Algebraizable logics. An exact reproduction of the text originally published in 1989 with an errata sheet prepared by the second author in 2014 (Q2812986)

In the sense of this monograph, an algebraizable logic is any deductive system \(S\) for which there is a class \(\mathrm{K}\) of algebras (so called equivalent algebraic semantics for \(S\)) such that (i) the consequence relation of \(S\) can be interpreted, in a natural way, in the semantical equational consequence relation of \(\mathrm{K}\), and (ii) an inverse interpretation also exists. Such a general and mathematically precise definition of the concept of algebraizable logic was given in (the first edition of) the monograph for the first time, and it made it possible to move from the study of various specific classes of algebras associated with specific logical systems, as in the classical algebraic logic, to the study of the process of algebraization itself and of relations betweeen metalogical properties possessed by the logical systems under consideration and the corresponding algebraic properties of the algebraic counterparts of these systems (abstract algebraic logic). In particular, only a precise definition of algebraizability enables one to prove that a logical system is not algebraizable. Two important results presented in the monograph are an intrinsic test, in terms of so called Leibniz operator, for algebraizability, and a characterization of algebraizability that is useful in practice.NEWLINENEWLINEFor a more detailed review of the content of the first edition of the book (published by the American Mathematematical Society in 1989), see [Zbl 0664.03042]. The errata sheet added to the present reprint (24 entries) contains corrections of various misprints. The most significant of them: page 39, line 7, replace `the previous lemma' by `Lemma 4.5'; page 51, line 12b, replace `Corollary 5.3' by `Theorem 5.1.(i).