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.