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# Logical Investigations, 2017, Vol. 23, No. 1.

NON-CLASSICAL LOGIC

L.Yu. Devyatkin. Non-classical Modifications of Many-valued Matrices of the Classical Propositional Logic. Part II

This paper constitutes the second part of the duology dedicated to many-valued matrices of the classical propositional logic regarded as a tool of construction and analysis of non-classical logics. There are many pairs of three-valued matrices which differ only in classes of designated values present in the literature. However, the majority of them induce non-classical consequence relations with respect to either one and two designated values. At the same time, there are matrices of non-classical logics, obtained from matrices of the classical logic by contraction or expansion of the class of designated values. The principal part of the paper is devoted to the two classes of matrices. The first class consists of matrices which would induce the classical consequence given D = {1, 2}, but are regarded as having D = {1, 2}. The second class is obtained by assuming D = {1, 2} in matrices inducing the classical consequence for D = {1, 2}. For the matrices in question I prove the maximality (in the strong sense) of paraconsistency or paracompleteness of logics they define, as well as analogues of Glivenko or Dual-Glivenko theorems. The matrices in classes under consideration form lattices with respect to functional embeddability relation. Some matrices obtained from matrices of the classical logic through modifications of their classes of designated values are shown to have equivalent formulations as functional extensions of matrices of the classical logic.

Keywords: many-valued logics, logical matrices, paraconsistency, paracompleteness

DOI: 10.21146/2074-1472-2017-23-1-11-47

E.F. Karavaev. One Way to Determine the Intervals in Hybrid Temporal Logic

This presentation discusses the opportunity of improvement of technical means of hybrid temporal logic through the introduction of time intervals. In the procedure of constructing intervals the author of the presentation follows ideas and development expressed and carried out by A.A. Markov in his article published in 1932. So the ‘Priorean paradigm’ of understanding of the logic (temporal qualification of judgments and the idea of hybrid logic) is complemented by a building of time metric based on the relation ‘earlier than’. It seems that the described improvement of the machinery of temporal logic allows, in particular, to perfect the approaches to the modelling of planning and strategic management.

Keywords: temporal logic, hybrid logic, satisfaction judgment, nominal-interval, discreteness of time, tree temporal structure, strategy, planning, management

DOI: 10.21146/2074-1472-2017-23-1-48-56

V.M. Popov. To the Problem of Characterization of Logic of the Vasiliev Type: on Tabularity I‹m, n› (x, y ∈ {0, 1, 2,… } and x < y). Part I

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‹m, n› 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 inequality m < n, is constructed logic matrix M(m, n), which is the finite characteristic matrix of logic I‹m, n›. 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. Before you the first part of the article, the second part of the article is planned to be published in the second issue of “Logical Investigations” for 2017.

Keywords: I-logic I‹m, n› (m, n ∈ {0, 1, 2,… } and m < n), the two-valued semantics of the I-logic I‹m, n› (m, n ∈ {0, 1, 2,… } and m < n), the cortege semantics of the I-logic I‹m, n› (m, n ∈ {0, 1, 2,… } and m < n)

DOI: 10.21146/2074-1472-2017-23-1-57-82

V.O. Shangin. A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics I‹α, β›

In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics I‹α, β›, α, β ∈ {0, 1, 2, 3,.. ω}, such that I0, 0 is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbert-style calculus H I‹α, β›, α, β ∈ {0, 1, 2, 3,.. ω}, for each logic in question, in order to construct examples of effects of Glivenko theorem’s generalization. Here we, first, propose a subordinated natural deduction system N I‹α, β›, α, β ∈ {0, 1, 2, 3,.. ω}, for each logic in question, with a precise definition of a N I‹α, β›-inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each α, β ∈ {0, 1, 2, 3,.. ω}, a Hilbert-style calculus H I‹α, β› and a natural deduction system N I‹α, β› are equipollent, that is, a formula A is provable in H I‹α, β› iff A is provable in N I‹α, β›.

Keywords: precise definition of inference, indirect rule, implication introduction rule, natural deduction, quasi-elemental formula, subordinated sequence

DOI: 10.21146/2074-1472-2017-23-1-83-104

PHILOSOPHICAL LOGIC

E. D. Smirnova The Nature of Logical Knowledge and Foundations of Logical Systems

In this paper, I address a wide range of problems related to the nature of apodictic knowledge and foundations of logic. In so doing, a primary focus is on ideal entities and connections, forming a ground for logical systems. The core component of the conception presented is a so called ‘generalized approach to semantics formation’, which presupposes that (i) every statement can be associated with a pair of sets representing ‘extension’ and ‘anti-extension’ of a statement, correspondingly; (ii) these sets consists of differently interpreted possible worlds. A variation of requirements for relations between these sets opens possibility to define logical consequence in a different way that, in turn, results in a variety of logical systems. An important consequence of a generalized approach is an identification of two types of epistemological presuppositions: those connected with conceptual apparatus of cognitive agent, and ontological commitments. The final section contains a discussion of perspective for logic and possible transformations of its subject-matter and methods.

Keywords: logical semantics, generalized approach, nature of logical knowledge, foundations of logic

DOI: 10.21146/2074-1472-2017-23-1-105-120

V.I. Shalack. Analytical Approach to Problem Solving

The work is devoted to the logical analysis of the problem solving. Typically, in each task we highlight the conditions and goals that we have to find or build. In this case the solution of the problem is seen as a kind of deduction from goals to the conditions. This representation of problems and their solutions is too narrow. In actual practice, a task or goal is often formulated in quite general terms as a wish. For example, the task to build a railway between the two cities. Sufficient conditions for the solution of this problem are initially unclear and should be found. For this kind of problems their solution can be represented as a gradual refinement of goals and their reduction to a simpler sub-goals. The methods by which we produce clarification of goals, we took from the theory of definitions. In this paper we construct a calculus in the form of analytical tables, which allows us to represent the whole process algorithmically.

Keywords: Problem solving, logical reduction, analytical tables, theory of definitions

DOI: 10.21146/2074-1472-2017-23-1-121-139

HISTORY OF LOGIC

S. Garin. Minimal Categorical System and Predication Theory In Porphyry

The article considers some problematic aspects of Porphyry’s typology of Aristotle’s categories and the theory of predication. Minimal (ἐλαχίστος) class of categories in Porphyry is revealed. The work has shed some light on the opposition between explanation and description (ἐξηγητικός / ὑπογραϕικός) within the framework of ancient categorical logic. A fourfold pattern of predication theory in Porphyry is described. The study aims to illuminate the development of Porphyry’s predication theory towards the archaic doctrine of quantifiers. Particular attention is paid to Porphyry’s account of semantic relation between sets. The paper represents Porphyry’s nine kinds of class / item relationships. The article focuses on the awakening of academic interest to the logical heritage of Porphyry.

Keywords: history of logic, categories, predication, Porphyry, Aristotle

DOI: 10.21146/2074-1472-2017-23-1-140-150

A.M. Pavlova. What Hamblin’s Formal Dialectic Tells About the Medieval Logical Disputation

In this paper we reconstruct a famous Severin Boethius’s reasoning according to the idea of the medieval obligationes disputation mainly focusing on the formalizations proposed by Ch. Hamblin. We use two different formalizations of the disputation: first with the help of Ch. Hamblin’s approach specially designed to formalize such logical debates; second, on the basis of his formal dialectics. The two formalizations are used to analyze the logical properties of the rules of the medieval logical disputation and that of their formal dialectic’s counterparts. Our aim is to to show that Hamblin’s formal dialectic is a communicative protocol for rational agents whose structural rules may differ, thus, varying its normative character. By means of comparing Hamblin’s reconstructions with the one proposed by C. Dutilh-Novaes we are able to justify the following conclusions: (1) the formalization suggested by Hamblin fails to reconstruct the full picture of the disputation because it lacks in some the details of it; (2) Hamblin’s formal dialectic and the medieval logical disputation are based on different logical theories; (3) medieval logical disputation, represented by the formalization of C. Dutilh-Novaes, and the two ones of Hamblin encode different types of cognitive agents.

Keywords: formal dialectics, game, medieval disputes of obligationes, dialogue logic, argumentation, Hamblin, belief revision

DOI: 10.21146/2074-1472-2017-23-1-151-176

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