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The history of Finnish-Soviet Logic Colloquium. The milestones of the object theory formation in the course of 19th century discussions in philosophy of logic are considered. The view, that the process mentioned was typical first of all for the Austrian tradition in logic and philosophy, is exposed. The hypothesis of the possible impact of that kind of approaches on the development of Frege’s logical ideas is examined. Ключевые слова: object theory, content and object of concept, school of Brentano, Frege. This paper sketches two approaches to the color exclusion problem provided by model-theoretical and gametheoretical semantics. The case study, modeling the experimentally confirmed perception of ‘forbidden’ (e.g., reddish green and bluish yellow) colors, is presented as neuropsychological evidence for game-theoretical semantics. Ключевые слова: invariance criterion, permutation invariance, color exclusion problem, binary colors, opponent-processing model, overdefined games, non-strictly competitive games, payoff independence. Gödelian sentences are self-referential first-order sentences in the language of arithmetics. Perhaps the most celebrated one is the sentence which asserts its own unprovability. It is well known that this sentence is neither provable nor refutable in PA (Peano Arithmetics). Some logicians and philosophers have complained that such a sentence is difficult to grasp given its ‘meta-theoretical’ content and they started to look for undecidable arithmetical statements which have a combinatorial content. One such sentence is a variant of Ramsey’s sentence: the Paris-Harrington theorem asserts its undecidability. In the present paper I shall argue that such a sentence is not first-order expressible and thereby it does not provide the desired example of a combinatorial, undecidable arithmetical sentence. Instead I shall argue that it is expressible in Independence-friendly (IF) logic. Ключевые слова: Peano arithmetics, Gödel’s incompleteness theorem, undecidability, Ramsey theorem, IF logic. In this paper von Wright’s truth-logic T′′ is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap’s four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright–Segerberg’s tense logic W, and so on. Ключевые слова: Wright’s truth-logic, De Morgan four-valued logic, twin truth operators, tetravalent modal logic TML, truth logic Tr, modal logic V2, von Wright–Segerberg’s tense logic. This paper traces the development of history of logic in Ukraine in the 19th century and early 20th century. The author particularly discusses and compares the logical concepts of representatives of Kyiv philosophies, who made their contribution to the development of logic as a science and academic discipline. Some of them had sunk into oblivion for a long time and their names are still unknown in the logic community. Ключевые слова: logic, history of logic in Ukraine, Kyiv Theological Academy, St. Vladimir University of Kyiv. We consider the logic QCTL, a first-order extension of CTL defined as a logic of Kripke frames for CTL. We study the question about recursive enumerability of its fragments specified by a set of temporal modalities we use. Then we discuss some questions concerned axiomatizability and Kripke completeness. Ключевые слова: non-classical logic, temporal logic, branching time logic, first-order logic, recursive enumerability, Kripke completeness. In 1926, Ernst Mally, an Austrian logician, has introduced a system of deontic logic in which he has proposed three fundamental distinctions which proved to be important in the context of the further development of the logic of norms. It is argued that in his philosophical considerations Mally has introduced a number of important distinctions concerning the very concept of norm, but by getting them confused in introducing the subsequent formalisms he failed to formally preserve them. In some of his philosophically made distinctions Mally apparently foresaw contemporary trends in logic of norms. To some extent this particular feature of Mally’s system open wide opportunities to reconstruct –– with the corresponding renovations — his illformed Deontik into many nowadays known systems of logic of norms and thus provides a fertile ground for this kind of research. Ключевые слова: deontic logic, Mally, agency, ought, obligation. In this paper we discuss a question about the trends in non-classical logic that were exactly anticipated by Nikolai Vasiliev. We show the influence of Vasiliev’s Imaginary logic on paraconsistent logic. Metatheoretical relations between Vasiliev’s logical systems and many-valued predicate logics are established. We also make clear that Vasiliev has developed a sketch of original system of intensional logic and expressed certain ideas of modal and temporal logics. Ключевые слова: Nikolai Vasiliev, imaginary logic, syllogistic, para- consistent logic, many-valued logic, intensional logic, modal logic, temporal logic. The ideas of Russian logician Nikolai Vasiliev concerning the status of the law of contradiction are discussed in this article. The arguments presented in his article ‘Logic and meta-logic’ are deeply explored bringing to light the weakness of his philosophical theory. His ‘imaginary’ logic is a system that describes not the system of the laws of reason, but relations in which objects of some ontology stand to each other. Comparing the fundamental idea of Vasiliev to the classical concepts of reason brings us to a better understanding of the fact that philosophical intention of Vasiliev has been left unfulfilled. Ключевые слова: laws of reason, laws of logic, Nikolai Vasiliev’s logic. In the paper technical systems with counters are considered as logical models. The questions of formalization in temporal logic and automatic analysis via computational tree transformations are discussed. Ключевые слова: temporal logic, verification, formalization, proof constructing. Abstract forms of Kolmogoroff’s complexity, Chaitin and Gödel’s theorems are stated. They are used to analyze numerous methodological issues: Kant’s Third antinomy, Parkinson’s law of committee, cooperative creative activity, multilanguage programming, benevolence to other’s views, dilemma of deism–atheism. Ключевые слова: Kolmogoroff complexity, Chaitin theorem, Gödel theorem, Kant antinomy, Parkinson law. Science is highly successful in making empirical predictions and guiding our practical actions. This paper defends the so-called ‘ultimate argument for scientific realism’ by claiming that this empirical and pragmatic success of scientific theories would be a miracle unless they are true or truthlike. This argument is abductive in Charles Peirce’s sense, as it appeals to inference to the best explanation. Ключевые слова: abduction, explanation, fallibilism, inference to the best explanation, scientific realism, truthlikeness. This short paper presents a new domain of logical investigations. Ключевые слова: paralogic, paracomplete logic, paraconsistent logic, paranormal logic, intuitionistic propositional logic. In this paper I will compare several solutions to a well known puzzle: Monty Hall. This will enable us to illustrate varitous styles of logical reasoning, and in particular to compare dynamic logic with game-theoretical approaches. Ключевые слова: game-theoretical semantics, IF logic, Monty Hall, dynamic epistemic logic, conditional probabilities. The article offers a look at the combinatorial logic as the logic of signs operating in the most general sense. For this it is proposed slightly reformulate it in terms of introducing and replacement of the definitions. Ключевые слова: combinatory logic, semiotics, definition, logic foundations. The paper introduces a non-standard analysis of intensional contexts on the ground of generalized approach to semantics construction. The principles of building such kind semantics are considered. As far as I can see it is an idea on domains and anti-domains that lays in the ground a semantics of intensional contexts. Intensional contexts differ from extensional by ascription of specific values to intensional predicates (operators) and, what is more important, by a way of their combination with arguments. Thus constructing operations play the leading role in proposed analysis. The peculiarities of IPL: any expression including intensional predicates and operators has an intension as well as an extension. Ключевые слова: generalized semantics, domains and anti-domains, propositional concept, operation of abstraction. One of the basic question we can ask about truth in a formal setting is what, if anything, we gain when we have a truth predicate at disposal. For example, does the expressive power of a language change or does the proof strength of a theory increase? Satisfaction classes are often described as complicated model theoretic constructions unable to give useful information toward the notion of truth from a general point of view. Their import is narrowed to a dimension of pure technical utility and curiosity. Here I offer an application of satisfaction classes in order to show that they can have a relevant role in confronting proof theoretical equivalent theories of truth. Ключевые слова: truth, satisfaction classes, axiomatic theories of truth, expansions, conservativity. In this paper I propose a formalization of protoentailment relation introduced by V. Shalak by means of RS logic. The first section clarifies the idea and formal developments of RS logic, which is the logic of Rational Subject. In the second section I will very briefly introduce the conception of proto-entailment as it was promoted in Shalak’s writings. The third section contains the formal account for proto-entailment and axiotimatization of resulting logic. Ключевые слова: proto-entailment, logic of rational subject, generalized truth-values.
Addition In this paper the procedure is presented that allows to determine in finite number of steps if consequence relations in two finite-valued logical matrices for propostional language L are equal. Ключевые слова: product of logical matrices, consequence relation, equality of matrices. Quasi-matrix logic is based on the generalization of the principles of classical logic: bivalency (a proposition take values from the domain {t (truth); f (falsity)}); consistency (a proposition can not take on both values); excluded middle (a proposition necessarily takes some of these values); identity (in a complex proposition, a system of propositions, an argument the same proposition takes the same value from domain {t; f}); matrix principle — logical connectives are defined by matrices. As a result of our generalization, we obtain quasi-matrix logic principles: the principle of four-valency (a proposition takes values from domain {tn; tc; fc; fi} or three-valency (a proposition takes values from domain {n; c; i}); consistency: a proposition can not take more than one value from {tn; tc; fc; fi} or from {n; c; i}; the principle of excluded fifth or fourth; identity (in a complex proposition, a system of propositions, an argument the same proposition takes the same value from domain {tn; tc; fc; fi} or domain {n; c; i}); the quasi-matrix principle (logical terms are interpreted as quasifunctions). Quasi-matrix logic is a logic of factual modalities. Ключевые слова: quasi-matrix logic, semantic completeness, decision problem, Kalmar’s method. This article deals with the problem of translations. It covers the history of translation in linguistics and analyzes peculiarities and role of translation in logic. Moreover, the article contains typical examples of embedding operations in terms of different logical theories. Ключевые слова: logic, translation, embedding, embedment, operation, language, calculi, theory. For an arbitrary fixed element β in {1; 2; 3; ...; ω} both a sequent calculus and a natural deduction calculus which axiomatise simple paracomplete logic I2;β are built. Additionally, a valuation semantic which is adequate to logic I2;β is constructed. For an arbitrary fixed element γ in {1; 2; 3;...} a cortege semantic which is adequate to logic I2;γ is described. A number of results obtainable with the axiomatisations and semantics in question are formulated. Ключевые слова: paracomplete logic, paraconsistent logic, cortege semantics, valuation semantics, sequent calculus, natural deduction calculus. This paper proves that sets of closed functional classes in 3-valued logics of Bochvar B3 and Hallden H3 contains a continuum of different closed classes. It is also proven that both of these logics contain a closed functional class which has no basis. Ключевые слова: Bochvar’s logic, Hallden’s logic, closed class, continuum, cardinality. In this paper implicative fragments of natural threevalued logic are investigated. It is proved that some fragments are equivalent by set of tautologies to implicative fragment of classical logic. It is also shown that some natural three-valued logics verify all tautologies of classical propositional logic. Ключевые слова: three-valued logis, natural implication, classical logic, set of tautologies. Recently some elaborations were made concerning the game theoretic semantic of Łℵ0 and its extension. In the paper this kind of semantics is developed for Dishkant’s quantum modal logic ŁQ which is also, in fact, the specific extension of Łℵ0. As a starting point some game theoretic interpretation for the SŁ system (extending both Łukasiewicz logic Łℵ0 and modal logic S5) was exploited which has been proposed in 2006 by C. Fermuller and R. Kosik. They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of Łℵ0 in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. Ключевые слова: Łukasiewicz’s logic, quantum loigic, dialogue games, risk value.
Наши авторы: Черноскутов Юрий Юрьевич — к.филос.н., доцент, кафедра логики, Санкт-Петербургский Государственный Университет. Подписной индекс в каталоге "Пресса России" 42046. 2-е полугодие 2012 г.
Публикуемые материалы прошли процедуру рецензирования. Журнал включен в новый перечень периодических изданий, рекомендованных Высшей аттестационной комиссией РФ для публикации материалов кандидатских и докторских диссертационных исследований в области философии (с 1 января 2010 г.).
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