Logics preserving degrees of truth from varieties of residuated lattices

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Félix Bou
Francesc Esteva
Josep Maria Font
Àngel J. Gil
Lluís Godo
Antoni Torrens
Ventura Verdú
Title: Logics preserving degrees of truth from varieties of residuated lattices
Journal: Journal of Logic and Computation
Volume 19
Number 6
Pages: 1031-1069
Year: 2009


Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e. 1 is the only truth value preserved by the inferences of the logic. In this article, we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences.

We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.

There is a Corrigendum for this paper in volume 22 (2012), pages 661–665, of the same journal. In it, a wrong argument in the proof of the last implication of Theorem 4.4 is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.