Correction to the paper “ Logics preserving degrees of truth from varieties of residuated lattices ”

A wrong argument in the proof of one of the main results in the paper is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction. The proof of the last implication in Theorem 4.4 of the referenced paper [1] is wrong. At a certain point, it performs a construction on an arbitrary algebra but the properties used in its development implicitly assume that the algebra is in fact a residuated lattice, which it need not be. Here we present a correct proof done by working only in the formula algebra, following the same ideas and performing essentially the same construction, modulo a certain crucial lemma that characterizes the supremum operation in the lattice of theories of one of the logics considered in [1]. Let us recall the necessary background. Let K be an arbitrary variety of commutative, integral residuated lattices. The paper [1] studies two finitary logics associated with each such K , which are denoted by their consequence relations: The first one, denoted by `K , is the truth-preserving logic determined by the algebras in K when their maximum 1 (which is also the unit of the monoid structure of the fusion operation ? ) is taken as representing truth; the second one, denoted by |=K , is the logic preserving degrees of truth determined by the ordering relation of the algebras in K (which are always lattices). More precisely: φ0, . . . ,φn−1 `K ψ ⇐⇒ ∀A ∈ K ,∀v ∈ Hom(Fm,A) , if v(φi) = 1 for all i < n, then v(ψ) = 1. / 0 `K ψ ⇐⇒ ∀A ∈ K ,∀v ∈ Hom(Fm,A) , v(ψ) = 1. φ0, . . . ,φn−1 |=K ψ ⇐⇒ ∀A ∈ K ,∀v ∈ Hom(Fm,A) ,∀a ∈ A , if v(φi)> a for all i < n, then v(ψ)> a. / 0 |=K ψ ⇐⇒ ∀A ∈ K ,∀v ∈ Hom(Fm,A) , v(ψ) = 1.