TABLE OF CONTENTS
HISTORY OF LOGIC
Valentin A. Bazhanov. Jean van Heijenoort as historian of logic.
The article attempts to provide a rather concise overview of the life and work of Jean van Heijenoort (1912-1986) in the field of the history of logic. Information of a biographical nature is given, in which the scientist's attitude towards Marxism is noted and his close cooperation with L.D. Trotsky in a certain period; disappointment in the key principles of the Marxist doctrine and especially its practical implementation, but the preservation of the general interest in this political trend. Based on some of the unpublished materials obtained from the American Mathematics Archive in Austin, Texas, van Heijenoort's works in the field of the history of logic described. It is noted that he received a mathematical education as a geometer and a topologist, but his interests were transferred to logic, which, by the nature of its search for proof and the canons of reasoning, were close to geometry. Particular emphasis placed in the article on the circumstances of the work on the fundamental anthology of the development of logical thought "From Frege to Godel", compiled and commented by van Heijenoort and a number of his colleagues. Attention paid to the fact that the algebraic direction of the development of mathematical logic remains outside the anthology. We describe the works of van Heijenoort relating to the incompleteness theorem, and the first two volumes of the collected works of K. Godel in the early 1980s, interrupted by the death of van Heijenoort. The assessments of his colleagues on van Heijenoort's legacyin the history of logic provided.
Keywords: Jean van Heijenoort, L.D. Trotsky, history of logic, quantification theory and algebraic traditions in logic, G. Frege, K. Godel.
PHILOSOPHY AND LOGIC
Angelina S. Bobroya. How to make tautologies clear?
The paper shows how the first part of Peirce's Existential Graphs theory answers Wittgenstein's question: "how must a system of signs be constituted in order to make every tautology recognizable as such in one and the same way?" Existential Graphs theory or Graphs theory is a diagrammatical system. Its basic unit is a graph or diagram that reminds Euler's diagrams. The first part of the theory, which is alpha, corresponds, approximately, to classical propositional logic. The theory provides graphic or iconic syntax. So, it is clear why Wittgenstein's problem is also solved in an iconic way. Graphs let observe tautologies. No transformations are required to identify a formula type. The possibility to observe tautologies is due to not only the diagrammatical syntax peculiarities but also its minimalism. The cut (it is the boundary of a diagram) is the only sign of the alpha-graphs. It plays both technical and logical functions. The theory is even more concise than approaches with NAND or NOR operators. In light of the talk about tautologies, the paper concerns the problem of cut evolution. The cut is treated as negation, but it is a generated implication. Thus, implication but not negation and conjunction or disjunction is a primitive and most analytic sign. At first glance, it might look strange as the implication is the most complex logical connective. However, the implication tracks the idea of logical consequence and reflects its main properties, such as antisymmetry and transitivity.
Keywords: Existential Craphs theory, Logical diagrams, Peirce, Wittgenstein, tautology.
Natalia V. Zaitseva. The riddle of paradeigma.
The article continues the study of paradeigma (parallel reasoning, reasoning based on an example). Paradeigma was considered in detail by Aristotle in Prior Analytics, and in a rhetorical manner — in "Rhetoric" as being one of the two modes of argument. In previous papers, the emphasis was made on cognitive-epistemological interpretation of corresponding cognitive procedure. This paper zeros on the logical characteristics of paradeigma. The first section contains the analysis of the relevant fragments of the Aristotle's text and brief summary of the cognitive-phenomenological interpretation of paradeigma. The aim of the next section was to identify the logical form of reasoning based on an example. It is shown to be non-reducible to other types of plausible (non-deductive) reasoning, such as inductive generalization, analogy and abduction. On this basis, a reasonable assumption is made that pradaeigma is a special independent kind of plausible reasoning. In the final part of the article, the place of the paradeigma and the underlying cognitive procedure in the logical and philosophical views of Aristotle is considered. Special attention is paid to the corresponding cognitive procedure of the first principle grasping, as described by Aristotle in the Posterior Analytics.
Keywords: paradeigma, plausible reasoning, Aristotle, universal cognitive mechanisms of reasoning.
Anastasia O. Kopylova. Empty terms in W. Ockham's logic: what is the reference for chimaeras.
The paper is devoted to the problem of supposition of terms in the propositions about imaginary objects and the conditions of their truth values in the doctrine of William of Ockham who was a leading figure of the scholastic nominalism. His rather radical onto-logical position acknowledges the existence of no more than two types of essences: unitary substances and qualities. Being devoid of the universals, the Ockhamist doctrine implied the transformation of the previously elaborated semantic theories, including the theory of supposition. In the reconstruction of Ockham's thought that became classical, the supposition closely approached the reference; however, in 2000s C.Dutilh-Novaes proposed the interpretation of supposition as a theory of propositional meanings. This approach brings forth the understanding of supposition as an intensional rather than extensional theory. One of the crucial arguments for this reconstruction is based on the application of supposition in the propositions about imaginary objects. According to our view, this argument is not free from some drawbacks. The term that makes the reference to the imaginary objects can have only simple or material supposition but not a personal one. W. Ockham names imaginary objects impossible objects.Chimaera is an impossible object, because it is considered as something which is combined of parts of different animals.That's why it should contain several substantial forms, which leads to contradiction with the metaphysical principle of the uniqueness of the substantial form. In Ockham's doctrine affirmative propositions about imaginary objects are always false since chimeras do not possess real existence. This observation implies that propositions about imaginary objects are more adequately squared with the extensional rather than intensional interpretation of supposition.
Keywords: supposition, nominalism, fictional objects, truth-conditions.
V.L. VASYIJKOV. Quantum categories for quantum logic.
The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum topo-sophical approach to "abstract quantum logic" was proposed based on the topos of functors [E, Sets] where E is a so-called orthomodular preorder category — a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself — so called quantos — which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C*-algebras, which in turn are studied in their quantum habitat of noncommutative C*-algebras — more fundamental structures than commutative C*-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra — orthomod-ular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quan-tos of functors [E, QSets] is constructed where QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.
Keywords: Quantum logic, quantum conditional, quantos, polynomial exponentiation, quantum sets.
Igor A. Gorbunoy. Finite axiomatizability of quasi-normal modal logics.
Quasi-normal modal logics are logics in a modal language that contain the logic K, are closed according to the modus ponens rule, and for which is not postulated Godel's rule. Until recently, little attention was paid to these logics, despite the fact that among the first systems of modal logics formulated by C.I. Lewis, there were also quasi-normal logics. In this paper, we consider the question of finite axiomatizability of quasi-normal modal logics. As is well known, the quasi-normal partner of the logic K does not have a finite axiomatiz-ation. In addition, there are other modal normal finitely axiomatizable logics, whose quasi-normal partners have no finite axiomatization. (An example of such logic is the logic D.) Therefore, the question of the finite axiomatizability of a particular modal quasi-normal logic is not trivial. Note that the well-known special criteria for the finite axiomatizability of quasi-normal logics concern only quasi-normal partners of normal modal logics. In this paper, a generalization of these particular criteria is obtained for the case of arbitrary quasi-normal modal logics. Thus, we obtain a special criterion of finite axiomatisability applicable both for quasi-normal partners of normal logics and for quasi-normal logics which are not a quasi-normal partner of any normal logic. In addition, a method for constructing a possible finite axiomatization of these quasi-normal finitely axiomatizable logics is given. We also present an algorithm that gives an absolute axiomatization of the logic L according to the available relative axiomatization of the quasi-normal logic L over the quasi-normal partner of the logic K. Separately, the axiomatization of extensions of the logic K4 is considered. A special criterion for the finite axiomatizability of extensions of this logic is formulated. We present an algorithm that gives an absolute axiomatization of the logic L by the available relative axiomatization of the quasi-normal logic L over the quasi-normal partner of the logic K4.
Keywords: quasi-normal modal logics, quasi-normal partners, absolute axiomatization, relative axiomatization, finite axiomatizability.
G.K. Japaridze. Computability logic: Giving Caesar what belongs to Caesar.
The present article is a brief informal survey of computability logic (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems — understood in their most general, interactive sense — are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics.
Keywords: Computability logic; game semantics; constructive logic; intuitionistic logic; linear logic; interactive computability.
N.Nepejyoda. Deformalization as the immanent part of logical solving.
Deformalization is the part of logical process least investigated and studied. It is often non-trivial and hard task because of subjective and objective complexities. Subjective complexities connected with logic. Deformalization is needed to present results of logical investigations to outsiders. Outsiders usually use languages and formalisms very far from logical ones. Their thesaurus usually barely intersects with logical one. Thus formulations on logical language cannot be appreciated and comprehended by outsiders and formulation of results needs to be completely replaced by non-logical. This task often is like to translating from one natural language into another with radically different semantic structure and system of notions (e.g. from Russian into Chinese and vice versa). Subjective complexities connected with roles. Systems of values of the problem solver and the decision consumer is radically different. Many aspects which were important during solution are out of scope of interests of the consumer. Many aspects which were "important" for the consumer are to be negligible for the solver but they are to be restored in presentation of the decision. This side of deformalization leads a bridge to the objective complexities. Objective complexities. Methods applied during formalization and solving induce "dual" methods are to be applied during deformalization. General conclusions and propositions. After analyzing whole process of logical solving in its unity it is possible to make some conclusions how logic can take a place which it is worth both in scientific analysis and in education. Interesting in more detailed speculations of this matter are addressed to the Russian variant.
Keywords: Applied logics, formalization, deformalization, translation from logical languages, methodological conclusions, pedagogic conclusions.