A Note on Probabilistic Validity Measure in Propositional Calculi

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. 1 The central point of this short note is the treatment of a propositional calculus by means of probability truthfulness of arbitrary formula in the context of considered calculus. Roughly speaking, the idea is to extend the propositional language by a kind of probability operators and then add the corresponding probability axioms to some propositional or modal logic. Simply, we adapt the Keisler's axioms (see 2]), related to the probability quantiiers, for use in the case of propositional language and extend a propositional non-classical logic by those new axioms. In such a way the logical system obtained may be considered as a kind of polymodal logic. In this article we present the basic deenitions and results regarding that approach and their extensive version can be found in 1]. Our system is based on the language consisting of the usual symbols for logical connectives, denumerable set of propositional letters, parentheses and two types of probability operators: r and r , for each r 2 S, where S is a nite subset of the real interval 0; 1], such that 0; 1 2 S and S is closed for +, where, e.g., in the case when r + s > 1, for r + s we take 1 (or, more generally, S can be any nite lattice with zero 0, unit 1 and join operation denoted by +). The set of formulae is deened inductively as the smallest set containing propositional letters and closed under formation rules: if A and B are formulae, then (:A), (A ^ B), (A _ B), (A ! B), r A and r A are formulae. The ususal set theoretic symbols, as well as the logical symbols &,), or, not, 8 and 9, will be used in the metalanguage with the meaning they have in classical logic. By L we denote any propositional superintuitionistic logic, i.e. the extension of the Heyting propositional calculus, and by L p { the logic having the following axiom-schemes: (0) All formulae provable in L.