The Suszko operator. I
The development of algebraic logic during the last two decades has demonstrated both the importance and usefulness of the so-called operator approach in metalogical research, in particular in the study of various intensional logics. Thus, protoalgebraicity, which is the weakest, in a sense, general property that makes a class of logics accessible to methods of universal algebra, and has been thoroughly investigated for this reason, is intimately related to the Leibniz operator. However, not all reasonable logics are included in the hierarchy of protoalgebraic logics: for example, a two-valued logic (i.e. one that is complete w.r.t. two-element truth tables) is protoalgebraic iff it includes either the implicational or the equivalential fragment of classical logic. The Suszko operator is a tool which can help to classify all deductive logics according to a general scheme. Recall that the Suszko operator for a logic \(C\) is the largest of all operators for \(C\) that is both monotone and compatible with all \(C\)-theories; it coincides with the Leibniz operator if \(C\) is protoalgebraic. The paper under review is intended to be an attempt (and, actually, is the first such attempt) of a systematic study of the Suzsko operator as well as of developing a general theory of non-protoalgebrical logics. The present Part I includes the necessary background on sentential logics, matrices, deductive filters, congruences and homomorphisms as well as the Leibniz operator and protoalgebraic logics. Its main body deals with the general properties of the Suszko operator and and with Suszko-reduced models of logics (i.e. matrices in which the Suszko congruence is the identity relation). In particular, continuity of the Suszko operator and the structure of the class of Suszko-reduced models of a logic are investigated. For example, every Suzsko-reduced model is shown to be isomorphic with a subdirect product of a family of Leibniz-reduced models for the same logic.
scientific article; zbMATH DE number 2042713
scientific article