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# Logical Investigations, 2015, Vol. 21, No. 2.

CONTENTS

SYMBOLIC LOGIC

In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.
Keywords: combinatory logic, definitions, definitional embedding, predicate logic, computable functions

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.
Keywords: elementary theory, algebraic theory, groups, Abelian groups, definitions, definitional embedding, predicate logic

In the present paper we introduce and elaborate some basic elements of the methodological approach characterized by the application of topological means in the analysis and representation of entities through the actualization of some explicit differentiation scheme (scheme of discernment). The latter is understood as a set of rules — termed differentiation criteria — that individuate particular invariants (symmetries) of the entity under examination. We introduce the notion of differentiation invariants as symbolic representatives of the latter, and show that, in all instances of differentiation, topological structure can be induced on the set of such invariants. Given that, we proceed in describing the theoretical framework within which objects of logical theories and systems, as well as properties and interrelations between them, can be represented and treated by formal means of topology and characterized in terms of topological properties. Exposition of the proposed method is given through its application, resulting in topological representation of material implication and the rule of inference modus ponens.
Keywords: differentiation, differentiation invariants, Kronecker delta function, topological space, topological representation, topological invariants, material implication, modus
ponens

NON-CLASSICAL LOGIC

In 1979 D.E. Over proposed game theoretical semantics for first-degree entailment formulated by Anderson and Belnap. In order to extend this approach to include other systems of relevant logc (e.g., R) we have two promoting facts. Firstly, there is Routley-Meyer’s situational semantic for system R of relevant logic. Secondly, this semantics shows some resemblance with Wójcicki’s situational semantic of non-fregean logic for which the situational game semantics was developed by author exploiting essentially the notion of non-fregean games. In the paper an attempt is done to give a partial account of these results and some conception of situational games developed which laid down into foundation of the game theoretical semantics of relevant logic R.
Keywords: situational semantics, relevant logic, dialogue games, situational games

In 1995, Sette and Carnielli presented a calculus, I1, which is intended to be dual to the paraconsistent calculus P1. The duality between I1 and P1 is reflected in the fact that both calculi are maximal with respect to classical propositional logic and they behave in a special, non-classical way, but only at the level of variables. Although some references are given in the text, the authors do not explicitly define what they mean by ‘duality’ between the calculi. For instance, no definition of the translation function from the language of I1 into the language of P1 (or from P1 to I1) was provided (see [4], pp. 88–90) nor was it shown that the calculi were functionally equivalent (see [13], pp. 260–261). The purpose of this paper is to present a new axiomatization of I1 and briefly discuss some results concerning the issue of duality between the calculi.
Keywords: weakly-intuitionistic logic, paracomplete logic, I1, paraconsistent logic, Sette’s calculus, P1

The general aim of the present paper is to provide the analysis of the connection between proof-theoretical and functional properties of certain logical matrices. To be more precise, we consider the class of three-valued matrices that induce the classical consequence relation and show that their operations always constitute a subset of one of the maximal classes of functions, which preserve non-trivial equivalence relations. We use a matrix with the single designated value as a sample for in-depth analysis, and generalize the results to suit other cases. Furthermore, on the basis of obtained results we conclude the paper with methodological considerations concerning the nature and interpretation of the truth-values in logical matrices.
Keywords: classical propositional logic, three-valued logics, consequence relation, maximal classes of functions, logical matrices

In this work we offer table-analytical axiomatizations of a row of logics. These logics are such expansions of known paraconsistent and paracomplete logic Par from [1] which are paralogics, that is paraconsistent or/and paracomplete logics. According to [2] there are only four paralogics including logic Par. For each of these the paralogic we describe simply arranged table-analytical axiomatization convenient for the organization of search of the proof. Rules of a reduction in all these axiomatizations same, as well as the principles of creation of analytical tables. Calculations differ from each other only in definition of the closed set of the marked formulas. Table-analytical constructions are carried out in style of Fitting (see [4]). Following [4], we consider two markers for formulas. These markers — T and F. The main difference of a set of the rules of a reduction offered here from a set of the rules of a reduction used in [4] consists that we use along with usual rules of a reduction which delete separate logical connectives, rules of a reduction deleting the whole complexes of logical connectives. So, all logics are investigated here, language of each of which is the propositional language L defined below, and each of which includes known paranormal logic of Par and is paraconsistent or/and paracomplete logic. Our aim — for any such logic to describe an adequate table-analytical calculation convenient for search of a proof.
Keywords: paralogics, paranormal logic, paraconsistent logic, paracomplete logic, analytical table, marked formula, closed set, table-analytical axiomatization

PHILOSOPHICAL LOGIC

In this paper the possibility of interpreting imperatives as sentences that are used by rational agents to impel or motivate other rational agents to act in a desired way investigates by the author. We claim that such impelling strategies, that ground practical reasoning, can be pictured formally by means of stit-logic. We introduce the basic semantical ideas of stit-logic and discuss the most acceptable way of formal representation of imperative and its corresponding impelling effect. Special attention is paid to embedded stit-formulas. We demonstrate that such formulas cannot serve as an appropriate way of imperative formalization since an agent-addressee lacks the possibility of choice, and the agent-sender becomes “omnipotent” about the future course of agent-addressee’s action. We show that this unwanted situation can be eliminated, and basic imperative properties can still be expressed with embedded stit-formulas, if different kinds of indexes are used in semantic definitions for different kind of stit-operators. Finally we put an assertion that using such formulas, evaluated with respect to different kind of indexes, leaves no room for usual paradoxes of imperative logic.
Keywords: stit-logic, imperative, logic if imperatives, logic of action

In this paper truth in two game-theoretical approaches is considered, namely: in dialogue logic of Paul Lorenzen and Kuno Lorenz and game-theoretical semantics proposed by Jaakko Hintikka and developed by Gabriel Sandu. In the course of the article the principal features of the semantic conceptions of game-theoretical semantics and dialogue logic are revealed and compared. Thus, two concepts of truth are considered, that is, truth in game-theoretical semantics and truth in dialogue logic. In both cases truth is defined as an existence of a winning strategy for the player defending the formula. The connection between those two consists in a possibility to transform the winning strategy for the player in one system into the winning strategy for the corresponding player in the other one following the exact and finite algorithm. The result of the comparison makes it possible to get a certain understanding of the relation between model-theoretical and proof-theoretical approaches.
Keywords: dialogue logic, game-theoretical semantics, truth, intuitionistic dialogues with hypotheses, truth in a model, validity

HISTORY OF LOGIC

Lewis Carroll was an author of original syllogistic theory which is different from Traditional syllogistic. Carroll’s system contains term negation, so it made him possible to eliminate o-type propositions (SoP) treating them as a kind of i-type propositions (SiP′). We set out the following axiom schemes for Carroll’s syllogistic: (MaP & SaM) ⊃ SaP, SiP ⊃ PiS, SiP ⊃ SaS, SaP ⊃ SiP, SeP ≡ ⌉SiP, SaP ≡ (SeP′ & SiS), SaP′ ≡ (SeP & SiS), SiSS′iS′. We prove that this system embeds into the Predicate calculus by the following interpretation (equivalent to Carroll’s understanding) of categorical propositions: SaP → (∀x(Sx ⊃ Px)& ∃xSx), SiP → ∃x(Sx & Px), SeP → ∀x(Sx ⊃ ⌉Px).
Keywords: syllogistic, negative terms, categorical propositions, embedding function

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 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

With the development of symbolic logic in the foreign literature in the XX century, new approaches to Boethius treatise “On the hypothetical syllogism” and attempts to introduce a hypothetical system of Boethius as a variant of propositional logic appear. As a result of these attempts to modernize the teaching of Boethius there was a discussion about the question whether Boethius developed Stoic logic, which is considered to be the forerunner of propositional logic, or logic of peripatetics, built on different principles. The article reviews the approaches of K. Durr, W. Kneale and M. Kneale, Jonathan Barnes, Eleonore Stump, C. Martin, A. Speca, J. Marenbon, and others. The objective of the article is to compare the main points of view of this scientific discussion, and the author’s goal is to clarify the nature and specifics of the teachings of Boethius’ hypothetical syllogisms.
Keywords: logic of Boethius, Aristotle’s logic, the logic of the Stoics, the doctrine of hypothetical syllogism, propositional logic, the doctrine of terms

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