Logics Preserving Degrees of Truth from Varieties of Residuated Lattices

Abstract

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 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.

Exploring Paraconsistency in Degree-Preserving Fuzzy Logics

“…Since the notion of truth is interpreted by a designated set of values in the algebras (often just one designated value), it appears that only these truth-values are relevant as regards to consequence. An alternative approach that has recently received some attention is based on the degree-preservation paradigm (see [3,4]), in which a conclusion follows from a set of premises if for all evaluations its truth degree is not lower than that of the premises. It has been argued that this approach is more coherent with the commitment of manyvalued logics to truth-degree semantics because all values play an equally important rôle in the corresponding notion of consequence (see e.g.…”

On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics

“…As regards axiomatization, the logic L ≤ admits a Hilbert-style axiomatization having the same axioms as L and the following deduction rules [5]:…”

“…In other words, the defining requirement in the truth-preservation paradigm for an inference to be valid is, actually, that every algebraic evaluation that interprets the premises as completely true, will also interpret the conclusion as completely true. An alternative approach that has recently received some attention is based on the degree-preservation paradigm (see [15,5]), in which a conclusion follows from a set of premises if, for all evaluations, the truth degree of the conclusion is not lower than those of the premises. It has been argued that this approach is more coherent with the commitment of many-valued logics to truth-degree semantics because all values play an equally important rôle in the corresponding notion of consequence (see e.g.…”

“…On the other hand, L ≤ is complete with respect to the class of all matrices of type (A, F ) where A is an MV-algebra and F is a lattice filter of A, see e.g. [15,5]. Besides, it is proved in [5] that L ≤ is also complete with respect to the restricted set of all matrices ([0, 1] MV , F ), where F is a lattice filter of [0, 1], i.e.…”

“…[15,5]. Besides, it is proved in [5] that L ≤ is also complete with respect to the restricted set of all matrices ([0, 1] MV , F ), where F is a lattice filter of [0, 1], i.e. intervals either of type [a, 1] with a > 0 or of type (a, 1], for a < 1.…”