The Algebraic Significance of Weak Excluded Middle Laws

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.