Probability and Infinity
In the preceding chapter the relations between the epistemic concept of probability, as a logic of partial belief, and the measure-theoretic concept of probabilitywere developed for objects — sentences — of finite complexity. The account showed that there are nice relations among (i) probabilities on sets of sentences, (ii) probabilities on the Tarski-Lindenbaum Algebras associated with those sets of sentences, (iii) the transparency of partial belief, and (iv) a condition formulated there, based upon the concept of coherence, called the simple sum condition. That account seems satisfactory as far as it goes, but, leaving aside the question to what extent it can be made less relativistic, that is to say, in what ways appropriate logics T can be more precisely specified, it remains obviously incomplete in one important formal respect: The functions there defined apply only to finitely complex objects, and, in particular, are additive only over finite disjunctions of incompatible sentences, or over finite unions of disjoint sets.
KeywordsBoolean Algebra Finite Subset Chapter Versus Individual Constant Basic Sentence
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