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  Logical Investigations, 2016, Vol. 22, No. 1.
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Logical Investigations, 2016, Vol. 22, No. 1.

TABLE OF CONTENTS 

 

NON-CLASSICAL LOGIC

In this paper Sobochiński’s four-valued modal logic V2 (extension of S5) is considered. The emergence of that logic, some its interesting properties and different equivalent formulations are presented. Its algebraic models are of particular interest: as the extension of De Morgan algebra by boolean negation ¬ and as the extension of Boolean algebra by the endomorphism g, which is interpreted then as the propositional truth operation T. The logic corresponding to the last case is denoted by Tr. The attention is paid to the application of Tr in Fitting’s theory of truth. The axiomatization of Tr in language (→; ¬; T) is considered. The completeness of logic Tr is proved with use of Sahlqvist’s powerful theorem, which gives the sufficient condition of Kripke completeness for normal modal logics. Algebraic completeness of logic Tr is also proved.

Keywords: modal logic V2, De Morgan algebra, boolean algebra, endomorphismus, logic Tr, Fitting’s theory of truth, Kripke completeness, Sahlqvist’s theorem, algebraic completeness

I study here logics of Vasiliev‘s type were found in the process of explication of some of the ideas of the Russian logician and philosopher Nikolai Alexandrovich Vasiliev underlying his “imaginary logic”. This article demonstrates how to construct a simple and convenient search for proof of sequent axiomatization of I-logics of Vasiliev‘s type and how to build intuitively clear two-valued semantics, adequate I-logics of Vasiliev‘s type. In the present paper are defined by I-logics of Vasiliev‘s type, built their sequent axiomatization, provides the necessary semantic definitions and prove a theorem about the justification of HIα,β-proofs (theorem 5) and theorem about the completeness of HIα,β-proofs (theorem 6). The work concludes with a number of corollaries of these theorems and the announcement of a solution to the problem of existence of finite characteristic matrix for the logics of Vasiliev‘s type.

Keywords: I-logic of Vasiliev’s type, sequent axiomatization of I-logics of Vasiliev’s type, semantics of I-logics of Vasiliev’s type

In the paper we state a non-standard semantics for positive syllogistic language, where the validity of atomic formulas (the forms of categorical propositions) is defined in terms of relevant entailment. This idea is realized within the bounds of V.I. Shalack’s approach
to the construction of syllogistic semantics [3]: the formulas of propositional logic are assigned to the subjects and the predicates as their meanings, and the validity definitions for syllogistic formulas base on the relation of classical deducibility. In the paper we change this relation for the entailment relation of relevant system FDE. Interpretative function δ assigns a formula of the propositional language with primitive connectives ¬, ∧ and ∨, for every universal term. We postulate the following validity conditions for syllogistic formulas under the interpretation : SaP is valid iff δ(S) entails δ(P) in FDE; SeP is valid iff δ(S) entails ¬δ(P); SiP is valid iff δ(S) doesn’t entail ¬δ(P); SoP is valid iff δ(S) doesn’t entail δ(P); for complex formulas they are usual. Syllogistic calculus formalized the set of logical valid formulas, contains the following postulates: classical tautologies, axiom schemes (MaP ∧ SaM) ⊃ SaP, (MeP ∧ SaM) ⊃ SeP , SeP ⊃ PeS, SaS, SiP ≡ ¬SeP , SoP ≡ ¬SaP, and the only rule modus ponens. The soundness and completeness theorems are proved.

Keywords: syllogistic, categorical propositions, relevant entailment, semantics, calculus, soundness theorem, completeness theorem

The history of the notion of many-valuedness is considered in unified manner as the history of negation. The main attention is being paid to N.A. Vasiliev logic, its connections with indian logic of syadvada, with Łukasiewicz logics and logic of D. Bochvar. Many-valued nature of Vasiliev logic and it’s formal model in topoi are cleared. A new notion of negation “from some viewpoint” is introduced in topoi. Then Vasiliev set of statement types appears as the set of elements forming a truth values distributive lattice of a topos. These elements translates one into the other by negation “from correspondent viewpoint” and Vasiliev negation understanding corresponds to lattice negation (pseudo-compliment). Vasiliev understanding of an excluded n-th rule and paraconsistency is discussed. Vasiliev considered his excluded n-th rule as disjunction of lattice forming elements, as disjunction of different negated “from some viewpoint” statement construction ways, and consistent rule as conjunction of a statement and its negation “from some viewpoint”. The conditions of these rules modeling in categories are specified.

Keywords: Vasiliev logic, many–valued logic, categorical semantics

In this paper natural deduction systems for four-valued logic FDE (first degree entailment) and its extensions are constructed. At that B. Kooi and A. Tamminga’s method of correspondence analysis is used. All possible four-valued unary (⋆) and binary (◦) propositional connectives which could be added to FDE are considered. Then FDE is extended by Boolean negation (∼) and every entry (line) of truth tables for ⋆ and ◦ is characterized by inference scheme. By adding all inference schemes characterizing truth tables for ⋆ and ◦ as rules of inference to the natural deduction for FDE, natural deduction for extension of FDE is obtained. In addition, applying of correspondence analysis gives axiomatizations of implicative extensions of FDE including BN4 and some extensions by classical implications.

Keywords: correspondence analysis, natural deduction, rst degree entailment, Belnap-Dunn logic, four-valued logic, implicative extensions, classical implication

 

SYMBOLIC LOGIC

In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don’t impose restrictions on the size of their models, can be reduced to pure logic.

Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others.

It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 ≠ x′.

Keywords: definition, definability, predicate calculus, theory, logicism

In this paper I investigate the phenomenological approach to foundations of mathematics. Phenomenological reflection plays the certain role in extension of mathematical knowledge by clarification of meanings. The phenomenological technique pays our attention to our own acts in the use of the abstract concepts. Mathematical constructions must not be considered as passive objects, but as categories are given in theoretical acts, in categorical experiences and in our senses. Phenomenology moves like a category theory from formal components of knowledge to the dynamics of constitutive process.

Keywords: Phenomenology of mathematics, infinite structures, category theory, abstract objects, movement, visibility

 

HISTORY OF LOGIC

In the article on the basis of archival documents describes the initial stage of the Renaissance of teaching and learning logic in the USSR in the first half of the 1940s. Considered: a conversation of Director of the Institute of Philosophy P.F. Yudin with Stalin about creating logic tutorial in 1941, the course and outcome of the discussion of logic textbooks by V.F. Asmus and E.J. Colman at the Institute of Philosophy of the Russian Academy of Sciences in 1943, the discussion on the re-release of logic tutorial by G.I. Chelpanov in the Institute of Philosophy of the Russian Academy of Sciences in 1943, job training Courses for teachers of logic in the universities and schools of higher education of the USSR in 1946.

Keywords: logic, Soviet philosophy, Institute of Philosophy, Stalinism

 

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