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Abstract:
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The thesis consists of seven chapters and two appendixes. The Chapter 1 and the appendixes contain known notions and properties from probability logics. In Chapter 2 some propositional probability logics are introduced and their languages, models, satisfiability relations, and (in)finitary axiomatic systems are given. Object languages are countable, formulas are finite, while only proofs are allowed to be infinite. The considered languages are obtained by adding unary probabilistic operators of the form P≥s. Decidability of the logics is proved. In Chapter 3 some first order probability logics are considered while in Chapter 4 new types of probability operators are introduced. The new operators are suitable for describing events in discrete sample spaces. It is shown that they are not definable in languages of probability logics that have been used so far. A propositional and a first-order logic for reasoning about discrete linear time and finitely additive probability are given in Chapter 5. Sound and complete infinitary axiomatizations for the logics are provided as well. In Chapter 6 a probabilistic extension of modal logic is studied and it is shown that those logics are closely related, but that modal necessity is a stronger notion than probability necessity. In Chapter 7 decidability of these logics is shown by reducing the corresponding satisfiability problem to the linear programming problem. Finally, two automated theorems provers based on that idea are described. |