It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...) algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

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)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

In [4], Caicedo and Cignoli study compatible functions on Heyting algebras and the corresponding logical properties of connectives defined on intuitionistic propositional calculus. In this paper we study some aspects of compatible functions on the algebras associated to positive propositional calculus and successive extensions of it: intuitionistic calculus itself, the modal symmetric propositional calculus of Moisil and n-valued Łukasiewicz propositional calculus.