Philosophy Colloquium Talk
David Miller, Professor, University of Warwick

If You Must Do Confirmation Theory, Do It This Way

In this talk I begin to draw together, and package into a coherent philosophical position, a number of ideas that in the last 25 years I have alluded to, or sometimes stated explicitly, concerning the properties and the merits of the measure of deductive dependence q(c | a) of one proposition c on another proposition a; that is, the measure to which the (deductive) content of c is included within the content of a. At an intuitive level the function q is not easily distinguished from the logically interpreted probability function p that may, in finite cases, be defined from it by the formula p(a | c) = q(c′ | a′), where the accent represents negation, and indeed in many applications the numerical values of p(c | a) and q(c | a) may not differ much. But the epistemological value of the function q, I shall maintain, far surpasses that of the probability function p, and discussions of empirical confirmation would be much illuminated if p were replaced by q.

Each of q(c | a) and p(c | a) takes its maximum value 1 when c is a conclusion validly deduced from the assumption a, and each provides a generalization of the relation of deducibility. But the conditions under which q and p take their minimum value 0 are quite different. It is well known that if a and c are mutual contraries, then p(c | a) = 0, and that this condition is also necessary if p is regular. Equally, if a and c are subcontraries (a ∨ c is a logical truth) then q(c | a) = 0, and this condition is also necessary if p is regular. It follows that q(c | a) may exceed 0 when a and c are mutually inconsistent. The function q is therefore not a degree of belief (unless a positive degree of belief is possible in a hypothesis that contradicts the evidence). But that does not mean that q may not be a good measure of degree of confirmation. Evidence nearly always con- tradicts (but not wildly) some of the hypotheses in whose support it is adduced.

The falsificationist, unlike the believer in induction, is interested in hypotheses c for which q(c | a) is low; that is, hypotheses whose content extends far beyond the evidence. I shall provide an economic argument (reminiscent of the Dutch Book argument) to demonstrate that q(c | a) measures the rate at which the value of the hypothesis c should be discounted in the presence of the evidence a.