A Note on Probabilistic Validity Measure in Propositional Calculi

Abstract

The propositional language extended by two families of unary propositional probability operators and the corresponding list of probability measure axioms concerning those operators is the basis of the system preseted here. We describe a Kripke-type possible worlds semantics covering such a kind of logical systems.

Intuitionistic propositional probability logic

“…Finally, two extensions of intuitionistic logic are presented in [6,7], but in those papers, the probability is modeled using two functions, where one is subadditive and the other is superadditive. The reason for that is philosophical in nature and should reflect the growth of knowledge.…”

Tableau for the logic ILP

“…In this paper we adapt the prefixed tableau system for intuitionistic logic [3,4] and propose a system for the logic ILP which formalizes intuitionistic reasoning about probabilities. The paper [2] presents an intuitionistic logic with probabilistic operators and a complete axiom system for intuitionistic Kripke models in which each possible world is equipped with two partial functions representing inner and outer probability measures with a finite range. Probabilistic logics based on intuitionistic logic, but with probabilistic operators obeying laws of classical logic (e.g., the probability of an uncertain proposition should be either greater or equal to some r or less than r) are proposed in [8][9][10].…”

“…• in contrast to [2] probability measures (and not inner and outer probabilities) are considered, and the range of probability measures is the unit interval of rational numbers (and not a finite subset of that interval), and • probabilistic operators, in contrast to [8][9][10], follow the laws of intuitionistic logic.…”