F. Esteva | Searcharxiv

arXiv2026

Degree-preserving Godel logics with an involution: intermediate logics and (ideal) paraconsistency

In this paper we study intermediate logics between the degree preserving companion of Godel fuzzy logic with an involution and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts. Although these degree-preserving Godel logics are explosive with respect to Godel negation, they are paraconsistent with respect to the involutive negation. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between the degree-preserving n-valued Godel fuzzy logic with an involution and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.

cs.LO↗

arXiv2009

Logics preserving degrees of truth from varieties of residuated lattices

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 paper 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 purely inferential 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.

math.LO↗