Relevant Consequence Relations: An Invitation

TL;DR

This paper extends the concept of consequence relations in logic to incorporate relevance, using multisets and the use criterion, and provides foundational definitions and results for these relevant consequence relations.

Contribution

It introduces a generalized framework for consequence relations that accounts for relevance and multisets, expanding the standard abstract logic approach.

Findings

Definitions of relevant consequence relations in single and multiple conclusion settings.

Results showing the definitions capture the intended relevance intuitions.

Demonstrations of the framework's applicability to logical derivations and theories.

Abstract

We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the \emph{use criterion}, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be \emph{used} in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining between \emph{multisets}. We motivate and state basic definitions of relevant consequence relations, both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases).