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# Department of Logic  Many-valued Extensions of the Classical Propositional Logic  / L. Devyatkin — M.: IPh RAS, 2018. — 96 p. — ISBN 978-5-9540-0340-6.   Many-valued logic is an important area of modern logical research. One of the most useful tools for constructing many-valued logics are logical matrices. However, as noted by many authors, not every many-valued matrix induces a many-valued logic. In fact, there is an infinite class of many-valued matrices that define the classical propositional logic. At first glance, such matrices are redundant with respect to the two-valued matrix of classical logic, and therefore are of little interest. However, it is not the case. Many-valued matrices of classical logic are of great importance for the design and analysis of a genuinely many-valued logics.   In this regard, there are two possible lines of investigation. First, one can obtain a non-classical logic by adding one or more non-classical operations to a matrix of classical logic. An example is Bochvar’s logic, which utilizes two types of operations, classical «external» and nonclassical «internal». Second, one can leave the operations of the classical matrix unchanged, but modify the class of values that are interpreted as «true». In particular, this is how Gödel’s matrices are obtained. In the literature, there are enough examples of individual matrices of the first and second types and of their classes. However, their systematic study, as far as the author knows, has never been conducted. The aim of this work is to lay the foundation for such a study.    Chapter 1 is devoted to functional extensions of classical logic matrices. Its task is to demonstrate the scale of a class of logics that can be constructed in this way. Here we describe examples of such matrices known from the literature, construct new examples, and show that a lot of many-valued matrices whose authors proceeded from completely different premises have equivalent formulations in the form of modifications of matrices which induce classical logic. The main result of the Chapter is presented in its final section. It is shown that there are infinitely many pairwise different functional extensions of many-valued matrices of classical logic for each fixed number of values exceeding two.   Chapter 2 deals with a new approach to the construction of nonclassical logic based on many-valued matrices of classical logic. In the literature, there are many pairs of three-valued matrices differing only in classes of designated values. But the vast majority of them induces a nonclassical consequence with either one designated value or two. However, there are non-classical logics obtained from the matrices of classical logic by contraction or expansion of the class of designated values.   Keywords: propositional logic, many-valued logics, paraconsistency, paracompleteness, classification of logics, logical matrices Essays on the foundations of logic / V. Shalack – M.: IPh RAS, 2017. – 111 p. – ISBN 978-5-9540-0320-8.   Logic is an instrument of intellectual cognition. It forms the basis of all sciences. In the first essay, we analyze Zeno's aporia “Arrow” and show the errors to which ignorance of logic leads. The conventional wisdom is that paradoxes of Zeno fixed inadequacy of our concepts of space, time and movement to describe the phenomena of the world. Physicists use Zeno's “Arrow” paradox for interpreting the results of quantum mechanical experiments. It is shown, that to solve the aporia it suffices to use elementary means of Aristotelian logic. It is believed that the transition from traditional logic to modern logic is a transition from the logic of properties to the logic of relations. An implicit assumption is that we should describe the world around us in terms of relationships. In the second essay, we analyze the ontological status of relations and functions and show that with equal success the world can be described in terms of functional dependencies between phenomena and measurements. The goal of the program of logicism was the reduction of mathematics to logic. This program has failed because it contained internal contradictions. Nevertheless, this does not mean that no fragments of mathematics cannot be represented by definitions in terms of pure logic. The necessary and sufficient condition for reducing some theory to logic is the existence of a degenerate one-element model of the theory. For philosophers, interesting examples are the theory of symmetrical relations, group theory, topos theory, combinatory logic. In the theories of the arithmetic and the geometry, the full reduction to logic is prevented by the existential axiom that asserts the existence of more than one object in the subject area. All other axioms are reducible to logic. The laws of classical Newtonian mechanics do not contain existential presuppositions and therefore are reducible to logic. The prerequisites for the emergence of a science of logic are found at the lowest level of sign systems. They are the interconnections of the phenomena of the surrounding world, when certain objects and phenomena are natural carriers of partial information about other objects and phenomena. Logic arises when we ignore the specific content of the information and make the object of study the fact of its transfer. Known systems of logic are the result of acceptance of specific ontological and epistemic assumptions about the structure of the surrounding world and the attitude of language to it. In this sense they are theories occupying an intermediate position between the theories of concrete sciences and real logic. In this situation it is necessary to make a transition to the use of logic based on the theory of signs. The formal theory of signs can be regarded as the logic of operating with signs, free from ontological and epistemic assumptions. Keywords: aporia, Zeno, syllogistics, epistemic assumptions, ontological assumptions, functions, relations, logicism, theory of definitions, foundations of logic, foundations of mathematics, theory of signs, index signs, iconic signs, symbol signs, protologic Bochvar's three-valued logic and literal paralogics / A.S. Karpenko, N. Tomova – M.: IPh RAS, 2016. — 110 p. — ISBN 978-5-9540-0314-7.   Paraconsistent and paracomplete logics are areas where interest continues to grow. One reason for this may be due to their simplicity and to the wide range of their applications (in computer science, artificial intelligence, and other areas). One crucial factor behind the development of paraconsistent logic is the belief that in certain circumstances we may find ourselves in a situation where our theory is inconsistent and yet we are required to draw inferences in a sensible fashion. The monograph is devoted to the study of the class of propositional literal paralogics. Literal paralogics are logics in which the paraproperties such as paraconsistence, paracompleteness and paranormality, occur only at the level of literals; that is, formulas that are propositional letters or their iterated negations. We examine the functional properties of logics, such as the functional inclusion of one logic into another, and the functional equivalence of logics. Analysis of logics based on these properties can lead to surprising results concerning the functional equivalence of logics having different axiomatizations and different meta-logical properties, as shown in the case of paraconsistent logic P1 and paracomplete logic I1.We begin by analyzing Bochvar’s three-valued nonsense logic B3. Its functional properties are determined by the union of different types of connectives – internal and external – and this fact accounts for B3 being “emergent” within a huge variety of three-valued logics. And this logic includes two isomorphs of the propositional classical logic C2. The combination of these two isomorphs leads to the construction of two famous paralogics P1 and I1, which are functionally equivalent. Moreover, each of these logics is functionally equivalent to the fragment of logic B3 consisting of external formulas only.A four-element lattice of three-valued paralogics with respect to the possession of paraproperties is presented at the end of the second chapter. In the closing chapter, we consider generalizations and hierarchies of literal paralogics and construct a semilattice of four-valued literal paralogics with respect to the relation of functional inclusion one paralogic to another.Keywords: Bochvar’s logic B3, isomorphs, extended formulas, paraconsistent logics P1 and P12, paracomplete logics I1 and I12, paranormal logic TK1, strong and weak modus ponens, lattice of paralogics, functional properties of paralogics Within the Boundaries of Three-valuedness / L. Devyatkin, N. Prelovskiy, N. Tomova – M.: IPh RAS, 2015. – 136 p. – ISBN 978-5-9540-0296-6.   The book "Within the Boundaries of Three-Valuedness" consists of three chapters, each of which contains the new, sometimes entirely unexpected results in the field of three-valued logics.In the chapter "Three-Valued Matrices for the Classical Propositional Logic", written by L.Yu. Devyatkin, the study of the three-valued isomorphs of the classical propositional logic C2 is taken to the new level. The main result is the proof of the theorem concerning the necessary and sufficient conditions, which an arbitrary three-valued matrix M, must satisfy in order to be an isomorph of C2. Here, the notion of an isomorph for the first time gains the rigorous logico-algebraic foundations. We will also note the following unexpected result: there are three-valued isomorphs of C2, which, with respect to the power of their closed subclasses, are continual!In the chapter "Continuality of some weak three-valued logics and the problem of the power of the precomplete classes", written by N.N. Prelovskiy, are presented the supplemented proofs of the theorems concerning the continual power of the sets of the closed classes of Bochvar’s three-valued logic B3 and Hallden’s three-valued logic H3. The following result of N.N. Prelovskiy sets one thinking about the mysterious properties of the weak (not functionally complete) three-valued logics: there can exist three-valued closed classes of functions, in which the number of precomplete classes is infinite. It is known that as far back as in 1965, A.V. Kuznetsov has shown that for each k there exists only a finite number of precomplete classes in a functionally complete logic. The results of the second chapter indicate the extraordinary complexity of the functional properties in weak three-valued logics. Hence, it is appears significant to have some well-grounded classification of the most interesting three-valued logics.In the closing chapter "On the Extension of the Class of Natural Three-Valued Logics: the New Classification", written by N.E. Tomova, the restriction is lifted that the rule modus ponens must be strict, i.e. preserving the designated value. As the result, the possibility emerges to define the class of so called "weakly" natural implications and to consider the extended class of logical matrices. The end result of the classification of three-valued logics proposed in this chapter is as follows: there are 10 distinct classes of functions, each of which is obtained by means of extending the weak regular Kleene’s logic Kw3 with appropriate implications. These 10 classes are lattice ordered with respect to functional inclusion of one class into another. The study of the obtained lattice shows that it holds the tremendous heuristic power for the achievement of non-trivial results in the field of three-valued logics.The book is introduced with the extensive foreword "The Variety of Three-Valuedness", written by A.S. Karpenko, with the further perspectives of research laid out. The material of the book is supplemented with subject and author indices. Contents Introduction to logic for philosophers / V. Shalack, N. Tomova – M.: IPh RAS, 2014. — 191 p. — ISBN 978-5-9540-0274-4.   The textbook is designed to be used in studying the introductory course course logic by philosophy students. It is different from the other available textbooks in the approach to the presentation of logic. It is presented not as an already given science but is gradually constructed on the basis of the accepted philosophical assumtions concerning the structure of the outside world. The book may be of interest not only to the students of logic, but also to its teachers. Prelovskiy N.N. Bivalent semantics: logico-philosophical analysis — M.: IPh RAS, 2013. — 138 p. — ISBN 978-5-9540-0255-3.   The monograph is devoted to the logico-philosophical analysis of the bivalent semantics of the multi-valued logics in connection to Suszko's Thesis. The various well-known algorhythms for construction of the bivalent semantics, which are the attempts of constructive realization of Suszko's Thesis, have been considered and the limits of their applicability have been shown. The class of non-Carthesian logics has been investigated. Tomova N.E. Natural three-valued logics: functional properties and relations — M.: IPh RAS, 2012. — 89 p. — ISBN 978-5-9540-0229-4.   The monograph is devoted to the functional properties of three-valued logics. Original approach to the problem of the relation between different three-valued logics is proposed. Different classes of three-valued logics are presented as the lattices w.r.t. relation of functional inclusion one logic to another. The lattices of implicative extensions of regular Kleene’s logics and the lattices of p-logic are presented. Łukasiewicz J. On Aristotle's principle of contradiction. Critical Investigation (trans. from Polish in Russian). Ed. A.S. Karpenko with introductory article and comments. — М. – SPb: CGI, 2012. — 256 p. — ISBN 978-5-98712-038-5. For the first time the famous book by J. Łukasiewicz "On Aristotle’s principle of contradiction" (1910) is published in Russian. In this book the firstly in the world a thorough critique of the principle of contradiction, defended by Aristotle in the “Metaphysics", is given. This edition also presents a newtranslation of the famous article "On determinism". There is a biography of Jan Łukasiewicz and introductory article that shows a painful internal struggle Łukasiewicz in connection with his proposed revolution in logic. Devyatkin L.Yu. Three-valued semantics for the classical propositional logic – M.: IPh RAS, 2011. – 108 pp. – ISBN 978-5-9540-0203-4. The monograph is devoted to the study of the properties of the three-valued semantics for the classical propositional logic. The wholeclass of the three-valued implicative-negative matrices for the classical propositional logic is described. The classification based on the functional properties of the basic operations of such matrices is devised. Also, the class of matrices with the classical set of tautologies but a non-classical consequence relation is investigated. It is shown that some important properties of the classical propositional logic are only present in the case of two-valued semantics.   PDF (in Russian)>> Karpenko A.S. The Development of Many-valued Logic. – Moscow: LKI Publishers, 2010. – 444 pp. – ISBN 978-5-382-01217-9. The book considers the development of multi-valued logic from Aristotle to present day. Due to the special role which multi-valued logic plays in computer science and its various applications the particular attention is given to its theory of the functional properties. Also worth mentioning is the complex problem of the interpretation of the truth-values which ultimately leads to the idea of their structuring. This book can serve as a reference work on many-valued logic with accurate preservance of the chronology of its development and a large list. of references   The book is aimed at the widest circle of readers and does not require any preliminary knowledge of the subject.   Preface >> Contents >> References >> Antinomies and paradoxes (ed. Gerasimova I.A.). – M.: "Canon+" ROOI "Rehabilitation", 2008. – 432 p. – ISBN 978-5-88373-116-6.   Duality, polarization, complementarity, paradox, antinomy - those are the key notions of this book. The questions of cognition of the opposing principles, contradictions and paradoxes are presented in the numerous aspects: historico-cultural, methodological, cognitive, linguistic. The sudy of the cultural contexts has led to the individualization of analytical, dialectic, dualistic and holistic strategies of cognition. The new theme is focused on the analysis of the induality principle in the philosophical traditions of East and West, its meaning for the holistic world view of the modernity. One of the leading themes has become the theme of a paradox and paradoxicality in scientific cognition and cultural practices. The book will be of interest for the specialists in cognition theory, theory of argumentation, methodology of science, philosophy of language, culturology, intercultural communications. Karpenko A.S. Fatalism and contingency of futures (logical analysis) – 2nd edition – M.: LKI Publishers, 2008. – 216 p. – ISBN 978-5-382-00410-5.   The book is concentrated around the problem area concerning the interrelations between Truth, Time and Necessity. The significance of the subject becomes evident due to the fact that there have been constructed the various fatalistic arguments and their refutations. The emphasis is made on emergence and development of discussions about Logical and Theological Fatalism. An intrinsic connection is established between these varieties of fatalism, on one hand, and Totalitarianism, on the other hand. A considerable attention is paid to the attempts of the refutation of Fatalism with the aid of some logical tools which led to various systems of non-classical logic. In the conclusion, various approaches are summed up toward the problem of logical status of propositions about future contingents, which resulted in developing an idea of structuralization of truth-values themselves. Shalack V.I. On the notion of logical consequence –  М.: IPh RAS, 2007. – 168 p. – ISBN 978-5-518-33416-8.   The monograph is devoted to the analysis of the philosophical foundations of logic. In first place, it concerns the notion of truth and the notion of the logical consequence. The author shows the limitation of the philosophical premises of modern logic and proposes the alternative approach to its construction. The book is aimed at everyone interested in the problems of the modern philosophical logic.   PDF (in Russian)>> Karpenko A.S. Lukasiewicz's Logics and Prime Numbers –  Luniver Press, 2006 – 168 p. – ISBN 978-0-9551170-3-8.   Is there any link between the doctrine of logical fatalism and prime numbers? What do logic and prime numbers have in common? The book adopts truth-functional approach to examine functional properties of finite-valued Łukasiewicz logics Łn+1. Prime numbers are defined in algebraic-logical terms (Finn's theorem) and represented as rooted trees. The author designs an algorithm which for every prime number n constructs a rooted tree where nodes are natural numbers and n is a root. Finite-valued logics Kn+1 are specified that they have tautologies if and only if n is a prime number. It is discovered that Kn+1 have the same functional properties as Łn+1 whenever n is a prime number. Thus, Kn+1 are 'logics' of prime numbers. Amazingly, combination of logics of prime numbers led to uncovering a law of generation of classes of prime numbers. Along with characterization of prime numbers author also gives characterization, in terms of Łukasiewicz logical matrices, of powers of primes, odd numbers, and even numbers. Shalack V.I. Logical analysis of the Internet – М.: IPh RAS, 2005. – 96 p. – 5-9540-0047-6.   The monograph is devoted to the problems of the analysis and construction of the logical models of the Internet for the more comlete extraction of the information it contains. The special emphasis is laid on the possibility of obtaining not the factual information as it is being done nowadays, but the analytical one, which is not explicitly presented in the global network. PDF (in Russian)>> Gerasimova I.A. Formal grammar and intensional logic – М.: IPh RAS, 2000. – 156 p. – ISBN 5-201-02024-0.   The strategy proposed by the American logician Richard Montague is utilised for the construction of the formal grammars of the Russian language. The systematical presentation of the method is given. The polymodal intensional logic adapted for the representation of the modal and epistemic contexts of the Russian language is constructed. The analysis of the problem of the semantic status of the musical event is given. The book is aimed at the logicians, linguists, mathemathicians, programmers. PDF (in Russian)>>

Papers

Publications at the website http://www.vaal.ru

Leonid Devyatkin

Natalya Tomova