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)

Multiple -conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations. This is a sustained study of the subject and is certain to stimulate further research. Part I reworks (...) the fundamental ideas of logic to take account of multiple conclusions, and investigates the connections between multiple - and single - conclusion calculi. Part II draws on graph theory to discuss the form and validity of arguments independently of particular logical systems. Part III contrasts the multiple - and the single - conclusion treatment of one and the same subject, using many-valued logic as the example; and Part IV shows how the methods of 'natural deduction' can be matched by direct proofs using multiple conclusions. (shrink)

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)