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SKEPTICISM AND PRIOR PROBABILITIES KEITtl LEHRER To avoid skepticism, one must assume that common sense con- tentions are more probable on the basis of our experience than their skeptical competitors. I shall argue in this paper that if such claims are more probable than their skeptical competitors on the evidence of experience, then the prior probability of the former must be greater than the latter. Thus, common sense claims must be assumed to be more probable than their skeptical competitors a priori to sustain the common sense position. The very important contribution of Jaakko Hintikka to Bayestian probability theory incorporating prior probabilities is thus shown to be germane to the avoidance of skepticism. A brief exposition of Bayesian probability theory will be useful. The Bayesian assumes that conditional probabilities, the probability of H on E, symbolically, p(H/E), is derived from prior probabilities, the probability of E, p(E), and the probability of the conjunction of H and E, p(H & E). The axiom for the computation of conditional probability is p(H & E) (a) p(H/E) = p(E) From this axiom we derive Bayesian theorems such as the following: p(E/H)p(H) (i) p(H/E) = p(E) and (ii) p(H/E) = p(E/H)p(H) p(EtH) p (H) + p (E/~H)p (~H) Theorem (ii) follows from (i) by p(E) = p(E/H)p(H) + p(E/~H)p(~H) 89

Skepticism and prior probabilities

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SKEPTICISM AND PRIOR PROBABILITIES

KEITtl LEHRER

To avoid skepticism, one must assume that common sense con- tentions are more probable on the basis of our experience than their skeptical competitors. I shall argue in this paper that if such claims are more probable than their skeptical competitors on the evidence of experience, then the prior probability of the former must be greater than the latter. Thus, common sense claims must be assumed to be more probable than their skeptical competitors a priori to sustain the common sense position. The very important contribution of Jaakko Hintikka to Bayestian probability theory incorporating prior probabilities is thus shown to be germane to the avoidance of skepticism.

A brief exposition of Bayesian probability theory will be useful. The Bayesian assumes that conditional probabilities, the probability of H on E, symbolically, p(H/E), is derived from prior probabilities, the probability of E, p(E), and the probability of the conjunction of H and E, p(H & E). The axiom for the computation of conditional probability is

p(H & E) (a) p(H/E) = p(E)

From this axiom we derive Bayesian theorems such as the following:

p(E/H)p(H) (i) p(H/E) = p(E)

and

(ii) p(H/E) = p(E/H)p(H)

p(EtH) p (H) + p (E/~H)p (~H)

Theorem (ii) follows from (i) by

p(E) = p(E/H)p(H) + p(E/~H)p(~H)

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K. LEHRER

which follows from the axiom (a) and the addition axiom. It says that if H and K are logically incompatible then

po t or K) = p(H) + p(K).

This yields the theorem

p(E) = p(E & H) + p(E & ~H)

from which the result in the denominator is (ii) is obtained from axiom (a) by algebraic manipulation.

Formulas (a), (i) and (ii) provide a theory of the probabilistic reasoning from the evidence, and they also reveal our prior prob- ability assignments. To show how they reveal prior probabilities, we shall first illustrate the general form of the reasoning. Imagine that we have two principles P and P' generating the same experiential expectations, and let E represent the latter. If E is the total experi- ential content implied by the principles, E summarizes the experi- ential content of both P and P'. Suppose, moreover, we are con- vinced that p(P/E) is greater than p(P'/E) . What we are assuming about prior probabilities is elucidated by probability principle (i) above from which we obtain the following equalities:

p(E/Pp0 a) p (P/E) =

p(E)

and

p ( P ' / E ) = p(E/P')p (P')

p(E)

It is an axiom of probability theory that if E is deducible from P, then

p(P/E) = 1.

Hence the former two equations reduce to the following:

p(P) p (P/E) =

p(E)

and

p(P') p(P'/E) = p(E)

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SKEPTICISM AND PRIOR PROBABILITIES

What this shows is that p(P/E) is greater than p0a'/E) if and only if p(P) is greater than p (P ' ) . The crux is that the conditional prob- ability is determined by the prior probabilities so that the condi- tional probability of one principle is higher than the other if and only if the prior probability of that principle is higher than the other. Hence, if we start with assumptions about conditional probabilities, the axioms of the Bayesian reveal our prior commitments. Scientific reasoning about the world is based on conditional probabilities, and the axioms of probability theory show that those conditional probabilities presuppose certain prior probabilites.

We now turn to the application of the principles concerned with skepticism. As a skeptical conjecture, consider the Cartesian hypo- thesis that our present sensory experiences are caused by some deceptive god rather than by the material object, a sheet of paper for example, we assume to be their source. The demonic hypothesis we refer to as hypothesis D. The ordinary hypothesis that our sensory experiences are caused by the material object, the sheet of paper, we call hypothesis M. Both M and D provide us with the same experiential expectations, for, by the Cartesian hypothesis, the decptive god provides us with exactly the same experiences as we would have if there were material objects before us. If we let E de- signate those experiences, then the probability of those experiences occurring would be the same on the one hypothesis as the other, that is, p(E/D) = p(E/M) . It might be reasonable to so interpret the statements involved so that E follows deductively from either M or D. But p(D/E) :/: p(M/E) , for we side with common sense and hold that the material object hypothesis is more probable than the demonic hypothesis in relation to what we experience.

We are already, from probability theory, committed to the equalities

p(E/D)p(D) p(D/E) = p(E)

and

p(M/E) = p(EM)p(M)

p(E)

Since p(E/D) = p(E/M), it follows that p(M/E) is greater than p(D/E) if and only if p(M) is greater than p(D). We must, therefore, assign the material object hypothesis a higher prior probability.

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K. LEHRER

Thus we presuppose in our prior probability assignment that a world of material objects is the more probable source o f our experiences.

Let us now consider Humean skepticism about causality. We assume there are necessary connections between causes and effects, such as a billiard ball striking another and causing it to move, and symbolize it as C. The skeptic tells us that there are just accidental constant conjunctions between the events rather than necessary con- nections, and we symbolize the latter thesis as A. Hume success- fully argued that our expectations concerning experience, which I symbolize as E, would be the same on A as on C. The probability of E is the same assuming A as C, that is, p(E/A) = p(E/C). Surely, however, C is a more probable hypothesis than A on the basis of E. It is, o f course, logically consistent to suppose that the succession of events we experience involves no necessary connection between the one billiard ball striking the second and the movement of the second, that the movement of the second regularly succeeding the movement of the first is nothing more than an accident. We assume, nevertheless, that this is very improbable and that what is more probable is that the events we witness are causally related. In short, p(C/E) is much greater than p(A/E).

We thus arrive at the Bayesian conclusion that the assumption of causality is assigned a higher prior probability. Again we have the probabilities

p(E/C)p(C) p ( C / E ) - p(E)

and

p (A/E)= p(E/A)p(A)

p(E)

from the axioms of probabilities. We have the equality, p(E/C) = p(E/A), because, as Hume showed, we have the same expectation of what we shall experience on the hypothesis of causality as on the hypothesis o f regular succession. Thus, p(C/E) is greater than p(A/E) if and only if the p(C) is greater than the p(A). This is now a familiar Bayesian argument showing that C has a higher prior probability.

The second Bayesian formula is also of interest. It yields the following equalities:

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SKEPTICISM AND PRIOR PROBABILITIES

p(C/E) = p(E/C)p(C)

p(E/C) p (C) + p (E/~C) p ("4?)

and

p(AIE)= p(E/A)p(A)

p(E/A)p(A) + p(E/~A)p(~A)

Experiential expectation is the same on A as on C, so p(E/C) = p(E/A). Assume that p(C/E) is greater than p(A/E). Since p(H) + p(~H) = 1 for any H, it follows that p(C) determines p(~C) and p(A) determines p(~A). It is reasonable to say that p(E/~C) is greater than p(E/~A). Our expectations of experiencing what we do would be less were we to suppose the causal hypothesis to be false rather than true, and our expectation would be still less were we to suppose that there are not even accidental constant conjunctions. The result is that for p(C/E) to turn out higher than p(A/E), the prior probability p(C) must be sufficiently higher than p(A) to compensate for the fact that p(E/~A) is lower than p(E/~C) in the denominator of the fraction.

The argument could be reiterated to show that other common sense assumptions and their scientific counterparts which we assume to be more probable on our evidence than skeptical hypotheses con- structed to generate the same sensory expectations must be assigned a higher prior probability. This reveals the a priori commitments that lie beneath our empiricist reasoning from evidence in terms of probability. It also shows that the development of probability theory to which Hintikka has contributed so brilliantly is an essential component in the rational explication of empirical reasoning.

UNIVERSITY OF ARIZONA TUCSON, ARIZONA 85721

USA

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