Topics, Non-Uniform Substitutions, and Variable Sharing

TL;DR

This paper explores a strong variable sharing property called lericone relevance in relevant logics, analyzing its implications for classical logic fragments and related philosophical concepts.

Contribution

It introduces and characterizes lericone relevance, a strong variable sharing property considering negations and conditionals, and applies it to relevant and classical logics.

Findings

Lericone relevance holds in the logic $ extbf{BM}$.

A related property holds in logic $ extbf{B}$.

Largest fragments of classical logic with these properties are characterized.

Abstract

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties -- requiring that in valid entailments $A\to B$, some atom must appear in both $A$ and $B$ with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf{BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf{B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan's notion of hyperformalism, and for the very definition of a relevant logic itself.