Logic, probability, and coherence

How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just stren gth of (prudent) partial belief, for this presumes logical omniscience. Thi s paper proposes that the way in which probability lies always between poss ibility and necessity can be made precise by exploiting a minor theorem of de Finetti: In any finite set of propositions the expected number of truths is the sum of the probabilities over the set. This is generalized to apply to denumerable languages. It entails that the sum of probabilities can nei ther exceed nor be exceeded by the cardinalities of all consistent and clos ed (within the set) subsets. In general any numerical function on sentences is said to be logically coherent if it satisfies this condition. Logical c oherence allows the relativization of necessity: A function p on a language is coherent with respect to the concept T of necessity iff there is no set of sentences on which the sum of p exceeds or is exceeded by the cardinali ty of every T-consistent and T-closed (within the set) subset of the set. C oherence is easily applied as well to sets on which the sum of p does not c onverge. Probability should also be relativized by necessity: A T-probability assign s one to every T-necessary sentence and is additive over disjunctions of pa irwise T-incompatible sentences. Logical T-coherence is then equivalent to T-probability: All and only T-coherent functions are T-probabilities.