Russia’s government was suspicious toward philosophy and at the mid of XIX century ousted philosophy from University curriculum. Nevertheless ban of philosophy have no impact on logic which was taught by philosophers as well. Study of logic continued in all Russian Universities though the program was compiled by Moscow Spiritual academy and approved by Holy Synod. We discuss the feature of this program and stress crucial role of Orthodox Church in preserving logical traditions in Russia during XIX — turn of XX centuries.
Keywords:formal (traditional) logic, Orthodox сhurch, logical education.
There are shown logic and geometrical sense of infinitely large and infinitesimal values. For the sample letters of a genetic code are taken. Symbols of universal language are entered as signs on a maximum and a minimum of hereditary variability. In the Matrix of Complementary based on non-Kronecker (left) tensor square we have blocks by the second letters. The matrix consists of 4 colours of psychotypes, each of which is in a miniature, operating as the fractal multiplier. Universal symbols show isomorphism of genetic code tables and Jung’s mental types. The non-numeric positional principle in humanitarian area offers not smaller advantages, than positional numerical arithmetics. In universal language the functional words are made of one-letter operators to realize the macrolevel of a genetic code and emphasizes the necessity to turn from the molecular level to the anthropomorphic level — from molecules to characters.
Keywords: infinitely large and infinitesimal Complementary genetic code, positional principle.
The paper is devoted to the logical ideas and the biography of the prominent Russian thinker Alexander Ivanovich Vvedenskiy (1856–1925).
Keywords: A.I. Vvedenskiy, history of logic, philosophical logic
Logical pluralism prove to be much more intruiging phenomenon if we envisage its impact on elementary logical theories. Breaking the tenet of the unique (namely, classical) logical basis for those we find ourself in the realm of non-classical elementary logical theories based on the various non-classical logics. It is expecially important if we take into account that such theories underlie non-classical mathematics according to the apt slogan “there are as many mathematics as logics” — suffish it to recall relevant arithmetic, quantum set theory, fuzzy set theory, paraconsistent mathematics etc. In the paper non-classical axiomatic category theories are approached which are based on some non-classical categorical constructions.
Keywords: pluralism, non-classical categories, axiomatic category theory, category theory of categories.
In the theological tradition antinomy, taken as self-contradiction, connection to the thesis and antithesis into a single entity, is understood as the best way to express the integrity of Divine Truth. This conclusion is a common place or antinomic strategies of different religious traditions. Does the concept of logical contradiction express the meaning of antinomy? Analysis of Florensky’s texts leads to the idea ofdistiguishing between aspects of logic, reasoning and rhetoric. This distinction helps to investigate the religious texts employing on scientific methodology.
Keywords: Florensky, antinomy, logical contradicion, theology, logic, reasoning, rhetoric.
Relay and switching circuits is the first model of the realization of the logical operations in modern technology. The concept of the equivalent operator electric circuits played one of the important roles in electrical engineering and especially in communication engineering. Each functional element of this circuit correspondents with the definite mathematical formulation or mathematical operation (differentiation, integration etc.). Such method of the construction and transformation of structural schemes of the automatic control systems and algebra of the structural transformation was developed by academician B.N. Petrov.
Keywords: logic circuitry, relay and switching circuit, electric circuit theory, theory of servomechanisms, simulation modeling, nano systems engineering.
In this paper it is shown how four consequence relations defined in terms of designated and anti-designated values allow to produce a six-element bilattice on the basis of two arbitrary finite-valued logical matrices for a propositional language L.
Keywords: product of logical matrices, anti-designated values, consequence relation.
The following expressibilities are constructed: (1) implication of every logical matrix MPCont, MPCont(1) and MPComp(1) in terms of the operations of the logical matrix MLPF , (2) implication of every logical matrix MPCont, MLPF and MPComp(1) in terms of the operations of the logical matrix MPCont(1), (3) implication of every logical matrix MPCont, MLPF and MPCont(1) in terms of the operations of the logical matrix MPComp(1). Also it’s proven that the implications of the logical matrices MLPF ,MPCont(1) and MPComp(1) are not expressible in terms of the operations of the logical matrix MPCont.
Keywords: matrix, operation, implication, valuation, expressibilitiy, paraconsistent logic, paracomplete logic.
The paper “Lesniewski’s systems of logic” is devoted to three systems of logic of the polish logician St. Lesniewski. Lesniewski intended to construct the consistent foundations of mathematics, and then his strategy was fighting with the closure of the language and ability to control the system’s openness by the especial rules of the inference. In this paper we consider the systems of Protothetic, Ontology and Mereology, and also their peculiar features and Lesniewski’s contribution to the logic. We briefly describe the formation of Lesniewski’s systems and characteristic properties of each system.
Keywords: Lesniewski, protothetic, ontology, mereology.
Lessons of soviet constructivism and their relations to practice are displayed here.
Keywords: intuitionism, constructivism, realizability, computability, applications of constructivism.
A proof of interpolation theorem for simple paranormal logic Int0,ω is proposed.
Keywords: quasi-elementary formula, paranormal logic, interpolation theorem.
Two submaximal classes of 3-valued functionally complete iterative system are characterized in the paper. These two classes are functionally precomplete classes of the famous Łukasiewicz’s logic.
Keywords: Łukasiewicz logic, functional class, submaximal class, iterative system.
Definition of logic following in the doctrine of Boethius is considered in this article. The author investigates not available in Russian language treatises of Boethius «De hypotheticis syllogismis», «In Ciceronis Topica» etc. Following problems are solved in research: a parity of sillogistics of Aristotle and stoics, a parity of categorical and conditional judgements, definition of logic following at Boethius, the reason of occurrence of «wrong moduses» in the doctrine of Boethius.
Keywords: logical doctrine of Boethius, relation of logic, hypothetical syllogisms, «wrong moduses» of hypothetical syllogism.
The aim of this paper is formulation of a notion of functional consequence and it’s axiomatisation. Problem of completeness is still open.
Keywords: grounds of logic, functional consequence.
The work in logic of Charles Peirce is surveyed in light of the characteristics enumerated by historian of logic J. van Heijenoort as defining the original innovations in logic of Frege and which together are said to be the basis of what has come to be called the “Fregean revolution” in logic and which are said to constitute the elements of Frege’s Begriffsschrift of 1879 as the “founding” document of modern logic.
Keywords: history of logic, modern logic, algebraic logic, abstract algebraic logic, propositional logic, first-order logic, quantifier elimination, equational classes, relational system.
In this paper I propose a particular algorithm by means of which humans come to understand the meaning of a logical formula. This algorithm shows why it is that some formulae are intuitively easy to understand while others border on the impossible. It also shows that the natural propositional logic is intuitionistic logic, not classical logic.
Keywords: logic, semantics, compositionality, intuitionism.
The modern history of many-valuedness starts with Łukasiewicz’s construction of three-valued logic. This pioneering, philosophically motivated and matrix based construction, first presented in 1918, was in 1922 extended to n-valued cases, including two infinite ones. Soon several constructions of many-valued logic appeared and the history of the topic became rich and interesting. However, as it is widely known, the problem of interpretation of multiple values is still among vexed questions of contemporary logic. With the paper, which essentially groups my earlier settlements, from , ,  and , I intend to put a new thread into discussion on the nature of logical many-valuedness. The topics, touched upon, are: matrices, tautological and non-tautological many-valuedness, Tarski’s structural consequence and the Lindenbaum–Wojcicki completeness result, which supports the Suszko’s claim on logical two-valuedness of any structural logic. Consequently, two facets of many-valuedness — referential and inferential — are unravelled. The first, fits the standard approach and it results in multiplication of semantic correlates of sentences, and not logical values in a proper sense. The second many-valuedness is a metalogical property of inference and refers to partition of the matrix universe into more than two disjoint subsets, used in the definition of inference.
Keywords: three-valued logic, many-valuedness, matrix, tautology, consequence operation, structurality, logical two-valuedness, Suszko’s Thesis, non-fregean logic, logical three-valuedness, inferential many-valuedness, inferential value.
Bazhanov V.A. — Honoured Science Worker of Russian Federation, D.Sc. in Philosophy, Head of Chair of Philosophy of Ulyanovsk State University.
Bakhtiyarov K.I. — D.Sc. in Philosophy, Ph.D. in Technical Sciences, Professor at the Department of Higher Mathematics of the Goryackin State University of Agricultural Engineering.
Biryukov B.V. — D.Sc. in Philosophy, Professor, Head of Interuniversity Center for Research of Reading an Informational Culture (at the MSLU).
Biryukova L.G. — Ph.D. in Philosophy, Assistant Professor at the Department of Higher Mathematics of the Plekhanov Russian Economic University.
Devyatkin L.Yu. — Ph.D. in Philosophy, Research scientist at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Gerasimova I.A. — D.Sc. in Philosophy, Leading research scientist at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Gorokhov V.G. — D.Sc. in Philosophy, Leading research scientist at the Department of Interdisciplinary Problems in the Advance of Science and Technology.
Moskvitsova N.G. — Post-graduate student at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Nepeivoda N.N. — Doctor of Ph.-Math. Sci, Professor, Head of the Department of Theory and Methodology of Informatics of the Udmurt State University.
Popov V.M. — Ph.D. in Philosophy, Assistant Professor at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Prelovsky N.N. — Post-graduate student at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Shalack V.I. — D.Sc. in Philosophy, Senior research scientist at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Tomova N.E. — Ph.D. in Philosophy, Research scientist at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Tonoyan L.G. — Ph.D. in Philosophy, Docent at the Department of logic of St. Petersburg State University.
Vasyukov V.L. — D.Sc. in Philosophy, Head of the Department of History and Philosophy of Science of the Institute of Philosophy of Russian Academy of Sciences.
Anellis I.H. — Ph.D. in philosophy, Visiting Research Associate, Peirce Edition, Institute for American Thought, Indiana UniversityPurdue University at Indianapolis, Indianapolis, USA.
Kracht M. — Professor, Faculty of Linguistics & Literary Studies, University of Bielefeld, Germany.
Malinowski G. — Professor, Department of Logic, University of Lodz, Poland.
Znamenskaya N.A. — Post-graduate student at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
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