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Inconsistency lemmas in algebraic logic
Authors:James G Raftery
Institution:Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, , Hatfield, Pretoria, 0028 South Africa
Abstract:In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0001 is algebraized by a variety urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0002, then urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0003 has an inconsistency lemma—in the abstract sense—iff every algebra in urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0004 has a dually pseudo‐complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0005 has a classical inconsistency lemma; (2) urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0006 has a greatest compact theory and urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0007 is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0008 form a Boolean lattice; (4) the compact congruences of any urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0009 constitute a Boolean sublattice of the full congruence lattice of urn:x-wiley:09425616:malq201200020:equation:malq201200020-math-0010. 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.
Keywords:Deductive system  inconsistency lemma  protoalgebraic logic  deduction‐detachment theorem  algebraizable logic  pseudo‐complement  filtral variety  Primary: 03B22  03G27  Secondary: 03G25  08C15
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