The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

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)

This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass–the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for (...) both classes, demonstrating their usefulness and importance. (shrink)

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)