TABLE OF CONTENTS
HISTORY OF LOGIC Oksana Yu. Goncharko et al. Theodoros Prodromos’ logical works: “On the great and the small”. This is the second in our series of articles concerning the logical treatises of the twelfth century Byzantine author Theodoros Prodromos. The subject of this paper is his treatise “On the Great and the Small” addressed to Michael Italikos and written in the tradition of neoplatonic commentary. The purpose of the article is to familiarize today’s readers with Theodoros Prodromos’ logical ideas and to analyze some of his commentaries on the “Categories” by Aristotle and the “Commentary on Categories” by Porphyry, assessing their originality.
PHILOSOPHY AND LOGIC Vitaliy V. Dolgorukov, Anastasia O. Kopylova. The “ontological square” and modern type theories. This paper focuses on the connection between “fourcategory ontologies” (which are based on Aristotle’s ontological square) and modern typetheoretical semantics. Fourcategory ontologies make a distinction between four types of entities: substantial universals, substantial particulars, accidental universals and accidental particulars. According to B. Smith, “fantology is a doctrine to the effect that the key to the ontological structure of reality is captured syntactically in the ‘Fa’ ”. Smith argues that predicate logic cannot adequately describe these four types of entities, which are reduced to just two kinds the general (‘F’) and the particular (‘a’). B. Smith has criticized G. Frege’s predicate logic. He argues that Frege, being the father of modern logic, simultaneously became the father of “fantology” with its ontological commitments. Smith transforms the ontological square to the ontological sextet (which also involves universal and particular events) and proposes a set of predicates for different ontological relations connecting these six types of entities. However, Smith’s approach has a number of limitations: he suggests a theory that describes only predicates of different types as universals. We argue for another formalization for the ontological square’s entities. This approach is based on modern typetheoretical semantics, according to which, the difference between substantial universals and accidental universals can be expressed. In firstorder logic the sentences “Socrates is a man” and “Socrates is wise” share the same logical form. However, this fact is not consistent with “ontological square” metaphysics (“being a man” is a substantial universal and “being wise” is an accidental universal). Whereas, according to the typetheoretical approach, relations to accidental universals are expressed by judgments about type (a:A),but relations to accidental universals are expressed by predication (‘Pa’).
NONCLASSICAL LOGIC Natasha Alechina. Model checking for coalition announcement logic. This talk is based on joint work with Rustam Galimullin and Hans van Ditmarsh, published in the German Conference on Artificial Intelligence (KI 2018). First I will introduce background and motivation for the work. I will introduce multiagent Epistemic Logic (EL) for representing knowledge of (idealised) agents, Public Announcement Logic (PAL) for modelling knowledge change after truthful announcements, Group Announcement Logic (GAL) for modelling what kinds of changes in other agents’ knowledge a group of agents can effect, and Coalition Announcement Logic (CAL) which is the main subject of the talk. CAL studies how a group of agents can enforce a certain outcome by making a joint announcement, regardless of any announcements made simultaneously by the opponents. The logic is useful to model imperfect information games with simultaneous moves. It is also useful for devising protocols of announcements that will increase some knowledge of some agents, but also preserve other agents’ ignorance with respect to some information (in other words, preserve privacy of the announcers). The main new technical result in the talk is a model checking algorithm for CAL, that is, an algorithm for evaluating a CAL formula in a given finite model. The modelchecking problem for CAL is PSPACEcomplete, and the protocol requires polynomial space (but exponential time).
I CONGRESS OF RUSSIAN SOCIETY FOR HISTORY AND PHILOSOPHY OF SCIENCE. MATERIALS ON LOGIC Angelina S. Bobrova. What do diagrams teach? Reasoning and perception. My paper concerns Peirce’s Existential Graph Theory (Graph theory) and its basic units that are diagrams or graphs. Graph theory is a valued logical system. It is algebra demonstrated in geometrical order. The theory is divided into several parts. They approximately correspond to propositional logic, first order logic, and modal logic. I will discuss the philosophical peculiarities of the theory rather than the technical ones. Philosophical ideas of this diagrammatic approach and its graphical syntax open new prospects for looking at the objective of logic. Graph philosophy will be scrutinized through the lens of information exchange and the growth of new knowledge. Special attention will be paid to the conceptual grounds to apply graph theory in practice. I will argue that diagram construction and perception can develop students’ logical skills. The existential graph theory realizes Peirce’s claim that logic is another name for semiotic. It deals with signs. Diagrams are signs (icons). These syntactical structures can be supplemented with interpretation (the theory accepts Tarski style semantics, gametheory semantics, etc.). A work with graphs is a graphical equivalent to scientific experiments. In the course of such experiments, we are not only able to discover necessary conclusions, which are implicitly or explicitly given in graphs, but also discover new knowledge and to observe the processes of information exchange. In short, graph theory allows us to perceive the nature of propositions, concepts and reasoning. The latter is treated as graph transformations (modifications regulated with the rules).
Vladimir L. Vasyukov. Logic of nonclassical science. The first still unresolved problem of theoretical physics the problem of quantum gravitation presupposes building a theory unifying the theory of general relativity and quantum theory. All attempts to date have failed. However, it seems that there is one specific answer to this question, albeit by virtue of its nature it was not in the uppermost of the researcher’s mind. Analysis of logical issues in quantum theory leads to the detection of the same general logical foundations both of relativity theory and quantum theory. It turned out that the logical structure of both disciplines is based on the construction of an or thomodular lattice and not Boolean algebra, which is typical for classical physics. On the other hand, taking into account the fact that quantum theory and the theory of relativity are nonclassical disciplines, then the common structure underlying both quantum logic and causal logic of spacetime should be evaluated as the logical basis of nonclassical science in general. It might be considered to be a specific character of the nonclassical type of scientific rationality which allows us to consider the links between knowledge on a subject and the specificity of means and activity operations. The construction of an orthomodular lattice is also employed in relativistic quantum theory. In this case, as is shown e.g. in Mark Hadley’s paper “The Logic of Quantum Mechanics Derived from Classical General Relativity” [Hadley, 1997], reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic to quantum logic. In so doing, an orthomodular lattice is not assumed as the basis of the theory but is discovered during the process of inquiry under detailed analysis. It would be detected in quantum theory as well as in the theory of relativity. But should it be assumed as the basis of the unified formal apparatus for resolving the problem of quantum gravitation? It is most likely that the presence of an orthomodular lattice structure in the depth of quantum theory and the theory of relativity is a consequence of the nonclassical nature of these disciplines. The lack of a distinguished implication connective (the conditional) in orthomodular lattice (as demonstrated by G. Kalmbach [Kalmbach, 1974] there are five such conditionals) indicates the need to monitor the peculiarities of our observations, since the implicit choice of “implicative” link between quantum propositions can affect our observation results.
Leonid Y. Devyatkin. On a continual class of fourvalued maximally paranormal logics. The problem of contradictory or incomplete information has an important place in modern philosophical logic. The methods of manyvalued logic have been widely applied in this field. One promising direction is the study of fourvalued logics, which facilitate working with contradictory as well as incomplete information simultaneously. This work lies within this approach. This paper is devoted to a set of continuum cardinality consisting of maximum strong fourvalued paranormal logics. I describe the fourvalued matrix that induces the logic I1P1 and demonstrate that, although it is neither maximally paraconsistent nor maximally paracomplete in the strong sense, there are continuummany of its fourvalued linguistic extensions that possess such properties.
Vitaliy Yu. Ivlev, Yuriy V. Ivlev. From determinism to quasideterminism in logic and beyond logic. This article is concerned with transition from determinate causation in logic, social and natural sciences to indeterminate causation in these branches of scientific knowledge. Analysis of this transition results in formulation of the principle of quasifunctionality for logic and the principle of quasideterminism for social, natural and technical sciences. In cognition, nature and society, there is not only the relation of definite conditionality between phenomena, but also the relation of indefinite conditionality, i.e. some definite cause can induce not only a single specific consequence, but also, under the same conditions, in one case, one distinct consequence of several possible consequences, and in another case an other. In logic the principle of functionality was implemented through the representation of logical terms as functions, and the principle of quasifunctionality was implemented through quasifunctions. Quasifunction is a correspondence by virtue of which an object from a certain subset of a certain domain is related with a certain object from a certain subset of some set (from the range of the quasifunction). Special cases of quasifunctions are a functional relation and complete uncertainty (randomness). An example of quasifunctional logic is the minimal modal logic Smin. Other examples of such logics are quasimatrix threevalue S_{r} logic; fourvalue quasimatrix S^{−}_{a},...S^{+}_{I} logics. Based on the principle of quasifunctionality,the idea of constructing abstract and real quasiautomata has been proposed. If there is a functional dependence between the signal at the input and the signal at the output of the automaton, then this dependence is quasifunctional in the quasiautomaton. The system of quasiautomatic machines can express functional dependence. Other actual problems are the application of the principle of quasideterminism in biology to the description of contingency, the consideration from this point of view the functioning of neural networks, development in the social sphere and other areas of knowledge and objective reality. It is proposed to revise technical, natural sciences and social knowledge on the basis of the principle of quasifunctionality.
Elena B. Kuzina. On the concept of proof. The term “proof” is used to refer to the whole spectrum of intellectual procedures aimed at establishing the objective truth or at proving the truth of a certain sentence, the acceptability of the imperative, the fairness of evaluation, as well at convincing other people of its adequacy. In mathematics, a proof plays a central role, but at the same time, there is not a general concept of mathematical proof. There are some very different perspectives on the nature of mathematical proof, its objectives, criteria and ideals, and over time these criteria and ideals change. Proof in other sciences is seen as a process of research, verification and confirmation of certain provisions for the search and justification of truth objective or conventionally accepted. Here proof consists essentially in searching for supporting evidence, assessing it and establishing that it proves the hypothesis best. Demonstrating reasoning, which is considered proof in deductive sciences, does not need to be built in many other areas. In different areas of knowledge, the criteria of viability and acceptability of evidence are different. In some it is formaldeductive rigor, in others it is evidence of arguments and the intuitive clarity of reasoning, in a third it is the reliability and adequacy of supporting evidence. The main criterion for the admissibility of evidence is its credibility the ability to cause the recipient to accept the proof of the statement so that he/she is willing to convince others. The proof is always immersed in the sociohistorical context, therefore, common to all sciences and all times, the concept of proof not only does not exist but cannot exist.
Vladimir I. Markin. De re  de dicto dichotomy and apodeictic syllogistic. Aristotle’s syllogistic is a modal deductive system, and his assertoric syllogistic is only a narrow fragment of it. This modal logical theory drew objections from Aristotle’s ancient and medieval successors and commentators. Aristotle considered some “mixed” syllogisms with one apodeictic premise, one assertoric premise and apodeictic conclusion to bevalid. His pupils Theophrastus and Eudemus introduced the principle that the conclusion always has the same modal character as the weaker of the premises, thereby they rejected all mixed modal syllogisms. In medieval logic, a distinction was made between de dicto and de re modalities. It was demonstrated that propositions with de dicto and de re modalities have different deductive characteristics. Aristotle’s apodeictic syllogistic contains both: reasonings valid only under de dictointerpretation of modalities (e.g. the law of I conversion) and reasonings valid only under de reinterpetation (e.g. modus Barbara). When we accept the “principle of the weakest premise”, apodeictic syllogistic can be naturally interpreted as containing de dicto modalities. The eminent Polish logician Jan Lukasiewicz suggested that both modal syllogistic versions were incorrect. In his opinion all mixed modi formed from the valid categorical syllogisms (e.g. Barbara rejected by Aristotle) are also valid. Lukasiewicz justified these modi by means of his positive assertoric syllogistic and fourvalued modal logic, which contains some theorems unprovable in normal modal calculi. We set out two translations of apodeictic and assertoric propositions into the modal firstorder logic with equality (G.E. Mints’ modal system T): the first provides the validity of all the laws of Aristotle’s apodeictic syllogistic, the second one preserves the validity of all apodeictic syllogisms accepted by Lukasiewicz. So, the apparatus of modern quantified modal logic can be used for the “rehabilitation” of apodeitic fragments of Aristotle’s syllogistic as well as Lukasiewicz’ syllogistic.
Yaroslav I. Petrukhin. Analytic tableaux for intuitionistic First Degree Entailment. N.D. Belnap formulated a relevant logic called FDE (First Degree Entailment) which avoids socalled paradoxes of classical entailment: “any proposition follows from a contradiction” and “a tautology follows from any proposition”. FDE deals with formulas which have an implication as the main connective and its antecedent as well as consequent that contain only the following connectives: negation, disjunction, and conjunction. Since intuitionistic entailment has the same paradoxes as the classical one, the problem of the presentation of an intuitionistic analogue of FDE which avoids the paradoxes of intuitionistic entailment arises. Y.V. Shramko solved this problem and presented the logic IEfde. IEfde deals with both relevant and intuitionistic implications, because, in contrast to classical implication, the intuitionistic one is not expressed via negation, conjunction, and disjunction. Y.V. Shramko formulated an intuitionistic version of E.K. Voishvillo’s semantics of generalized descriptions of states originally developed for FDE. In this paper we present an adequate analytic tableaux in the style of M. Fitting for IE_{fde}, based on Y.V. Shramko’s semantics for this logic. We modify M. Fitting’s analytic tableaux for intuitionistic logic by adding two new types of signed formulas (TA (A is nottrue) and FA (A is notfalse)), reduction rules for them and adapting relevant definitions as well as the rules for TA and FA. A set of signed formulas S is called closed if it contains at the same time signed formulas of types TA and TA or FA and FA. A closed tableaux for {TA, TB} is called a proof of a formula A→B. In the rules, where in intuitionistic logic signed formulas of type FA are deleted, in IE_{fde} signed formulas of type TA are also deleted. Last, but not least, our analytic tableaux for IE_{fde} shows that this logic is decidable.
Nikolai N. Prelovskiy. Infinitevalued Lukasiewicz logic and Farey sequences. The paper explores the relations between MacNaughton’s criterion for infinitevalued Lukasiewicz logic, prime numbers and Farey sequences. The author gives a definition of prime numbers in terms of infinitevalued Lukasiewicz logic. According to MacNaughton’s criterion, the set of functions expressed in infinitevalued Lukasiewicz logic coincides with the set of certain continuous piecewise linear functions. The paper shows that natural number n is prime only if infinitevalued Lukasiewicz logic contains functions that the restriction to aproper finitely valued Lukasiewicz logic coincide with functions N_{1/n}(x). While every such function has piecewise linear counterparts, linear parameters for which may be obtained in certain Farey sequences. Therefore, it is possible to find all regarded linear functions in the point with coordinates (i/n,1/n). All such functions have equations f(x) =b+kx with integer parameters b and k, and 1/n=b+k(i/n), so it makes it possible to find the required parameters in certain Farey sequences.
Our approach to studying the relationship of various types of logical calculus is based on a study of evaluation as a morphism that preserves the structure of the algebra of formulas in the structure of the estimated values. The use of nonclassical logic in mathematics is currently limited. However, evergrowing and changing requirements for the mathematical apparatus used in formal models of complex objects and processes may significantly change this situation and lead to the development of mathematical theories based on the use of various types of nonclassical logic. Investigation of the interrelation between different types of logical calculus on the basis of evaluation is associated with the attraction of nonfinite methods of structure theory, to which one can associate the methods of generalized nonstandard analysis as a section of category theory. Development of the approach to the study of formal logic types based on the use of non finite methods of generalized nonstandard analysis allows us to consider the set of logic algebra formulas with the introduced equivalence relation as a factor  algebra with a certain structure. The use of methods applying modern mathematical theories allows us to reveal the mathematical structure of formal logic and to trace the relationship of different types of logical calculus, in other words, to identify the mathematical content of the considered type of logical calculus. The validity of the use of nonfinite methods in logical research is due to the fact that metamathematics is a theory that studies formalized mathematical theories. Formalized the ory is a set of finite sequences of characters (formulas and terms) and a set of operations on these sequences. Operations replace elementary steps of deduction in mathematical reasoning. In this statement, mathematical logic (metamathematics) itself becomes a branch of math ematics. That is, the logic itself in such a statement becomes the subject of mathematical research. This approach allows us to consider formal logic as a dynamic system, the development of which consists in the disclosure of a system of particular types of logical calculus, for the description of which it is proposed to use nonfinite methods of generalized nonstandard analysis.
Natalya E. Tomova. On fourvalued paranormal logics. The paper presents some results of the study of fourvalued paranormal logics. The properties of paranormal logics are such that they can be used for handling inconsistent and incomplete information, i.e. these logics are simultaneously paraconsistent and paracomplete. Logical systems are represented by logical matrices. The relation between paranormal matrices by class of tautologies and by class of valid consequence relation is investigated. Two fourvalued paranormal matrices, which are obtained by combining isomorphs of classical logic, contained in fourvalued logic of Bochvar B_{4} are considered. They are denoted as M_{15} and M_{16}. The matrices in question are literal, i.e. have the properties of paraconsistence and paracompleteness at the level of propositional variables and their iterated negations, or, what is the same, at the level of literals. We propose a method for proving the equivalence of these fourvalued paralogies in the class of tautologies. It is also indicated that the matrix M_{15} with designated value class D={1} coincides with the logic matrix V, which was suggested as a formalization of the imaginary logic of N.A. Vasiliev. We also consider two more fourvalued matrices that are characteristic for paranormal logics AVP and S^{4}. These matrices cannot be considered as a result of combining isomorphs of classical logic and differ from the matrices M_{V} and M_{15} only in determining the negation. It is established that the matrices M_{AVP} and M_{S4} relate to each other in a similar way as M_{V} and M_{15}. They are equivalent in tautologial class, that is, they specify the same paranormal theory, but they have different deductive properties. As a result, a further area for investigation is outlined; the question now arises of whether the matrices MV and MAVP specify the same paranormal theory, and what deductive difference can be established between pairs of matrices M_{15} and M_{S4}, M_{V} and M_{AVP}
Yury Yu. Chernoskutov. Logic and theory of science in the 19th century philosophy. In the paper I discuss some key milestones of the program, which strived, in the 19th century, to reduce the general theory of science to (formal) logic. Projects of this kind were inconsistent with the basic tenets of the Kantian theory of knowledge. Therefore, the former developed most under the traditions that were least influenced by the latter. Most attention is paid to the historical development of this program in Austria. We have shown that the basic principles of this approach were laid down by B. Bolzano, who identified the “Wissenschaftslehre” project with logic. The originality of the Bolzanian concept of logical form is analyzed. It is shown in particular that the Kantian opposition of form and content is not relevant to the sense in his doctrine. Further, I consider the reception and development of the Leibnizean project of “Characteristica universalis” by philosophers from the Bolzano circle, namely F. Exner and R. Zimmermann. Unlike the influential Trendelenburg interpretation, these two authors very firmly and decisively associated the project with progress in formal logic. Exner had in fact set the goal of creating a pure logical calculus which would be based on the Bolzanian method of variation of representations. Zimmermann, among other things, proposed that any kind of traditional category should not be used as a primitive conceptual basis for such a calculus; rather some special expressive means that can be used to construct structures of knowledge from units should serve for this goal. I also consider the role of R. Zimmermann’s textbook for gymnasiums titled “Formal Logic”, in particular the second edition. We try how he believes he can achieve the assumed purpose of logic, which, in his view, consists in elaborating the unification of the methods of science and the full ordering of knowledge.
Vladimir I. Shalack. Weak consequence relation between Aterms. The language of the λcalculus has many applications for solving different problems in logic, information technology, linguistics and artificial intelligence. The λcalculus is based on the basic relation between terms, which is called βconversion. In the presented report, we formulate a weaker relation between the λterms, which makes it possible to establish more subtle connections between logic and λcalculus. The basic idea is that when we assign a type α to a term X relative to the context Γ, which is written in the form Γ ˫X:α, the concept of context plays a role analogous to the concept of a model in logic. If in logic the expression M=A means that the formula A is true in the model M, then in the λcalculus with types the expression Γ ˫X:α means that in the context Γ the term X is assigned the type α, and this term has a value that can be computed. In logic, the relation of logical consequence between the formulas A and B is defined as A=B⇔∀M(M=A⇒M=B . If we transfer this scheme to the λcalculus, then the λ consequence relation between terms can be defined as X=_{λ}Y⇔ ∀Γ∈Ctx[∃α(Γ ˫ X:α)⇒ ∃β(Γ ˫ Y:β)] . The meaning of this relation is that in every context in which we can assign some type to the X, we can also assign some type to the term Y. In other words, if the function represented by the term X is computable, then the function represented by the term Y is also computable. The λconsequence between terms relation has many properties analogous to the classical logical consequence relation between formulas, as well as a number of new properties, characteristic for the λcalculus with types.
Taras A. Shiyan. Multiple meaning and typology of terms.
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