Logike sa integralima u uslovnim očekivanjima

It could say that the research of logical systems in the dissertation goes in two directions: the first one is related to probability theory i.e. measure theory, while the other direction is related to topology. The main subject of this dissertation is extending the classical logic to formal systems which will be adequate for describing and concluding in the mentioned mathematical environments. The first part represents follow-up of works by M. Rašković and R. Đorđević in the area of probability logics and especially in the area of biprobability logics. The focus is on the logics with integral operators and logics with conditional expectation operators. The topological class logic [11, 12], which is adequate for studying the notions of topological product and continuous functions on topological class-spaces, is presented in the second part of this dissertation. The introduction part of the dissertation describes history and significance of probability and topological logics and also gives a shorter review of infinitary logic Lω1ω and logics with generalized quantifiers L(Q). Then, the dissertation is divided into these seven chapters: (1) Nonstandard analysis, (2) Admissible sets, (3) Countable fragments of infinitary logic L∞ω, (4) Middle model, (5) Logics with integrals, (6) Logics with conditional expectation operators and (7) Topological logics. The first chapter, Nonstandard analysis, describes the basic notions of nonstandard analysis and two techniques that have the main role in constructing the expected models of represented logic systems: construction of Loeb measure and Loeb construction of Daniell integral. The second chapter, Admissible sets, describes elementary notions related to Kripke–Platek set theory and for its transitive models-admissible sets. Some special types of admissible sets which will be crucial for building the syntax of probability and topological logics are also described in this chapter, as well as relationship with classical computability theory. The third chapter, Countable fragments of infinitary logic L∞ω, is the representation of the countable fragment LA. It describes the way of formalizing the syntax an semantics in KPU theory, and consistent property, and it is proving completeness and compactness (Barwise compactness) for countable admissible fragments. All new logics, that will be described in this dissertation, will be infinitary logics an appropriate admissible sets. The fourth chapter, Middle model, represents o