Image of Soviet and Russian logic in the West. Latter Half of the XXth Century

The article attempts to overview Western scientific knowledge of research in mathematical logic and its history in the USSR and Russia in the first half of the 20th century. We claim that Western scholars followed and were generally aware of the main works of their Soviet and Russian colleagues on mathematical logic and its history. It was possible, firstly, due to the fact that a number of Western scientists knew the Russian language, and, secondly, because Soviet and Russian logicians published their works in English (sometimes in German) in the original journals of mathematical logic or Soviet publishing houses (mainly Mir Publishers) translated Soviet authors into English. Thus, the names of A.G. Dragalin, Yu.L. Ershov, A.S. Karpenko, A.N. Kolmogorov, Z.A. Kuzicheva, Yu.I. Manin, S.Yu. Maslov, F.A. Medvedev, G.E. Mints, V.N. Salii, V.A. Smirnov, A.A. Stolyar, N.I. Styazhkin, V.A. Uspensky, I.M. Yaglom, S.A. Yanovskaya, A.P. Yushkevich, A.A. Zinov’ev were quite known to their Western counterparts. With the dawn of perestroika, contacts of Soviet / Russian logicians expanded significantly. Nevertheless, the analysis of Western works on mathematical logic and the history of logic suggests that by the end of the 20th century the interest of Western scientists in the works of their Russian colleagues had noticeably waned.

Mathematical Logic of the Jones and Homfly Polynomials of Knotted Trivalent Networks

Two rational functions are defined logically for special type of knotted trivalent networks as state models of planar trivalent networks. The restriction of these two rational functions reduce to the Jones and Hom y polynomials for non oriented links. Also, these two models are used to define two invariants for this special type of knotted trivalent networks embedded in R3. Finally, we study some congruences of these two polynomials for periodic knotted trivalent networks this generalize the work of periodicity of the Jones and Hom y polynomials on knots to these two rational functions of knotted trivalent networks.

MATHEMATICAL PRACTICE AND MATHEMATICAL LOGIC: TO THE HISTORY OF RELATIONSHIP

The article annuls the role of practice in the development of mathematics in the 19th century in the formation of mathematical logic. It is shown that the revolutionary transformations of mathematics of the 19th century, which led to an increase in the abstractness of mathematical theories and concepts, was accompanied by an increase in uncertainty regarding the standards of proof, which led to the universal spread of anxiety (J. Gray) as an element of mathematical practice. It is argued that this element of practice was one of the sources of the emergence of mathematical logic, which claims to give rigor and accuracy to mathematics. The article argues that the socio- epistemological analysis of the practice of mathematics and the formation of mathematical logic will clarify the specifics of the development of relations between mathematics and mathematical logic.

Mathematical logic: construction of logic circuits from logical elements in Maple

В статье рассматриваются возможности применения библиотеки Logic системы компьютерной алгебры Maple в аспекте компьютерного моделирования логических схем в различных базисах. Смоделированы основные логические элементы в Maple. На конкретном примере детально представлен алгоритм построения логической схемы в различных базисах. The article discusses the possibilities of using the Logic library of the Maple computer algebra system in the aspect of computer modeling of logic circuits in various bases. Basic logic gates are modeled in Maple. On a specific example, an algorithm for constructing a logical circuit in various bases is presented in detail.

Sequent Calculi for the Propositional Logic of HYPE

In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Logic and the Basis of Theology: an Investigation into the Help Logic Can Give to the Understanding of Different Theologies, Following the Contributions to Both Disciplines of Arthur Prior, 1914-1969

<p>My thesis is that modem symbolic mathematical logics have an important contribution to make to theologies. I demonstrate this firstly in a 'theoretical section' (i) by showing what logics are and why they can be trusted; (ii) by showing how all theologies may be correctly treated as axiomatic systems; (iii) by outlining some modern logics which can assist theological thinking, including a logic I construct for this purpose called the Theologic. I demonstrate this, secondly, in an 'applied logic' section, by looking at (iv) the theology of one current branch of Christianity in detail, outlining its logical problems and the consequences of trying to avoid them; (v) 'post-modern' Christian theologies, firstly those that suggest that the word 'God' is a symbol rather than a name, and secondly at three feminist theologies two of which are logically quite radical; (vi) pantheism, in particular at Spinoza's ideas and Lovelock's Gaia; (vii) two religions, Buddhism and Confucianism, which, in their basic religious thinking, can be said to have no gods. I find that all religions I have studied - and they are representative of religions actual, proposed and imagined - have serious logical flaws, some known of old, others brought to light by the modern logics. The consequences of making the religions more logically sound are generally unacceptable to the members of the faiths. The suggestion that the gods use a different sort of logic to us is generally logically unacceptable. This does not leave abandoning religion as the only other possibility: the work of theologians in future, assisted by mathematical logic, may be (a) to bring about changes in basic beliefs, and (b) to assist in the birth of new, logically sound, religions. These investigations are carried out in the spirit of A N Prior, who came to logic through a Christian upbringing which gave him an interest in theology, a desire to make that theology more consistent, and as Professor of Philosophy at Canterbury College (as it then was) taught me. My upbringing was similar. We both, in the end, found conventional Christianity too illogical to believe. Time having past, I have been able to examine the logic of other, and newer, theologies.</p>

Logic and the Basis of Theology: an Investigation into the Help Logic Can Give to the Understanding of Different Theologies, Following the Contributions to Both Disciplines of Arthur Prior, 1914-1969

<p>My thesis is that modem symbolic mathematical logics have an important contribution to make to theologies. I demonstrate this firstly in a 'theoretical section' (i) by showing what logics are and why they can be trusted; (ii) by showing how all theologies may be correctly treated as axiomatic systems; (iii) by outlining some modern logics which can assist theological thinking, including a logic I construct for this purpose called the Theologic. I demonstrate this, secondly, in an 'applied logic' section, by looking at (iv) the theology of one current branch of Christianity in detail, outlining its logical problems and the consequences of trying to avoid them; (v) 'post-modern' Christian theologies, firstly those that suggest that the word 'God' is a symbol rather than a name, and secondly at three feminist theologies two of which are logically quite radical; (vi) pantheism, in particular at Spinoza's ideas and Lovelock's Gaia; (vii) two religions, Buddhism and Confucianism, which, in their basic religious thinking, can be said to have no gods. I find that all religions I have studied - and they are representative of religions actual, proposed and imagined - have serious logical flaws, some known of old, others brought to light by the modern logics. The consequences of making the religions more logically sound are generally unacceptable to the members of the faiths. The suggestion that the gods use a different sort of logic to us is generally logically unacceptable. This does not leave abandoning religion as the only other possibility: the work of theologians in future, assisted by mathematical logic, may be (a) to bring about changes in basic beliefs, and (b) to assist in the birth of new, logically sound, religions. These investigations are carried out in the spirit of A N Prior, who came to logic through a Christian upbringing which gave him an interest in theology, a desire to make that theology more consistent, and as Professor of Philosophy at Canterbury College (as it then was) taught me. My upbringing was similar. We both, in the end, found conventional Christianity too illogical to believe. Time having past, I have been able to examine the logic of other, and newer, theologies.</p>