Combining fragments of classical logic: When are interaction principles needed?

TL;DR

This paper characterizes when combining fragments of classical logic conservatively extends the logic and identifies the necessity of interaction principles for certain combinations, based on algebraic and semantic frameworks.

Contribution

It provides a precise characterization of when combined logic fragments produce the full classical fragment and when interaction principles are needed, using algebraic and semantic methods.

Findings

Interaction principles are required when the combined fragment is not fully characterized by the components.

The paper applies Post's clone theory, Rautenberg's axiomatization, and Avron's non-deterministic matrices to analyze logic combinations.

It offers a semantic and algebraic framework to understand the combination of classical logic fragments.

Abstract

We investigate the combination of fragments of classical logic as a way of conservatively extending a given Boolean logic by the addition of new connectives, and we precisely characterize the circumstances in which such a combination produces the corresponding fragment of classical logic over the signature containing connectives from both fragments given as input. If the thereby produced combined fragment is only incompletely characterized by the components given as input, this means that connectives from one component need to interact with connectives from the other component, giving rise to interaction principles. The main contributions strongly rely on the (well-known) description of the 2-valued clones made by Post, on the (not so well-known) axiomatization procedures for 2-valued matrices laid out by Rautenberg, and on Avron's non-deterministic matrices, which have (recently) been used to produce a significant advance on the understanding of the semantics of fibring.