Inconsistency lemmas in algebraic logic |
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Authors: | James G Raftery |
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Institution: | Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, , Hatfield, Pretoria, 0028 South Africa |
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Abstract: | 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 , then has an inconsistency lemma—in the abstract sense—iff every algebra in 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 is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in form a Boolean lattice; (4) the compact congruences of any constitute a Boolean sublattice of the full congruence lattice of . 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. |
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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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