The Algebraic Significance of Weak Excluded Middle Laws

TL;DR

This paper explores the algebraic implications of weak excluded middle laws in deductive systems, establishing conditions under which certain algebraic and logical properties hold, and characterizing these laws in various logical frameworks.

Contribution

It introduces a signature-independent abstraction of WEML, linking it to algebraic properties of quasivarieties and extending the concept to protoalgebraic logics and specific modal and relevance logics.

Findings

WEML implies every relatively subdirectly irreducible member has a greatest proper congruence

Super-intuitionistic logic has WEML iff it extends KC

Normal extensions of S4 with WEML extend S4.2

Abstract

For (finitary) deductive systems, we formulate a signature-independent abstraction of the \emph{weak excluded middle law} (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety $\mathsf{K}$ algebraizes a deductive system $\,\vdash$. We prove that, in this case, if $\,\vdash$ has a WEML (in the general sense) then every relatively subdirectly irreducible member of $\mathsf{K}$ has a greatest proper $\mathsf{K}$-congruence; the converse holds if $\,\vdash$ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends $\mathbf{KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of $\mathbf{S4}$ has a global consequence relation with a WEML iff it extends $\mathbf{S4.2}$, while every axiomatic extension of $\mathbf{R^t}$ with an IL has a WEML.