Categorical Abstract Algebraic Logic: Algebraic Semantics for Institutions

摘要

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π-institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N-rule based extension of an N-rule based π-institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok-Rebagliato-style characterization of those π-institutions with a mono-unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π-institution to possess an algebraic semantics is provided.