Probability quantifiers and operators. (English) Zbl 0933.03044

Series in Pure and Applied Mathematics. Belgrade: Vesta Company. iv, 121 p. (1996).
This short monograph extends work by Keisler and others. The first four chapters present an integrated and quite elegant exposition of nonstandard analysis and measure theory, Keisler’s logic of probability operators, and logics with integrals and conditional expectations, also introduced by Keisler. Chapter 5 introduces a logic embodying a \({\sigma}\)-finite measure in place of Keisler’s general probability measure. Chapter 6 presents a logic appropriate to the study of structures with two measures. Chapter 8 introduces a second order probability logic. In chapter 9 we have an intuitionistic logic with probability operators; these operators are general in that they may apply to either classical or intuitionistic logic. Finally, in chapter 10 the notion of cylindric probability algebra is introduced, relating the theory of deductive systems of probability logic to a geometry associated with basic set-theoretic notions. The volume is not easy reading, but presents as full a treatment of logics with probability operators as there is.


03C80 Logic with extra quantifiers and operators
03B48 Probability and inductive logic
03H05 Nonstandard models in mathematics
60A10 Probabilistic measure theory
28E05 Nonstandard measure theory
60B05 Probability measures on topological spaces
60A05 Axioms; other general questions in probability