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