Relational Semantics for Nonclassical Logics: Formulas are Relations

Possible world semantics of logical languages introduced by Kanger and Kripke around 1960 is the most widely used technique for formal presentation of nonclassical logics. In spite of some shortcomings connected with the incompleteness phenomenon, it provides an intuitively clear interpretation of the fact that a formula is satisfied in some possible worlds and it might not be satisfied in some of the others. With this interpretation formulas can be treated as those subsets of a universe of possible worlds in which they are true. In most of the possible world models the truth conditions for the intensional propositional operations are articulated in terms of properties of a binary or ternary accessibility relation between possible worlds. It follows that from a formal point of view possible world semantical structures are not uniform. The part responsible for the extensional fragment of a logic under consideration determines a Boolean algebra of sets, and the part responsible for the intensional fragment refers to an algebra of relations. Our main objective in the present paper is to develop a unifying algebraic treatment of both extensional and intensional parts of logical systems.

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