Two logical dimensions

Abstract

Relational formalization of nonclassical logics is provided by means of a relational semantics for the languages of these logics and relational proof systems. Relational semantics is determined by a class of algebras of relations, possibly with some nonstandard operations and constants, and by a meaning function that assigns relations to formulas. A formula is true under relational interpretation whenever its meaning is the unit element of the underlying algebra. Relational proof systems consist of Rasiowa-Sikorski style decomposition rules for every relational operator admitted in the respective algebras, and with specific rules that reflect properties of relational constants. Nonclassical logics are two-dimensional in the following sense: they consist of an extensional part and an intensional part. The extensional part carries a declarative information about knowledge states. The intensional part includes a procedural information, it represents transitions between knowledge states. Relational formalization exhibits these two dimensions in several nonclassical logics, including various modal logics, intuitionistic logic, Post logics, relevant logics. In the relational semantical structures of these logics the Boolean reducts of the underlying algebras of relations provide a formal counterpart of the first dimension. The monoid reducts are the counterpart of the second dimension. The two dimensions are also manifested in the relational proof systems. The decomposition rules for Boolean operators refer to the first dimension, and the rules for monoid operators correspond to the second dimension. In the paper the paradigm of two logical dimensions and their manifestation in relational formalisms is illustrated with examples of many-valued logics.