Information, Confirmation, and Conditionals

Abstract

Loosely speaking, a proposition adds the more information to corpus b, the greater the proportion of possibilities left open by b that it rules out. Plausible qualitative constraints lead to the result that any measure of information-added is a strictly decreasing rescaling of a conditional probability function sending 1 to 0. The two commonest rescalings are -log P and 1−P. In a similar vein, e is favourable evidence for hypothesis h relative to background b if h rules out a smaller proportion of the possibilities left open by b and e jointly, than left open by b alone. In terms of the underlying probability measure, this secures the familiar positive relevance conception of confirmation and that f is more favourable evidence for h than e iff h rules out a smaller proportion of the possibilities left open by b and f jointly than left open by b and e jointly.      In these terms, a measure of confirmation should be a function of the information added by h to b∧e and to b, decreasing with the first and increasing with the second. When e = h, the possibilities that drop out as we narrow the focus with e are exactly the possibilities left open by b but excluded by h. Thus the extent to which hconfirms h relative to b is a measure of the information h adds to b.      Given a measure I of information added, we can think of I(ac,b) − I(a,b) as a measure of the “deductive gap”, relative to b, between a and a∧c. When, I(a,b) = P(a|b), I(ac,b) − I(a,b) = -logP(c|ab), the amount of information the indicative conditional ‘if a then c’ adds to b on Ernest Adams' account of that conditional. When I(a,b) = 1−P(a|b), I(ac,b) − I(a,b) = I(a⊃c,b) where a⊃c is a material conditional. What, if anything, can be said in general about “information theoretic” conditionals obtained from measures of information-added in this way? We find that, granted a couple of provisos, all satisfy modus ponens and that the conditionals fall victim to Lewis-style triviality results if, and only if, I(a∧¬a,b) = ∞ (as happens with -logP(.|b)).