Inconsistency lemmas in algebraic logic



In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system inline image is algebraized by a variety inline image, then inline image has an inconsistency lemma—in the abstract sense—iff every algebra in inline image has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) inline image has a classical inconsistency lemma; (2) inline image has a greatest compact theory and inline image is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in inline image form a Boolean lattice; (4) the compact congruences of any inline image constitute a Boolean sublattice of the full congruence lattice of inline image. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false.