Containment logics: algebraic completeness and axiomatization

TL;DR

This paper explores the algebraic and semantic foundations of containment logics, providing completeness results and axiomatizations for these consequence relations.

Contribution

It introduces a matrix-based semantics for containment logics and develops a completeness theorem and Hilbert-style axiomatization.

Findings

Established a matrix-based semantics for containment logics

Proved a completeness theorem for a broad class of containment logics

Provided a Hilbert-style axiomatization for these logics

Abstract

The paper studies the containment companion of a logic $\vdash$. This consists of the consequence relation $\vdash^{r}$ which satisfies all the inferences of $\vdash$, where the variables of the conclusion are \emph{contained} into those of the (set of) premises. In accordance with our previous work on logics of left variable inclusion, we show that a different generalization of the P\l onka sum construction, adapted from algebras to logical matrices, allows us to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.