cit., 251; Keith Lehrer, `Reason and Consistency', in Analysis and Metaphysics (Dordrecht: D. Reidel, 1975), 65; and John N. Williams, `Inconsistency and Contradiction', Mind 90 (1981), 601602. (4) In saying that A_{1}, ..., A_{n} are logically independent I mean not just that no statement implies any other in the set, but also that no statement implies, or is implied by, any contingent truth functional compound of any of the other statements. I also assume here that the probability of a contingent statement is greater than 0 and less than 1. This assumption is later examined. (5) See, for example, Pollock, op. cit., 24244, and R.M. Chisholm, Perceiving: A Philosophical Study (Ithaca: Cornell University Press, 1957), 13. (6) This proposal was suggested by L. Jonathan Cohen's treatment of what he calls ‘inductive probability’; Cohen denies that the probability of a statement and its negation are complementary. See The Probable and the Provable (Oxford: Clarendon Press, 1977), 37. (7) Note that throughout this paper, an assumption has been made which is necessary for the epistemic probability argument, as well as the analogous arguments, namely, that there is a degree of confirmation less than 1 which is sufficient for justified belief. If this assumption were not granted, then, of course, no conclusion concerning warranted belief would follow. But the same arguments would show that, in the cases presented, there is strong confirmation that there is a natural therapy which cures AIDS (or that someone has exhibited psychic powers). And this conclusion is itself quite absurd. It can be avoided, however, only by rejecting the fit between degree of confirmation and the calculus. (8) This model for conjunction is endorsed by Pollock, op. cit., 24849, and Cohen, op. cit., 221. ^{} 