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  Logical Investigations, 2017, Vol. 23, No. 2.
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Logical Investigations, 2017, Vol. 23, No. 2.





I.A. Gorbunov. Deductive Logics and Their Relation to Intuitionistic Logic

R. Wojcicki introduced the notion of well-defined logic [5]. A propositional logic is called well-determined if it satisfies conjunction property and weak deduction theorem. The weak deduction theorem has the following form: ⊢ ⇔ ⊢ → . Well-determined logics are interested because their logical consequence may be\certainly represented by means of the logic.

We consider well-determined logics for which the following deductive theorem holds: for any set of formulas X and any formulas and it is true that X; ⊢ ⇔ X ⊢ → . Logics with this property we call deductive. We call a set of formulas Lstrongly deductive if there exists a deductive logic C such that C(∅) = L.

In this paper we introduce an operation of adding of consequences under a theory and study some its properties. We prove that any theory under a deductive logic is closed under modus ponens. The notion of minimal deductive logic is introduced. The main results are a criterion of strong deductivity for a set of formulas and the proof that the set of tautologies of minimal deductive logic coincides with the conjunctive and implicative fragment of intuitionistic logic.

Keywords: deduction theorem, deductive propositional systems, strongly-deductive set of sentences, minimal deductive logic, intuitionistic logic

DOI: 10.21146/2074-1472-2017-23-2-9-24

V.M. Popov. To the Problem of Characterization of Logic of the Vasiliev Type: on Tabularity Ihx;yi(x; y 2{0, 1, 2,. . . } and x < y). Part II

In this article, continuing the work carried out in [1], the problem of tabularity of the I-logics of the Vasiliev type (propositional logic is called tabular if it has a finite characteristic matrix). The main result obtained in this article: for any non-negative integers x and y, the first of which is less than the second, the logic I⟨x;y⟩ is tabular (the class of all such logics is an infinite subclass of the class of all I-logics of the Vasiliev type). The proposed study is based on the use of the results obtained in [1], and on the use of the authors’ “cortege semantics”. To achieve the above main result, we show how on arbitrary nonnegative integer numbers m and n, satisfying the inequalitym < n, is constructed logic matrix M(m; n), which is the finite characteristic matrix of logic I⟨x;y⟩. Since the carrier of the logical matrix M(m; n) is some set of 0-1-corteges, the semantics based on this logical matrix is naturally called the cortege semantics. Important note: the article is published in two parts, which is due solely to external factors for this article. Before you, the second (final) part of the study, the first part of which was published in the first issue of “Logical Investigations” for 2017.

Keywords: I-logic I⟨m;n⟩(m; n 2 f0; 1; 2; : : :g and m < n), the two-valued semantics of the I-logic I⟨m;n⟩(m; n 2 f0; 1; 2; : : :g and m < n), the cortege semantics of the I-logic I⟨m;n⟩(m; n 2 f0; 1; 2; : : :g and m < n)

DOI: 10.21146/2074-1472-2017-23-2-25-59


M.N. Rybakov. Undecidability of Modal Logics of Unary Predicate

We consider first-order modal logics with unary predicate letters only. We show that any sublogic of QS5, QGLLin, or QGrz.3 is undecidable in the language with just one unary predicate letter (with or without Barcan formula). We also show that logics of finite Kripke models (with expanding or constant domains) for QK, QT, QD, QK4, QS4, QS5, QGL, QGrz, and some others are not recursively enumerable in the language with one unary letter. Nevertheless tabular logics and mlogics of Kripke frames with restrictions on the number of worlds accessible from any world are decidable in the language with infinitely many unary predicate letters.

Keywords: modal logic, first-order logic, decidability

DOI: 10.21146/2074-1472-2017-23-2-60-75


V.L. Vasyukov. Potoses: Categorical Paraconsistent Universum for Paraconsistent Logic and Mathematics

It is well-known that the concept of da Costa algebra [3] reects most of the logical properties of paraconsistent propositional calculi Cn, 1  n  ! introduced by N.C.A. da Costa. In [10] the construction of topos of functors from a small category to the category of sets was proposed which allows to yield the categorical semantics for da Costa's paraconsistent logic. Another categorical semantics for Cn would be obtained by introducing the concept of potos { the categorical counterpart of da Costa algebra (the name \potos" is borrowed from W.Carnielli's story of the idea of such kind of categories)

Keywords: paraconsistent logic, categorical semantics, potos, paraconsistent set the-ory, da Costa algebra

DOI: 10.21146/2074-1472-2017-23-2-76-95


V.I. Shalack. Some Remarks on A. Tamminga’s Paper “Correspondence Analysis for Strong Three-valued Logic”

In this note we present two remarks to the A. Tamminga’s paper. The first remark relates to incorrect Definition 1, and the second remark relates to the main theorem of the paper. We propose the necessary corrections.

Keywords: Tamminga, many-valued logic, correspondence analysis

DOI: 10.21146/2074-1472-2017-23-2-96-97




A.S. Karpenko Counterfactual Thinking

Counterfactual thinking is thinking about a past that did not happen. This often takes place in “if only...” situations, when we wish something had or had not happened. Counterfactual reasoning is basic to human cognition and universal in occurrence. Currently, the principles of couterfactual thinking and its results appear in research in various disciplines, such as logic, philosophy, psychology, cognitive processes, sociology, economics, history, political science etc. The special thing about the counterfactuals is that they are the mental imitations of the different variants of what could have happened in the past. It is observed that two uniquely human characteristics — counterfactual thinking (imagining alternatives to the past) and the fundamental drive to create meaning in life — are causally related. Gradually, we have come to understand that we are dealing with the phenomenon of exceptional importance.

Keywords: counterfactuals, possible world semantics, counterfactual thinking, metaphysical modalities, alternative reality

DOI: 10.21146/2074-1472-2017-23-2-98-122

A.A. Pechenkin. Quantum Logic and Probability Theory

The paper provides the review of the texts on quantum logic, the texts which are directly connected with the mathematical foundations of quantum mechanics. These are texts which discuss the theory of quantum probability. The development of the mathematical scheme of quantum mechanics is discussed along the following line: P.A.M. Dirack’s 1927 “The Principles of Quantum Mechanics”, J. von Neumann’s 1932 “Mathematische Grundlagen der Quantenmechanik”, G.Birkgoff-I.von Neumann’s 1936 “The logic of quantum mechanics”. It is shown that the further development of the mathematical foundations of quantum mechanics resulted in the construction of quantum theory of probability, the theory generalizing A.N.Kolmogorov’s classicalprobability and critically improving von Neumann’s 1932 mathematical scheme.

Keywords: lattice, boolean algebra, -algebra, frequency understanding of probability, mathematical justification

DOI: 10.21146/2074-1472-2017-23-2-123-139




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