Categorical Abstract Algebraic Logic: Structurality, protoalgebraicity, and correspondence

摘要

The notion of an I-matrix as a model of a given pi-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 all 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 pi-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 pi-institutions, i.e., those with an implication system, and of protoalgebraic pi-institutions, i.e., those with a monotone Leibniz operator, the present paper interjects the class of those pi-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.