W. J. Blok and Don Pigozzi set out to try to answer the question of what it means for a logic to have algebraic semantics. In this seminal book they transformed the study of algebraic logic by giving a general framework for the study of logics by algebraic means. The Dutch mathematician W. J. Blok (1947-2003) received his doctorate from the University of Amsterdam in 1979 and was Professor of Mathematics at the University of Illinois, Chicago until his death in (...) an automobile accident. Don Pigozzi (1935- ) grew up in Oakland, California, received his doctorate from the University of California, Berkeley in 1970, and was Professor of Mathematics at Iowa State University until his retirement in 2002. The Advanced Reasoning Forum is pleased to make available in its Classic Reprints series this exact reproduction of the 1989 text, with a new errata sheet prepared by Don Pigozzi. (shrink)

The general aim of this book is to provide an elementary exposition of some basic concepts in terms of which both classical and non-dassicallogirs may be studied and appraised. Although quantificational logic is dealt with briefly in the last chapter, the discussion is chiefly concemed with propo gjtional cakuli. Still, the subject, as it stands today, cannot br covered in one book of reasonable length. Rather than to try to include in the volume as much as possible, I have put (...) emphasis on some selected topics. Even these could not be roverrd completely, but for each topic I have attempted to present a detailed and precise t’Xposition of several basic results including some which are non-trivial. The roots of some of the central ideas in the volume go back to J. Luka siewicz’s seminar on mathematicallogi. (shrink)

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...) directions we tried to give an overview of the most recent results and research trends, too. (shrink)

This book brings together several directions of work in model theory between the late 1950s and early 1980s.

This superb collection of papers focuses on a fundamental question in logic and computation: What is a logical system? With contributions from leading researchers--including Ian Hacking, Robert Kowalski, Jim Lambek, Neil Tennent, Arnon Avron, L. Farinas del Cerro, Kosta Dosen, and Solomon Feferman--the book presents a wide range of views on how to answer such a question, reflecting current, mainstream approaches to logic and its applications. Written to appeal to a diverse audience of readers, What is a Logical System? will (...) excite discussion among students, teachers, and researchers in mathematics, logic, computer science, philosophy, and linguistics. (shrink)

This is a review of those aspects of the theory of varieties of Boolean algebras with operators that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.It begins with a survey of the duality that exists between BAO's and relational structures, focusing on the notions of bounded morphisms, inner substructures, disjoint and bounded unions, and canonical extensions of structures that originate in the study of validity-preserving operations on Kripke frames. This duality is (...) then applied to polymodal propositional logics having finitary intensional connectives that generalise the Box and Diamond connectives of unary modal logic. Issues discussed include validity in canonical structures, completeness under the relational semantics, and characterisations of logics by elementary classes of structures and by finite structures.It turns out that a logic is strongly complete for the relational semantics if the variety of algebras it defines is complex, which means that every algebra in the variety is embeddable into a full powerset algebras that is also in the variety. A hitherto unpublished formulation and proof of this is given that applies to quasi-varieties. This is followed by an algebraic demonstration that the temporal logic of Dedekind complete linear orderings defines a complex variety, adapting Gabbay's model-theoretic proof that this logic is strongly complete. (shrink)