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

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