A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

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)

A syntactic derivation of Cornish identity from the axioms of HBCK is presented which amounts to a syntactic proof of Wronski's conjecture that naturally ordered BCK-algebras form a variety.

This is an abstract of the paper presented at the seminar held by prof. A. Wronski at the Jagiellonian University. It was announced in [1] that T. Traczyk showed that any commuta- tive BCK-algebra such that any two elements have an upper bound is a distributive lattice. In this paper we give a generali.

It was shown in [1] that if X is a BCK-algebra then is a poset, and moreover if X is a commutative BCK-algebra, i.e. x = y holds in X, then is a lower semilattice. In this paper we consider properties of certain ideals in these BCK-algebras which are lower semilattices as referred to [1] and [2]. The following example shows that the class of BCK-algebras considered here is considerably wider than the class of commulative BCK-algebras.