Modular many-valued semantics for combined logics

TL;DR

This paper introduces a novel modular many-valued semantic framework for combined logics, enabling the construction of complex logics from simpler components while analyzing properties like axiomatic strengthening and decidability preservation.

Contribution

It presents the first modular many-valued semantics for combined logics, constructed via universal operations on partial non-deterministic matrices, applicable to multiple- and single-conclusion logics.

Findings

Preserves finite-valuedness in multiple-conclusion logics

Provides semantic characterizations for logic strengthening and combination

Establishes conditions for decidability preservation

Abstract

We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics whereas, unsurprisingly, it may be lost in the context of single-conclusion logics. Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism.