Discussing the relations between logic and probability, this book compares classical 17th- and 18th-century theories of probability with contemporary theories, explores recent logical theories of probability, and offers a new account of probability as a part of logic.

How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just strength of (prudent) partial belief, for this presumes logical omniscience. This paper proposes that the way in which probability lies always between possibility 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 neither exceed nor be exceeded by the cardinalities of all consistent and closed (within the set) subsets. In general any numerical function on sentences is said to be logically coherent if it satisfies this condition. Logical coherence 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 cardinality of every T-consistent and T-closed (within the set) subset of the set. Coherence is easily applied as well to sets on which the sum of p does not converge. Probability should also be relativized by necessity: A T-probability assigns one to every T-necessary sentence and is additive over disjunctions of pairwise T-incompatible sentences. Logical T-coherence is then equivalent to T-probability: All and only T-coherent functions are T-probabilities. (shrink)

Ramsey's “Facts and Propositions” is terse, allusive, and dense. The paper is far from easy to understand. The present essay is an effort, largely following Brian Loar's account,1 to say what Ramsey's goal is, to spell out what he took to be the means to accomplish it, and to show how those means, at least in the terms of F&P, cannot accomplish that end. I also contrast Loar's own account of judgment, explicitly modeled on Ramsey's view, with the latter. The (...) exercise is not at all academic. Loar makes clear the striking depth and originality of Ramsey's insights. (shrink)

The interpretation of the calculus of probability as a logic of partial belief has at least two advantages: it makes the assignment of probabilities plausible in cases where classical frequentist interpretations must find such assignments meaningless, and it gives a clear meaning to partial belief and to consistency of partial belief.

Ramsey's “Facts and Propositions” is terse, allusive, and dense. The paper is far from easy to understand. The present essay is an effort, largely following Brian Loar's account,1 to say what Ramsey's goal is, to spell out what he took to be the means to accomplish it, and to show how those means, at least in the terms of F&P, cannot accomplish that end. I also contrast Loar's own account of judgment, explicitly modeled on Ramsey's view, with the latter. The (...) exercise is not at all academic. Loar makes clear the striking depth and originality of Ramsey's insights. (shrink)

The main sources here are Hume, Husserl, and De Finetti. The problem is how phenomenological investigation has to do with partial or probabilistic judgment. Behavioristic, frequentist and subjectivistic views are briefly surveyed. A variant of Hume's account of the probability of chances is developed with the help of De Finetti's concept of exchangeability. The question of transcendental elements in or behind partial judgment is considered in the light of understanding disagreement and error in partial judgment.