Coherence and the Racehorse Paradox

Coherence and the Racehorse Paradox' KEITH LEHRER

his paper is concerned with a paradox revealing that a principle of rational T acceptance based on probability alone, or any combination of probability and informative content, will lead to unsatisfactory results. The lottery paradox illustrates that a principle directing us to accept what is highly probably will lead to the acceptance of a logically inconsistent set of statements. A number of philoso- phers, including most notably Levi, Hempel, Hintikka, Pietarinen, Hilpinen, and myself, attempted to avoid such inconsistency by proposing principles of rational acceptance that combined probability and informative content.' The paradox presented here shows that some factor other than probability and content must be brought in to yield a satisfactory principle of rational acceptance. The missing ingredient is coherence.

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There are three conditions of adequacy for principles of rational acceptance which, though not beyond controversy, provide us with familiar desiderata of 'rationality. They are as follows:

1. Consistency. The set of accepted statements should not be known to be a logically inconsistent set, that is, a set from which one knows a contradiction is logically deducible.

2. Closure. Any statement that is known to be a deductive consequence of accepted statements should also be accepted.

3 . Nonarbitrariness. One should not be arbitrary in what one accepts, that is, if one accepts a statement, then one should accept every other statement that is not known to differ from it in any relevant respect.

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184 KEITH LEHRER

Both the consistency condition and the closure condition have been contro- verted, most notably by K y b ~ r g , ~ and so some justification is appropriate. Note that the consistency requirement only disallows known inconsistency. I t may be reasonable for a person to accept an inconsistent set of statements because it is reasonable, though erroneous, for him to accept that the set is consistent. Such a person would not know that the set of statements he accepts is inconsistent. I t also may seem reasonable for someone who accepts a great deal t o accept that at least something he accepts is false. If he accepts this, then not everything he accepts can be true, but, as I have noted elsewhere: following Harman,' the set of statements he thus accepts need not be logically inconsistent. Moreover, if a person were to note than he accepts S,, S,, S,, and so forth, and then were also to accept that S, is false or S, is false or S, is false, and so forth, so that the set of statements he accepts is logically inconsistent, then he would be unreasonable. I t is, of course, highly probable that one or the other of the statements S,, S,, S,, and SO forth is false, but he may note that this is highly probable without accepting the statement that it is so. There are advantages of refusing to accept such a statement. Let US

assume that one embraces the objective of accepting what is true without accepting what is false. If one accepts the statement in question, this will guarantee that one fails in one's objective of not accepting what is false, for if S,, S,, S,, and so forth are all true, then the statement affirming that one or the other of these statements is false must itself be false. By accepting the latter statement one would convert the high probability of error which we naturally confront into an unwanted neces- sity. Moreover, by maintaining consistency we preserve the integrity and utility of what we accept for the purposes of prediction and explanation. Thus the suggestion that it is reasonable for a person to accept that at least some of things he accepts are false is erroneous. One should remain content with accepting that it is very probable, though not certain, that at least one or the other of things one accepts is false. *

Similar remarks are appropriate to the closure condition. I t may be reasonable for a person not to accept a deductive consequence of what he accepts because he may reasonably, though erroneously, accept that it is not a deductive consequence of what he accepts. But when one knows that something is a deductive consequence of what one accepts, then one should accept it. Harman6, objecting to closure, has noted that a person who discovers some untoward consequence of what he accepts may reasonably reject something formerly accepted rather than accept the consequence. That observation is compatible with the intended inter- pretation of the closure condition. That condition applies only to what a person accepts at one time. So if one accepts a set of statements at time t , , deduces a consequence at t , , and then decides to alter what one accepts at time t3 so that what one accepts does not have the consequence deduced at t,, that is perfectly compatible with the closure condition. Moreover, on this closure condition, as contrasted with others, there is no violation of the condition at t l when the person does not accept the deductive consequence, because at t l the person has not yet deduced the consequence and does not, we suppose, know that it is a consequence.

COHERENCE AND THE RACEHORSE PARADOX 185

Other objections to the closure condition rest on rejection of the consistency condition and on attempts to avoid including a contradiction in what one accepts. However, if one is adverse to accepting contradictions, that is because what one thus accepts is certainly erroneous. Such adversity to error implies, however, that one should accept the consistency condition.

The nonarbitrariness condition may seem the least controversial of the three. I t is not beyond controversy, however. Consider the lottery paradox. There are a million tickets in a fair lottery numbered in numerical order with one winning ticket. 1 know this. I hold ticket 124,318. 1 accept the hypothesis that my ticket will not win. If 1 conform to the condition of nonarbitrariness, then, for each ticket in the lottery, I should accept that it will not win. For there is no relevant difference between the ticket I hold and any other ticket in the lottery on the basis of the information I possess. But if I accept the hypothesis that each ticket will not win, the set of such hypotheses is inconsistent with what I know, namely, that one of the tickets will win. Thus the nonarbitrariness condition together with the consistency condition yields the result that one may not accept any of the hypotheses affirming that a specific ticket will not win in the lottery.

The lottery examples are two-dimensional because there are only two alter- natives for each ticket, either it wins or it does not. We shall now consider some paradoxes that are n dimensional since there are n possible outcomes for each number. The best realization of this is a horserace in which each horse can place first, second, third, and so forth. The paradox is presented in terms of horseraces. The horseraces, like the lotteries we discuss, are somewhat idealized, however. When thinking about the horseraces, all the information about horseraces should be disregarded except that supplied in the example.

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As a first illustration of such paradoxes, suppose that we use a high probability rule, one saying that any hypothesis that was more probable than 1/2 should be accepted. Consider, then, that one is informed that there is a horserace to be run with horses numbered 1, 2, 3, and that one of the following patterns will prevail, these being outcomes of previous races, and that each is equally probable.

Example A First 1 2 3 Second 2 3 1 Third 3 1 2

Notice that it is more probable than 1/2 that horse number 1 will beat horse 2: he does so in two out of three patterns, so you may accept that. I t is just as probable that 2 will beat 3, so we may accept that, and it follows deductively that 1 will beat 3. But, alas, it is probable that 3 will beat 1: he does so in two out of three patterns. So one should accept that 3 will beat 1. Thus one should accept, on the high probability rule, the statement that horse 1 will beat horse 3 and also that horse 3 will beat horse 1.

186 KEITH LEHRER

To those familiar with the lottery paradox, the foregoing result will not be unexpected, and the proper conclusion to be drawn is that the high probability rule is unsound. However, such horserace examples also generate difficulty for more sophisticated kinds of rules, namely, those that combine considerations of probability and informative content. Indeed, horseraces provide problems that cannot be solved by appeal to considerations of probability and informative content. Suppose that we are informed that the horses in the race are numbered 1, 2, 3 , 4 , 5, and that one of the following twenty-five patterns will prevail in the next race, these being the results of previous races, and that each is equally probable.

Example B

First 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Second 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5

Third 3 3 3 4 4 5 5 2 2 4 4 5 5 2 2 3 3 5 5 2 2 3 3 4 4

Fourth 4 4 5 3 5 3 4 4 5 2 5 2 4 3 5 2 5 2 3 3 4 2 4 2 3

Fifth 5 5 4 5 3 4 3 5 4 5 2 4 2 5 3 5 2 3 2 4 3 4 2 3 2

We should, in this example, accept that the number 1 horse will win the next race, for he virtually always does, but we should not accept anything about how the other horses will place since they place virtually randomly in the patterns supplied. Now let us consider the hypothesis that the number 1 horse will win. That is equivalent on our information to a di$unctive hypothesis saying that one of twenty- four patterns will prevail. That hypothesis is equal in probability to any disjunctive hypothesis saying that one of any twenty-four patterns will prevail. The patterns are equal in informative content: each tells us exactly how the race will come out, and so any disjunction of n distinct patterns is equal in content to any other. Each disjunctive hypothesis of twenty-four distinct patterns tells us on the basis of our information that the excluded pattern*will not prevail. So if we accept all such twenty-four pattern disjunctive hypotheses, the set of such hypotheses will exclude all the patterns. That is inconsistent with the information that one pattern will prevail. If we are restricted to probability and content as relevant factors, then either we must not accept the disjunctive hypotheses equivalent to the hypothesis that the number 1 horse will win, or, to avoid arbitrariness, we must accept all twenty-four pattern disjunctions and fall into inconsistency. Thus, if we are restricted to content and probability as relevant factors, we shall violate the condition of nonarbitrariness or the condition of consistency if we accept the hypothesis that the number 1 horse will win. We may reinforce the argument by noting that if the fact that the number 1 horse wins in twenty-four of the twenty-five cases is not convincing enough, we could add one more horse and generate 121 patterns in which the number 1 horse wins in 120, or two more, and generate 721 patterns in which the number 1 horse wins in 720, and so forth.

There is a reply to this line of argumentation, derived from Levi', which has some plausibility. I t is that the measure of content is generated from a question


posed and the possible answers to the question. Thus, if the question is - will horse 1 win or not? - thenfhe hypothesis that horse number 1 will win, and the hypothesis that he will not win, are equal in content. The hypothesis that he will win being as probable as it is, it becomes reasonable on the basis of the content and probability to accept that the number 1 horse will win. One reply, which is by itself adequate, is that how the other horses pattern themselves in a horse-race is germane to whether the number 1 horse will win, and, therefore, content should be measured in terms of the patterns cited above. Each pattern is equal in content to each other since each tells us exactly what will happen. Measuring content in that way, the hypothesis that the number 1 horse will win is equivalent to a disjunction of twenty-four patterns, whereas the hypothesis that the number I horse will not win is equivalent to the single other pattern. The latter is much more informative, since it tells us exactly how the race will turn out, whereas the hypothesis that the number 1 horse will win is much less informative since it tells us only that one or the other of twenty-four different patterns will prevail. Moreover, it will not do to insist that if what interests us is whether the number 1 horse will win or not, then the information about how the other horses will place is otiose. For we are interested in how all the horses will place. It is just that, given our information, we cannot tell how any of the horses will place, except horse 1, and we can tell that he will win.

Moreover, even if it be allowed that questions generate content measures, so that we may accept that the number 1 horse will win because it is equal in content to its denial, this solution will beget another paradox. For suppose we allow that if horse number I wins in all patterns but one, where the number of patterns is sufficiently large, then, if one asks whether number 1 will win or not, it is reasonable to accept that he will. By parity of reasoning, then, we should subscribe to the principle that, if the nth horse comes in nth in all but one of a sufficiently large number of races, then if one asks whether the number n horse will come in nth or not, it is reasonable to accept that it will. For, there is no relevant difference between finishing first and finishing n th .

Now suppose that we are given the information that there are as many consecutively numbered horses in the race as you care to imagine and that one of the following patterns will prevail:

Example C

First 1 2 3 4 5 . . . Second 2 1 2 2 2 . . . Third 3 3 1 3 3 . . . Fourth 4 4 4 1 4 . . . Fifth 5 5 5 5 1 . . . . . . . . . . . . . . . . . . . . . . . .

The paradox is immediately obvious. Suppose we allow that if the n numbered horse places n in four out of five patterns, then if we ask whether the nth

188 KEITH LEHRER

horse will place n t h , it is reasonable to accept that it will. When we ask whether the number 5 horse will come in fifth, it will be reasonakle to accept that it will. Simi- larly, when we ask whether the number 4 horse will come in fourth, the number 3 horse third, the number 2 horse second, it will be reasonable to accept that they will. So for each of the horses 5 , 4 , 3, and 2, we will accept that it will place accord- ing to number. Closure commits us to accepting that the number 1 horse will come in first. But when we ask whether the number 1 horse will come in first, we find that in 4 out of 5 patterns it does not come in first, and so we accept that it will not come in first. Therefore, once again, we violate the consistency condition.

It might be replied that the answer we gave to each question was consistent, and that is alI that can be required. We have argued, however, that the consistency condition should be satisfied for the total set of statements one accepts and not just for answers one gives to individual questions. A rational person does not think in logic tight compartments, refusing to consider the answer he gave to one question when answering another.

What should we say about what it is reasonable to accept in the last exampIe? Notice, first of all, that we can add as many horses to the race as we wish by ex- panding the pattern, and in a thousand horse race, the number n horse, for all n greater than I, will finish nth in 999 of the patterns and finish otherwise only once. So the frequency can be made as great as the frequency with which the number 1 horse finished first in example B. Nevertheless, if we accept that the number n horse, for all n greater than 1, will finish nth in example C , closure leads us to also accept that the number 1 horse will finish first in example C, which is extremely unlikely and must not be accepted. Therefore, in this example, example C, we should not accept the hypothesis to the effect that the number n horse will finish nth, or we shall be led to paradox. 1n.cxample B, however, we should accept that the number 1 horse will finish first. The problem is to formulate a principle that allows US to accept that the number 1 horse will win in example B but not allow that we accept that the n horse will fin& nth in example C.

As a first step toward formulating a solution of the problem formulated above, I should like to consider a rule I had formulated earlier to deal with the lottery paradox.* This principle does not cope with the paradox formulated above, but a modification of the principle does so. The principle is based on the idea here defended that reasonable acceptance is to be explicated in terms of prevailing over competition. The problem is to specify the relations of competition and prevalence. Competition, here referred to as r-competition, is defined as follows. A statement H r-competes with K on E if and only if K is negatively relavant to H on E, that is, the probability of H is greater on E than on the conjunction of K and E. On this notion of competition, H may r-compete with K on E even though they are logically consistent with each orher. Moreover, in the lottery example, the hypothesis that the number 1 ticket will not win r-competes with the hypothesis that the number 2 ticket will not win, and in the horserace examples, any disjunction of all patterns except pattern i r-competes with any disjunction of all patterns except j , where i # j .

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Now consider the rule that tells us that we should accept H on E if H is more probable than any K that H r-competes with on E. This rule also fails in example B to allow us to accept that the number 1 horse will win. For the statement that one of the first twenty-four patterns will prevail is an r-competitor of the statement that the number 1 horse will win, and both are equally probable. Moreover, the example may be reinforced by supposing that each of the patterns after the first two is ten or fifty times as probable as the first two, those two being equally probable. Again the statement that the number 1 horse will win is equivalent on our information to the statement that one of the twenty-four patterns after the first will prevail, and the statement that either the first pattern or one of the twenty-three patterns after the second will prevail is equally probable to the former statement. Thus the rule fails to allow us to accept that horse 1 will win when it is even more obvious that it will. The rule avoids unwanted acceptance in example C, but it is too restrictive in example B.

The solution to the problem before us is to recognize the importance of coherence. The hypothesis that the number 1 horse will win in example B race is more reasonable than any other twenty-four pattern because it is more coherent. On the other hand, in example C, the hypothesis that the number 2 horse will come in second is not more reasonable than the hypothesis that the number 3 horse will come in third, and not all hypotheses telling us that the n-numbered horse will come in nth can be accepted. To accept all such hypotheses would lead to incoherence. It is no more coherent to suppose that the number I horse will come in first in example C than that he will finish in any other place.

The crux is that it will not do to consider hypotheses about how a certain horse will finish in isolation from hypotheses about how other horses will finish. In deciding whether to accept a hypothesis about how a given horse will finish, one must consider what to accept about how every horse will finish. Of course, one might be able to accept hypotheses about how one horse will finish and not accept any about how others will finish. But, to obtain a coherent account, one must accept the consequences that follow from the statements one accepts about how any specific horse will finish. So if there are five horses and if one accepts that number 5 will finish fifth, number 4 fourth, number 3 third, number 2 second, then to avoid incoherence, one must accept that number 1 will finish first. One must not do that in case C, however. In short, when you consider the strategy of accepting that the number n horse will come in nth, you must consider where a coherent use of that strategy will lead you in the resulting system or set of accepted statements.

If it is granted that it is more reasonable in case B to accept the hypothesis that the number 1 horse will win than to accept any other twenty-four pattern disjunction, and that it is not more reasonable in case C to accept that horse 5 will come in fifth than that horse 4 will finish fourth, then we can formulate a rule articulated in terms of reasonableness that yields the result that it is reasonable to accept the hypothesis that the number 1 horse will win in case B but not to accept any of the hypotheses about how horses will finish in case C.

190 KEITH LEHRER

The rule is as follows:

Accept H on E if it is more reasonable to accept H on E than any statement K that H r-competes with on E.

This rule differs from the preceding rule by requiring that the accepted hypothesis be more reasonable rather than just more probable than its competitors. This is crucial. For, as we noted above, the hypothesis that the number 1 horse will win in the B case is no more probable than any other twenty-four pattern disjunction with which it competes, nor, for that matter does it have higher content. I t gives us a more coherent account of what will happen than do the other disjunctions, since they do not allow us to draw any conclusion about how any horse will finish. This proposal resolves the paradox. I t does so in terms of a notion of reasonableness that cannot be explicated in terms of probability and content. The notion of coherence, though unexplicated, is not entirely mysterious. Three of its components are consistency, closure, and nonarbitrariness discussed above. These are objectives of reason, desiderata of epistemic rationality. Their satisfaction increases the coherence of the account we give and, in that way, makes it more reasonable.

The conclusion I propose is that the reasonableness of accepting a hypothesis depends on systemic considerations, on what sort of.overall account one can give. Two hypotheses that’are equal in content and probability may differ in an im- portant way. One may be such that if we accept it and are not arbitrary and do not violate closure, we shall be led to inconsistency. Examples A and C illustrate such situations, as does the original lottery paradox. In other examples, such as B, we may accept a hypothesis without inconsistency though we observe closure and are not at all arbitrary. I conclude that reason mandates consideration of the system of statements or hypotheses we accept and’ does not permit us to decide what to accept in terms of the intrinsic features of hypotheses considered in isolation. Rational acceptance is a matter of coherence within a system.

Notes

1.

2.

I am greatly indebted to Glenn Ross, Donald Hubin, Lee Carter, and W. V . 0. Quine for their remarks and proposals pertaining to this paper.

1. Levi, Gambling with Tnrth (New York, 1967); C. G. Hempel. “Deductive-Nomologi- cal vs. Statistical Explanation,” vol. 111 in Minnesota Studies in the Philosophy of Science, ed. H. Feigl and G. Maxwell (Minneapolis, 1962), pp. 98-169; J. Hintikka and J . Pietarinen, “Se- mantic Information and Inductive Logic,” in Aspects of Inductive Logic (Amsterdam, 1966), pp. 96-1 12; R. Hilpinen, Rules of Acceptance and Inductive Logic, Acta Philosophica Fennica, Fasc. 22 (1968); and K. Lehrer, “Truth, Evidence, and Inference,” American Philosophical Quarterly (1974): 74-92.

3 . H. E. Kyburg, Jr., “Conjunctivitis,” in Induction, Acceptance, and Rational Belief, ed. M. Swain (Dordrecht, 1970), pp. 55-82.

4 . K. Lehrer, “Reason and Consistency,” in Analysis and Metaphysics, ed. K . Lehrer (Dor- drecht, 1975), pp. 57-74.

5. C. Harman made this observation at an A. P. A. meeting in his comments on an early version of the paper cited in note 4. His comments are, unfortunately, unpublished.

COHERENCE AND THE RACE HORSE PARADOX 191

6 . 83-99.

7. Levi, Gambling with Truth. 8 .

G. Harman, “Induction.” in Swain, Induction, Acceptance, and Rational Belief, pp.

K. Lehrer. Knowledge (Oxford, 1974) , pp. 192-97.

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Coherence and the Racehorse Paradox