N.L. Arkhiereev. Logical modalities as arithmetical functions
In the paper some new approach to the theory of logical modalities is considered. This approach presupposes construction of basic logical notions of logical necessity, logical contingency, logical impossibility by means of Restricted and Relatively Restricted Sets of State-Descriptions (RSSD and RRSSD respectively), which can be also treated as ordered sets of possible truth-values for variables. Recalculation of RSSD and RRSSD for each particular formula is provided by arithmetical functions of special type.
Keywords. logical necessity, possibility, contingency, possible world, model structure, restricted/relatively restricted set of state descriptions, arithmetical function.
B.V. Biryukov, I.P. Pryadko. The problem of logical contradiction and Russian religious philosophy
This article touches upon the logical ideas of two Russian philosophers of a beginning XX century — the discoverer of imaginary logic N.A.Vasiljew, and P.A. Florensky, author of the solid work. By instruments of the classic proposition logic make explication of the Florensky’s doctrine. The imaginary logic of Vasiljew is compared by logical comprehension of Florensky. It is demonstrated, a distinction of the theological contexts theirs logical doctrines.
Keywords: Florensky, Vasiljew logical contradiction, history of logic, non-classical logics.
V.N. Brushinkin. Logic and proof-search procedures
The article deals with the capacity of the proof-search theory for the simulating creativity and solving logico-philosophical problems. The ideas of prominent Russian logician Vladimir Smirnov are analyzed. The proof-search theory emergence is explained in the light of the Boris Gryasnov’s model of development of scientific theories. The explanation is based on a new interpretation of unexpected consequences (porism) of Hilbert’s proof theory, in particular from the constructing proofs bottom-up in sequential style logical systems, subformula principle and it’s interpretation in the field of automatic proof-search procedures.
V.L. Vasyukov. Metalogical pluralism and universal logic
Conception of logical pluralism claims that there is not one true logic but there are many. Conception of metalogical pluralism is based on the assumption that there is not one correct answer as to whether a given argument is deductively valid, but there are many. Since that leads to the interplay between logics and metalogics the question arises: what is the nature of this interplay? The Universal Logics approach gives us hints at some answers to this question. There are also some semantic keys to the issue under consideration.
А.S. Karpenko. Continuity of three-valued logic: problems and hypotheses
Functional properties of three-valued logics are considered. Among these logics ones with closed function classes set cardinality that is continuum. The problem of continuity of Bochvar’s three-valued logic is discussed and a hypothesis concerning the citeria of continuity of an arbitrary three-valued logic is proposed.
N.P. Kozachenko. Rationality criteria within belief revision: consistency
We consider some possible strategies in explicating the rationality criteria within various directions in belief revision. More specifically, we compare the so called AGM-approach (by Alchourron, Gärdenfors and Makinson) with the Paraconsistent Belief Revision advocated recently by Edwin Mares, as to dealing with inconsistent theories and believe bases. We conclude that although it is hardly possible to eliminate the principle of consistency, it still can be weakened by a suitable concept of coherence. It allows to effectively isolate contradictions and to work rationally with inconsistent believe bases.
V.I. Levin. Victor Ivanovich Shestakov and logic modelling
The scientific biography of the outstanding cybernetician V.I. Shestakov, one of pioneers in logic modelling in technics is stated and analysed. The short analysis of his basic works is given. His mutual relations with other scientists, features of his person are considered. The full bibliography of his scientific publications is presented.
Keywords: discrete circuits, logic modeling, logic design, the history of logic applications
N.N. Nepeivoda, A.P. Beltukov. Manifest of applied constructivism
Manifest of constructivism reborned. This constructive concept is in some extents more liberal and in some extents more strict than traditional ones. It is intended to join together best sides of intuitionism and Soviet constructivism and to apply the to practical problems of informatics and other domains.
Keywords: constructivism, applied constructivism, informatics.
V.M. Popov. Sequent-systems axiomatizing simple paralogics
The sequent systems axiomatizing some simple paralogics are presented and the solution of the decision problem for these simple paralogics is given. The connection of these logics with the classical propositional logic and the intuitionistic propositional logic is described.
Keywords: sequences, calculus, intersection of logics, paralogics.
М. Reznik. Structuralism and the identity of mathematical objects
The paper deals with the nature of mathematics from the point of view so called mathematical structuralism. The author proposes specific analysis of mathematical objects, and within the particular — relativistic standpoint — justify that mathematics is the science of patterns. He responded for the objections toward mathematical structuralism.
Keywords: structuralism, patterns, structural relativity, positions and the identity of logical and mathematical objects.
N.E. Tomova. Implicative extensions of regular Kleene logics
The family of regular 3-valued Kleene´s logics (strong, weak and intermediate) are considered as bases for other logics. Extentions of regular logics by implicative connectives (“natural” implication) are considered. In conclusion all exententions presented as a lattice.
Keywords: three-valued logics, kleene´s regular logics, implication, extensions of regular logics.
V.I. Shalack. Logic of functions vs logic of relations
It is proved that for any first-order theory with equality, the domain of interpretation of which contains at least two individuals, there exists mutually embeddable theory in language with functional symbols and only one-place predicate.
J.-Y. Beziau. What is a logic? Towards axiomatic emptiness
We first recall the original Greek sense of the word logic and how logic was developed on the one hand as an efficient way of reasoning by the use of reduction to the absurd and on the other hand as a useless system of logic by Aristotle. Then we discuss the changes of the modern conception of logic: the rejection of the principle of noncontradiction considered as fundamental by Aristotle and the structuralist move breaking the Aristotelian accident/essence dichotomy. Finally we explain why and how in universal logic — like in universal algebra — axiomatic emptiness prevails: a logical structure is a structure obeying no axioms.
W. Carnielli. Formal polynomials, heuristics and proofs in logic
This note surveys some previous results on the role of formal polynomials as a representation method for logical derivation in classical and non-classical logics, emphasizing many-valued logics, paraconsistent logics and modal logics. It also discusses the potentialities of formal polynomials as heuristic devices in logic and for expressing certain meta-logical properties, as well as pointing to some promising generalizations towards algebraic geometry.
Keywords: formal polynomials, algebraic proof procedures, heuristics in logical proofs, many-valued logics, modal logics.
Arkhiereev N.L. — Ph.D. in Philosophy, Assistant professor of the Department of Informational law, Informatics and Mathematics of The Russian Law Academy of the Russian Federation Ministry of Justice.
Gerasimova I.A. — Ph.D. in Philosophy, Professor, Leading research scientist at the Institute of Philosophy of Russian Academy of Sciences.
Markin V.I. — Ph.D. in Philosophy, Professor, Head of the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Subscription index under the catalogue «Press of Russia»: 42046. Second half-year period of 2010.
Price: 240 RUB.