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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 math formula is algebraized by a variety math formula, then math formula has an inconsistency lemma—in the abstract sense—iff every algebra in math formula has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) math formula has a classical inconsistency lemma; (2) math formula has a greatest compact theory and math formula is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in math formula form a Boolean lattice; (4) the compact congruences of any math formula constitute a Boolean sublattice of the full congruence lattice of math formula. 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.