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The thesis consists of six chapters. Chapter 1 contains the structures, in which the probability logics are realized, and the basic methods of nonstandard analysis which are used in the other chapters. In Chapter 2 the syntax and semantics of the following probability logics are presented: the logic with the probability quantifiers L_Ap, the logic with the integral operators L_A_∫ , the logic with the operator of conditional expectation L_AE and adapted probability logic L_ad. Moreover, the certain important results about these logics are given. The problems of Barwise’s completeness, completeness, compactness, the existence of analytic and hyperfinite models for biprobability logics LA_P1_P2, LA_∫_1_∫_2 and L_ad in absolute continuous and singular cases are solved in Chapter 3. The many-probability logic BC{L_AP_i:i∊I}, I∊A obtained by Boolean combinations of probability logics L_AP is introduced and some model-theoretical properties of that logic are given in Chapter 4. In Chapter 5 the probability logic L^2AP∀ of second order is introduced, which is motivated by Keisler’ s problems with L_AP∀ and some topological logics. The problem of completeness for the logic L^2AP∀ is proved. In Chapter 6 cylinder probability algebras are introduced and some possibilities to solve problems for these algebras (which are characteristics of standard cylinder algebras, as the representation, axiomatization and decidability) are presented. |
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