Reasons, Knowledge, and Probability Fred I. Dretske Philosophy of Science, Vol. 38, No. 2. (Jun., 1971), pp. 216-220. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28197106%2938%3A2%3C216%3ARKAP%3E2.0.CO%3B2-F Philosophy of Science is currently published by The University of Chicago Press.
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REASONS, KNOWLEDGE, AND PROBABILITY* FRED I. DRETSKE University of Wisconsin Though one believes that P is true, one can have reasons for thinking it false. Yet, it seems that one cannot know that P is true and (still) have reasons for thinking it false. Why is this so? What feature of knowledge (or of reasons) precludes having reasons or evidence to believe (true) what you know to be false? If the connection between reasons (evidence) and what one believes is expressible as a probability relation, it would seem that the only satisfactory explanation of this fact is that when one knows that P is true, the reasons or evidence one has in support of P are such as to confer upon P the probability of 1. It is shown by an application of Bayes' Theorem that any value smaller than 1 would permit having reasons to believe what one knows to be false. Hence, it would seem that knowledge requires conclusive reasons to believe (if reasons or evidence is required at all).
T wish to state what I take to be a fact and, then, to offer an explanation of that fact. The explanation I offer will strike many as objectionable enough t o warrant rejecting it as the explanation of the fact in question or, if no better explanation can be found, objectionable enough to justify a rejection of the fact itself. Nonetheless, I think both my 'fact' and my explanation of it will survive examination. There is nothing to prevent one from believing what one has reasons or evidence to think false. One may have better reasons for thinking it true. Or it may simply be a question of stubbornness, unyielding faith, neurotic fixation, or whatever. If S knows that P is true, however, S can have no reason, no evidence, to think P is false. There may be certain facts, F, with which S is acquainted which are such that if S did not know that P was true, he would (rightly) regard them as reasons for believing that P was false. Moreover, these facts may constitute reasons for others, those who do not know that P is true, to think that P is false. But they do not, indeed cannot, constitute reasons for S to think that P is false. This is the fact mentioned above. Let me say a few more words about it before attempting to explain it. Bill knows there are some cookies in the jar (he just now looked); Sam does not. Both watch a hungry child peer into the jar, replace the lid without extracting anything, and leave with a disappointed look on his face. Sam now has a reason to think the jar empty; Bill does not. For Bill, who knows there are cookies in the jar, the child's behavior is not a reason to think the jar empty; it is, instead, sonzethitzg to be explained (if he does not already know the explanation). Perhaps the child does not like peanut butter cookies. Perhaps he didn't see them when he looked into the jar. What makes the child's behavior something to be explained from Bill's point of view is, of course, the fact that the absence of cookies is not available to explain it. Bill knows there are cookies in the jar, and, hence, knows
* Received February, 1970.
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that the child's empty-handed departure cannot be explained by an absence of cookies. Bill could take the child's behavior as a reason to think the jar empty (whether o r not, as a result of this, he believed the jar empty is irrelevant) only if he could regard an empty jar as a possible, a more or less competitive, explanation for the child's behavior. This, though, is something he cannot do as long as he knows the jar is not empty. If he should take the child's behavior as evidence, as a reason, to think the jar empty (despite having just seen cookies in it) he can no longer be described as knowing that the jar has cookies in it whatever he might have known, or thought he knew, a moment ago. For to persist in saying he (still) knows the jar has cookies in it is to say something which is absurd: viz. that S is treating a hypothesis (no cookies in the jar) which he knows to be false as a possible, a more or less competitive, explanation for the child's behavior. Anyone who has reasons to believe not-P (whether or not he believes it), once he learns that P is the case, will describe his fornaer situation by saying that he had reasons for believing not-P. I-Ie no longer has these reasons although he still knows everything he previously knew which constituted his having had reasons. Anyone who has reasons for believing not-P, as long as he continues to have these reasons, cannot truly be said to remember that P; nor can he see that P, discover that P,or, in general, 4that P where " S 4s that P" entails " S knows that P." This is not because having a reason to believe not-P makes it difficult, psychologically or otherwise, to remember that P, see that P, or discover that P. It is, rather, because we give up calling something a reason (for him) to believe not-P as soon as we begin describing him, directly or indirectly, as knowing that P. Knocking at the door of a darkened house for several minutes without a response may be a good reason to suppose the occupants away. When I hear voices from within, and recognize these voices as the voices of the occupants, the reasons I had for thinking the occupants gone have vanished. Or, perhaps, they have become reasons for believing something quite different. If, however, I hear voices from within, but do not recognize them, I can still have my former reasons for believing the occupants away (the voices I hear might be burglars) although I might now concede that I also have some reason to think they might be home. With this brief attempt to establish my fact, let me turn to its explanation. Suppose S believes that P is the case and the reasons (grounds, evidence) he has for believing P to be the case (if he has reasons) are R. Let F be a consideration, fact, or whatever, which functions, or is capable of functioning, as a reason against P, as evidence for not-?. For example, S believes his mother-in-law will be arriving for a visit today (P), and he believes this on the basis of her recent letter in which she declared such an intention (R). S now learns that all airplane flights have been cancelled due to a snow storm (F) and, knowing she intended to come by plane, takes this as a reason to suppose her visit will be delayed, as a reason to believe she will not come today. Being a persistent woman she might come by other means, but the news concerning the restriction of air travel gives S some reason to believe she will not be arriving for a visit today. Let me try to express these rela-
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FRED I. DRETSKE
tionships in a standard way within the probability calculus. If we agree to read Pr(R : P ) as : the probability of P given R, then I think we can say that if F is to be a reason for S to believe not-P, if S is to have F as a reason for believing not-P, then necessarily
where R is understood to compreheiid all S's relevant background information. (1) simply says that if P i s to be a reason for S to believe not-P, the likelihood of P's being the case must, in some degree, be diminished by the addition of F to whatever other information S has relative to the truth of P. We need not understand (1) in any quantitative way; I think the way we speak about some facts making others more or less likely will itself justify the use of an inequality sign. Now, if Pr(R, F: P) cannot be 17zade smaller than Pr(R : P), then F cannot be a reason for S (for whom R represents the total information relevant to the truth of P ) to believe that not-P.l In general, Pr(R, P : P) can always be made smaller than Pr(R : P). There is one relevant (see note 1) exception. When Pr(R : P ) = 1, then Pr(R, F: P ) must also equal 1. This can be proved by a simple application of Bayes' Xi~versionTheorem :
=
Pr(R, P : F )
1 (by hypothesis) x Pr(R : F )
Since the probability of P, given R, is 1, the probability of F given both R and P (the numerator of the last term on the right) can be no smaller than the probability of F given simply R (the denominator). This is so because P adds no new information, information independent of R, which could diminish the likelihood of P relative to R. To shift our manner of characterizing R, F a n d P for the moment (i.e. from facts, states of affairs, or situations to classes) the class intersection of R and P is simply R since our hypothesis, Pr(R : P ) = 1, tells us that R cP. Hence, whatever proportion of Rs are F, the same proportion of Rs and Ps will be F: For example, if the probability of a man's being over six feet tall, given that he is a professional basketball player, is X, then the probability of a man's being over six feet tall, given that he is a professional basketball player and has a mother is still X even though most people who have a mother are not over six feet tall. The reason this new fact (he has a mother) does not diminish the probability of our man's being over six feet tall is that the information we already had (the person is a professional This is not strictly true since in the event that Pr(R: P ) = 0 it is clear that Pr(R, F: P) cannot be made smaller than Pr(R: P ) ; yet, since this is equivalent to Pr(R: not-P) = 1 we might wish to say that even though the probability of not-P (relative to R) was 1 we could still acquire additional (albeit superfluous) reasons in favor of not-P. In what follows I shall ignore this special case. I am interested in those situations in which P remains a possible, a more or less probable, alternative. That is, I am interested in those situations in which Pr(R : P) > 0 and, nlore particularly, 1 am interested in those situations in which Pr(R: P) > .5.
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219
basketball player) already included this 'additional' information in virtue of the fact that all professional basketball players (not to mention persons) have a mother. Siace, therefore, the numerator of the last term on the right can be iao smaller than the denominator (it cannot be larger without deriving a probability greater than 1 for Pr(R, F : P)), the last term on the right must be equal to 1 when PrfR: P) is 1. Therefore, when Pr(R : P) is 1, Pr(R. F': P) cannot be made snialler than Pr(R : P); and this is the only time (neglecting Pr(R : Pj = 0) it cannot be made smaller. This, then, I take to be the explanation of the {act with which we began. The reason one cannot have reasons for believing not-P when one knows that P i s that knowing that P involves having a degree of evidential support which, when expressed within the notation of the probability calculus, confers upon P the probability of 1. That is, knowing that P (on the basis of X) implies Pr(R : P ) = 1 and hence, no fact, F; can be such as to lower the probability of P (no fact, F, can be, such as to make PrfR, F: P) < Pr(R: P)). And this means that no fact, F; can serve as a reason against P. If we express Pr(R : P) = I by saying that R is a conclusive reason for P, the explanation for our opening fact is that knowing that P on the basis of R implies that R is a conclusive reason for P and no additional facts courzt against that for which one has conclusive reasons. It is analogous to a sitnation in deductive logic: when one has a deductively valid argument, no additional premises can diminish or in any way alter its validity. If this is the explanation of our fact, there is an important corollary: viz. if evidence, grounds, or reasons are required at all for knowing that P, then the connection between this evidence, these reasons, and P is not one of high inductive support, high confirmation, or any other relationship which admits the possibility of still higher degrees. If we wish to use this notation to express the relationship between oae's grounds for believing P and P itself when one is said to know that P, then we must express this as Pr(grounds :P) = I. Anything less than 1 would imply that we could reduce the probability of P relative to our evidence by additional pieces of information, and this is inconsistent with the facts already discussed." The probability calculus, the machinery it supplies for analyzing degrees of inductive support and degrees of confirmation, is largely irrelevant to the analysis of knowledge. Where knowledge is concerned, the grounds, evidesoe. or reasons in the sense of adinitting no one has (if they are required at all) must be co~zchrsil~e better support, There are no degrees. One final word. t e s t the conclvsion of this paper should strike sonle as necessarily involving sceptical consequences, 1 should add that P believe PrQR : P ) 1 can be realized without R entaili~gP. P believe that Pr(R : P) = I can justifiably be used to express the state of affairs which obtains when, as a nlatter cf fact, R would
-
There are, of course, other reasons for representing knowledge by a probability of 1. For example, if we should say that S knons that P (on the bask ol N) was to be represented by Pr(R : P) 2 .9, then we should encounter the situation in which S knew that P on the basis of R (Pr(K: P) = .9), knew thzt Q on the basis of M (Pr(M: Q ) = .9), but did not know that P nnd Q were the caye on the basis of R and M since, generally speaking, Pr(R,M: P and @) < .9.
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not be the case unless P were the case. For the latter statement implies that when we are given R, it is false that not-P might be the case and this, essentially modal expression, can only satisfactorily be rendered in the probability calculus by setting Pr(R : P ) equal to 1. Hence, sve can know that P (on the basis of R) even when this entails Pr(R : P ) = 1 though our reasons, R, are logically independent of P.
