Categorical Abstract Algebraic Logic: Structurality, protoalgebraicity, and correspondence

Abstract

AbstractThe notion of an ℐ ‐matrix as a model of a given π ‐institution ℐ is introduced. The main difference from the approach followed so far in Categorical Abstract Algebraic Logic (CAAL) and the one adopted here is that an ℐ ‐matrix is considered modulo the entire class of morphisms from the underlying N ‐algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed (N,N ′)‐logical morphism. The motivation for introducing ℐ ‐matrices comes from a desire to formulate a correspondence property for N ‐protoalgebraic π ‐institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π ‐institutions, i. e., those with an implication system, and of protoalgebraic π ‐institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π ‐institutions with the correspondence property, as applied to ℐ ‐matrices. Moreover, this work on ℐ ‐matrices enables us to prove many results pertaining to the local deduction‐detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)