On axioms and rexpansions

TL;DR

This paper explores methods to strengthen logical systems by adding axioms through rexpansion, providing both general and effective characterization techniques, and demonstrating how to develop analytic calculi for these enhanced logics.

Contribution

It introduces two characterization methods for rexpanding logics with axioms, including an effective approach applicable to many existing examples, and shows how to derive analytic calculi.

Findings

Two characterization methods for rexpansions are developed.

The effective method applies to a wide range of existing logics.

Analytic multiple-conclusion calculi can be constructed for the resulting logics.

Abstract

We study the general problem of strengthening the logic of a given (partial) (non-deterministic) matrix with a set of axioms, using the idea of rexpansion. We obtain two characterization methods: a very general but not very effective one, and then an effective method which only applies under certain restrictions on the given semantics and the shape of the axioms. We show that this second method covers a myriad of examples in the literature. Finally, we illustrate how to obtain analytic multiple-conclusion calculi for the resulting logics.