Contents:
N.L. Arkhiereev. Semantic of possible sets of truth-values for S5. Decision procedure.
The article deals with some peculiarities of the decision procedure for the semantic of Restricted Sets of State-Descriptions (RSSD) for propositional fragment of S5. Basis of the decision procedure is modified apparatus of analytical tableaux for classical propositional logic with truth-functional and special set-theoretic («volumetric ») rules.
Keywords: Set-theoretic semantics for modal logics, logical modalities, restricted set of state-descriptions, volumetric rules.
K.I. Bakhtiyarov. How computer may think in terms of boolean many-valueness.
How a computer may think? The non-classical manyvaluedness can be presented by means of classical logic. We use F = −1 as a negative number for false, and the logical zero is released for uncertainty N = 0 which is better to be accepted by a computer by default than a presumption of lie. Uncertainty N = 0 = 0/1 forms the
triad ∇_{N} : −1, 0/1 , 1 where false F gives −1/1 ∨ 0/1 = 0/1 . Nonsense B = 0′ = 1/0 forms the inverse triad′ △^{B} : 1, 0′,−1′ where F′ gives 1/−1 ∨ 1/0 = 1/0 . Two triads generate FIVE system, where N ∨ B = 0 ∨ 0′ = 0/1 ∨ 1/0 = 1/1 . Operations are done componentwise. The letters of a genetic code are taken for a sample: A (adenine) as a maximum, u (uracil) as a weak minimum, and also the inverted letters: V as a minimum, n as a weak maximum. The Matrix of Complementarity, columns of which represent complementary pairs, has been constructed under laws of creation of a living organism. The complete block Ψ characterizes dependence on Last Name-dominants A and V Isomorphism of matrices of a genetic code and mental types shows predisposition of characters, in many respects determined by their genetic code. It is offered the arithmetizaton of modal logic on the basis of the Matrix of Complementarity.
Keywords: Bilattice FIVE, false-nonsense, self-duality of inversion, complementary, genetic code, dominants, psychogenetics.
B.V. Biryukov, L.G. Biryukova. Alexander Ivanovich Vvedenskiy as logician. Part I.
The paper is devoted to the logical ideas and the biography of the promonent Russian thinker Alexander Ivanovich Vvedenskiy (1856–1925).
Keywords: A.I.Vvedenskiy, history of logic, philosophical logic.
V.L. Vasyukov. Paraconsistent categories for paraconsistent logic.
It is well known that the concept of da Costa algebra reflects most of the logical properties of paraconsistent propositional calculi C_{n} (1 ≤ n ≤ ω) introduced by N.C.A. da Costa. The construction of topos of functors from a small category to the category of sets was proposed which allows to yield the categorical semantics for da Costa’s paraconsistent logic. Another categorical semantics for C_{n} would be obtained by introducing the concept of potos — the categorical counterpart of da Costa algebra (the name “potos” is borrowed from W.Carnielli’s story of the idea of such kind of categories). In the paper the potos completeness of da Costa logics (i.e. in respect to the potoses) is proved.
Keywords: da Costa algebra, paraconsistent categories, potoses, validity, foundations of category theory, complementary closed categories.
A. A. Vladimirov. On a concept of non-infinite set in constructive mathematics.
It is known that the concept of non-infinite set is nonmonolithic from the point of view of A.A. Markov’s constructive mathematics. In the paper we present one general approach to constructivist conception of «type of non-infiniteness».
Keywords: Constructive mathematics, Markov constructivism, finite set.
I.А. Gorbunov. Well-defined logics.
Some questions concerned the deduction theorem for consequence operations and sentential calculi are considered in the present paper.
Keywords: Well-determined logic, deductive set of formulas, deduction theorem.
L.Y. Devyatkin. On some functional properties of three-valued logical matrices for classical logic.
In this paper a number of functional properties of implicativenegative three-valued logical matrices with the classical cosequence relation is described.
Keywords: Classical propositional logic, three-valued logics, logical consequence relation, logical matrices.
D.V. Zaitsev, О.М. Grigoriev. Bipartite truth — one logic.
In this paper we develop a new conception of two-component truth-values combining ontological and epistemological strands. In so doing we first present philosophical justification for this approach and then consider algebraic and semantical (valuational) systems of twocomponent truth.
Keywords: Generalized truth values, first degree relevant logic, Dunn-Belnap logic, bipartite truth, truth values reduction.
А.А. Ilyin. Traditional syllogistic with negative terms.
We set out the following axiom schemes for traditional syllogistic with negative terms: (MaP &SaM) ⊃ SaP, SiP ⊃ PiS, SaS, SaP ⊃ SiP , SeP ≡ ¬SiP , SoP ≡ ¬SaP, SaP ≡ SeP ′ , SiP ≡ SiP ′′. We prove that this system embeds into the predicate calculus by the interpretation of categorical propositions made by M. Bejanishwili and L. Mchedlishwili.
Keywords: Traditional syllogistic, negative terms, axiomatization, categorical propositions.
А.S. Karpenko. Towards protologic.
This paper is devoted to the problem of the foundations of logic. The path of development of logic towards generalization and abstraction is traced. The central notions here are minimal logic, basic logic, abstract logic, universal logic, protologic.
Keywords: Minimal logic, basic logic, abstract logic, universal logic, protologic.
E.E. Lednikov. Knowledge and belief: its role in science and communication.
The paper exposes notions of knowledge and belief specific traits of contexts, discerned in accordance with philosophical tradition. Perspectives of construction of rational theory of discussion about beliefs are evaluated.
Keywords: Knowledge, belief, truth, communication, rational discussion.
V.I. Markin. Singular extensions of Lukasiewicz syllogistic.
We implement two systems of syllogistic with singular terms that are the extensions of Lukasiewicz syllogistic. In the language of the first system singular terms can place only the subject position and singular propositions are of special type. In the language of the second one singular terms can place the subject as well as the predicate position and there are no special syllogistic constants for singular propositions. We prove the embedding of these singular syllogistics into the classical predicate calculus.
Keywords: Syllogistic, singular terms, calculus, semantics, formalization, embedding translation.
N.N. Nepeivoda. Constructive mathematic: review of progress, lacks and lessons. Part I.
The whole manifold of constructivist conceptions of last century is analyzed. They are systematized and some aspects and lessons for naive attempts of applying are highlighted.
Keywords: Constructivism, realizability, Kolmogorov interpretation, topological interpretations, approximations, incorrect problems, finite information principle.
V.М. Popov. Sequential axiomatization of paranormal logic PContPComp.
The sequent calculas axiomatizing paranormal logic PContPComp is presented.
Keywords: Sequent, calculus, paraconsistent logic, paracomplete logic, paranormal logic.
V.М. Popov. Sequential axiomatization of propositional Nelson’s type logics.
The sequent systems axiomatizing four Nelson’s type logics are presented.
Keywords: Sequent, calculus, paraconsistent logic, paracomplete logic, paranormal logic
N.N. Prelovsky. A non-Cartesian extension of system E.
In this paper a non-Cartesian extension of relevant system E (of entailment) is considered.
Keywords: Suszko's thesis, relevant logic, non-Cartesian logic, extension of E.
N.E. Tomova. Natural p-logics.
In this paper the notion of p-logic is generalied and the class of natural p-logics and it’s functional properties are considered. It is shown that paraconsistent logic P^{1} and paracomplete logic I^{1} are funclionally equivalent. In conclusion natural p-logics are presented as a lattice.
Keywords: Three-valued logics, p-logics, natural p-logics, lattice of natural p-logics, functional properties of natural p-logics.
V.I. Shalack. Two approaches to the construction of logic.
The aim of this paper is a comparative analysis of two conceptually different approaches to the construction of logic. The first approach, going from Aristotle, called traditional, the second can be called semiotic.
Keywords: Foundations of logic, ontological assumptions, epistemic assumptions, semiotics.
V.I. Shalack. About genetic method.
The aim of this paper is to answer the question what is the genetic method for constructing theories. The distinctive feature of genetic theories is the use of functional languages, and proof in these theories is essentially based on the structure of terms.
Keywords: Genetic method, genetic reasoning, Euclidean geometry, recursive arithmetic, functional language, relational language.
V.O. Shangin. Natural deduction systems of some logics with truth-value gluts and truth-value gaps.
In the paper we present Fitch-style natural deduction systems of some logics with truth-value gluts and truth-value gaps. We show that natural deduction systems of the logics in question can be set up with different formulations of either the disjunction elimination rule or the negation introduction rule. We give a constructive proof that for
each natural deduction system N of a logic α, a formula A is N-provable iff A is a α-theorem.
Keywords: Proof theory, natural deduction, non-classical logic, logics with truth-value gluts and truth-value gaps, disjunction elimination rule.
Our authors:
Arkhiereev N.L. — Ph.D. in Philosophy, Assistant professor at the Department of Informational law, Informatics and Mathematics of The Russian Law Academy of the Russian Federation Ministry of Justice.
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.
Gorbunov I.A. — Ph.D. in Ph.-Math. Sci, Assistant professor at the Department of Algebra and Mathematical Logic of the Tver State University.
Grigoriev O.M. — Ph.D. in Philosophy, Assistance Lecturer at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Ilyin A.A. — Senior Lecturer at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Karpenko А.S. — D.Sc. in Philosophy, Head of the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
Lednikov E.E. — D.Sc. in Philosophy, Professor at the Department of Philosophy of the Lomonosov Moscow State Academy of Fine Chemical Technology.
Nepeivoda N.N. — Doctor of Ph.-Math. Sci, Professor, Head of the Department of Theory and Methodology of Informatics of the Udmurt State University.
Markin V.I. — D.Sc. in Philosophy, Professor, Head of the Department of Logic of the Faculty of Philosophy of the Moscow 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.
Shangin V.O. — Ph.D. in Philosophy, Assistance Lecturer at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
Tomova N.E. — Ph.D. in Philosophy, Research scientist at the Department of Logic of the Institute of Philosophy of Russian Academy of Sciences.
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.
Vladimirov A.A. — Ph.D. in Ph.-Math. Sci., Senior research scientist at the Dorodnicyn Computing Centre of the Russian Academy of Sciences.
Zaitsev D.V. — Ph.D. in Philosophy, Assistant Professor at the Department of Logic of the Faculty of Philosophy of the Moscow State University.
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