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Abstract: Let K be a variety of commutative, integral residuated lattices. Thesubstructural logic usually associated with K is an algebraizable logic thathas K as its equivalent algebraic semantics, and is a logic that preservestruth, i.e., 1 is the only truth value preserved by the inferences of thelogic. In this paper we introduce another logic associated with K, namely thelogic that preserves degrees of truth, in the sense that it preserves lowerbounds of truth values in inferences. We study this second logic mainly fromthe point of view of abstract algebraic logic. We determine its algebraicmodels and we classify it in the Leibniz and the Frege hierarchies: we showthat it is always fully selfextensional, that for most varieties K it isnon-protoalgebraic, and that it is algebraizable if and only K is a variety ofgeneralized Heyting algebras, in which case it coincides with the logic thatpreserves truth. We also characterize the new logic in three ways: by a Hilbertstyle axiomatic system, by a Gentzen style sequent calculus, and by a set ofconditions on its closure operator. Concerning the relation between the twologics, we prove that the truth preserving logic is the purely inferentialextension of the one that preserves degrees of truth with either the rule ofModus Ponens or the rule of Adjunction for the fusion connective.

Author: F. Bou, F. Esteva, J. M. Font, A. Gil, L. Godo, A. Torrens, V. Verdú

Source: https://arxiv.org/

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