A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.Keywords: Algebraic logic, finitely algebraizable logic, abstract algebraic logic, quasivariety.Our paper A Survey of Abstract Algebraic Logic [Su] presents, among other notions, the concept of algebraizability of a logical system. This concept was introduced by W. Blok and D. Pigozzi in [23]. In their monograph only finitary logics are considered, and their algebraic counterparts are restricted to quasivarieties; as a consequence, the interpretations involved in the original notion of an algebraizable logic are assumed to be finite. Later on all these finiteness restrictions have been removed in the works of other scholars such as Herrmann, Dellunde or Czelakowski. Accordingly the term "algebraizable" in the literature has come to be used for a more general notion, and some adjectives qualifying "algebraizable" are now in common usage to refer to several stricter versions, among them one incorporating the finitary original character. Section 3.3 of [Su] introduces the notion of algebraizable logic by describing Blok and Pigozzi's original approach, but using the new terminology and without explicitly assuming that the logics are finitary, in a way that is not consistent with later usage in the same paper. Moreover, we have detected a linguistic confusion when cross-referencing two conditions that share the same label. Since the Survey intends to give compact and reliable references to notions and results that were scattered in the literature or were simply part of the folklore of the field at the time of publication, we believe it may be useful to publish some precisions.The context where the algebraizability of logics is discussed in [Su] is the following. We reproduce here the main definitions and notations involved, as they appear in pages 38-39 of [Su]:
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the seventh publication in the Lecture Notes in Logic series, Font and Jansana develop a very general approach to the algebraization of sentential logics and present its results on a number of particular logics. The authors compare their approach, which uses abstract logics, to the classical approach based on logical matrices and the equational consequence developed by Blok, Czelakowski, Pigozzi and others. This monograph presents a systematized account of some of the work on the algebraic study of sentential logics carried out by the logic group in Barcelona in the 1970s.
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.
This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.
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