The algebraic significance of weak excluded middle laws

Abstract

For (finitary) deductive systems, we formulate a signature-independent abstraction of the 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 $K$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of $K$ has a greatest proper $K$-congruence; the converse holds if ⊢ 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 $\mathrm{KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of $S4$ has a global consequence relation with a WEML iff it extends $S4.2$, while every axiomatic extension of $R^t$ with an IL has a WEML.