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Inconsistency lemmas in algebraic logic
Author(s) -
Raftery James G.
Publication year - 2013
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201200020
Subject(s) - congruence relation , mathematics , lemma (botany) , finitary , boolean algebra , pure mathematics , discrete mathematics , semilattice , variety (cybernetics) , algebraic number , algebra over a field , congruence (geometry) , semigroup , mathematical analysis , ecology , statistics , geometry , poaceae , biology
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.