We deal with monotone structural deductive systems in an unspecified propositional language \ . These systems fall into several overlapping classes, forming a hierarchy. Along with well-known classes of deductive systems such as those of implicative, Fregean and equivalential systems, we consider new classes of unital and weakly implicative systems. The latter class is auxiliary, while the former is central in our discussion. Our analysis of unital systems leads to the concept of Lindenbaum–Tarski algebra which, under some natural conditions, is (...) a free algebra in a variety closely related to the deductive system in focus. (shrink)

Alexander Vladimirovich Kuznetsov, also known to the second generation of Soviet logicians as Sasha Kuznetsov, was born in Moscow on the 28 th of October, 1926. He lived a short yet fruitful life and died of cancer 1 in Chişinău, Moldova, on the 24 th of July, 1984.

On the 28th of October, 2006, Alexander Vladimirovich Kuznetsov, so is his full name, would have turned 80. Although belated, the editorial board of Logic and Logical Philosophy, we, the editors and contributors of the present issue, and other members of the logic community mark this event with the present issue. Most of those who contributed to it knew Kuznetsov in person and/or were influenced by him or by his ideas, which very often resided in somebody else’s papers or became (...) part of a scientific folklore. But, to be sure, there are even more mathematicians and logicians who have been witnesses to his scientific activity and/or have taken advantage of its results. (shrink)

ABSTRACT This is a continuing attempt in a series of papers [KM 93, Mur 93, Mur 96] to show how computer-represented knowledge can be arranged as elements of an effectively represented semantic (or algebraic) domain in the sense of [GS 90]. We present a direct deductive description of the domain, which was defined semantically in [KM 93], via the Scott's notion of information system. Also, the internal structure of the continuous ampliative operations coordinated with the domain's effective basis is established. (...) Though we always remain in the paradigm of the toleration of contradictory information described in [Bel 75, Bel 76], the approach in question could be extended to include domains for consistency knowledge bases. (shrink)

This paper is devoted to the concepts of consequence and rejection, formulated as operators on a nonempty set of sentences, which may initially be unstructured. One of the issues that we pay attention to is the “cyclicity” of these concepts when they are defined one through the other. In addition, we explore this cyclicity, when the set of all sentences acquires some structure, or we can assume some structure of sentences in the sense that the operation of substitution can be (...) applied to them. (shrink)

We discuss nonmonotonic reasoning in terms of consequence relations and corresponding operators. Based on the matrix consequence that gives the monotonic case, we define a restricted matrix consequence that illustrates the nonmonotonic case. The latter is a generalization of the relation of logical friendliness introduced by D. Makinson. We prove that any restricted single matrix consequence, although it may be nonmonotonic, is always weakly monotonic and, in the case of a finite matrix, the restricted matrix consequence is very strongly finitary. (...) Further, by modifying the definition of logical friendliness relation formulated specifically in a proof-theoretic manner, we show a possibility of obtaining other reflexive nonmonotonic consequence relations, for which a limited result towards finitariness is proved. This leads to numerous questions about nonmonotonic consequence relations in the segment between the monotonic consequence relation based on intuitionistic propositional logic and logical friendliness. (shrink)

We seek to trace how the assertion–rejection dichotomy arose, as well as in what forms it was realized in logical discourse. From this viewpoint, we observe the approaches to the concept of rejection by Łukasiewicz, Carnap, and Słupecki. We also explore the controversy between rejection and negation. Our main observation is that for a correct understanding of this dichotomy, it is necessary to distinguish between the object language and metalanguages of different levels.

We will outline the contributions of A.V. Kuznetsov to modal logic. In his research he focused mainly on semantic, i.e. algebraic, issues and lattices of extensions of particular modal logics, though his proof of the Full Conservativeness Theorem for the proof-intuitionistic logic KM (Theorem 17 below) is a gem of proof-theoretic art.