Philosophy Colloquium

David Miller

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 ($a \vee c$ 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 contradicts (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 $a$.

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