Conditionals and Conditional Probabilities: Three Triviality Theorems

Abstract

Talk of conditionals rather than hypotheticals, and of conditional probabilities rather than degrees of confirmation. Also, write ‘A → B’ for ‘If A, then B’, ‘P(A, B)’ for ‘The conditional probability of A on (or given) B’, and ‘T’ for ‘¬(A & ¬A)’, A here some fixed but arbitrary statement. The conditionals Jeffrey advocates then are those which issue from, and hence for which P meets, this constraint

$$P(A \to B,C) = P(B,A\& C),$$

(1)

hence this one

$$P(A \to B,T) = P(B,A).$$

(2)

We might then define a sense of ‘if’ by saying that the degree of confirmation of a hypothetical on evidence e is the degree to which e, conjoined with the antecedent, confirms the consequent.

—Richard C. Jeffrey¹

“If,” Journal of Philosophy 61, pp. 702–703, 1964.