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.

C. I. Lewis

Clarence Irving Lewis (b. 1883–d. 1964) is arguably the most important philosopher bridging the pragmatism of the golden age of William James and Charles Sanders Peirce and the analytic quasi-pragmatism of philosophers like W. V. Quine, Nelson Goodman, Wilfrid Sellars, and Hilary Putnam (the first three of whom were taught by him). Lewis’s philosophy as a whole reveals a unified systematic development from his dissertation in 1910, his early work in logic, the development of his epistemology in the 1920s and 1930s, his account of value theory in the 1940s and 1950s, culminating in his work in ethics, which occupied him until his death. Along the way he offered a devastating critique of American absolute idealism and offered a rich epistemology grounded in a Peircean kind of pragmatism. Early in his career Lewis wrote the first the history of logic in English, and, critical of the paradoxes of material implication, he developed an account of strict implication and a set of successively stronger modal logics, the S systems becoming the father of modern modal logic. Lewis was the most influential American philosopher from the mid-1930s until after his retirement in the 1950s. His work helped shape American philosophy as an academic endeavor and contributor to the growing acceptance of rigorous philosophical analysis and European logical empiricism. Lewis spent practically his entire career at Harvard University, bridging the Harvard of James and Royce and the modern department of Quine and Goodman. During his career he wrote six books and a hundred or so papers and reviews. A student of Josiah Royce, William James, and Ralph Barton Perry, a contemporary of Hans Reichenbach, Rudolf Carnap, and the logical empiricists of the 1930s and 1940s, and the teacher of Quine, William Frankena, Goodman, Roderick Chisholm, Roderick Firth, Sellars, and others, he played a pivotal role in shaping the marriage between pragmatism and empiricism that has come to dominate much of current analytic philosophy. Despite his significant contributions, his work soon became neglected and misinterpreted, lost in the influx of interest in Wittgenstein and the philosophy of language. Fortunately, this neglect has begun to wane.

Completeness: From Husserl to Carnap

In his Doppelvortrag (1901), Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize non-extendible manifolds and axiom systems (different from Hilbert’s axiom of completeness), an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete (decidable). Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic.

Abu Nasr Al-Farabi’s Science of Logic

In modern conditions of dynamically developing knowledge, the demand for correct thinking remains an urgent problem. Unfortunately, the course of logic, and especially the history of logic is excluded from the educational program of preparation of many specialties. Although the knowledge of logical science, its laws, techniques and operations in the practical and theoretical work of not only the humanities, but also representatives of technical, natural and mathematical specialties can hardly be overestimated. In this article, the authors' idea is aimed at filling the existing gap and presenting an analysis of the history of its formation in the format of the history of culture and the history of philosophy, using a comparative approach. In the history of logical science, a significant place is occupied by the logic of al-Farabi, who in the Middle Ages left a bright, indelible mark in the field of many sciences with his original ideas. The article analyzes the contribution made to the development of the science of logic by al-Farabi, which in turn contributes to the formation of a culture of thinking and comes to the following conclusions: First, al-Farabi closely links formal logic with the science of language. However, he does not completely identify thinking and language, noting that language is ethnic, while thinking is universal. Secondly, the Second Teacher considers dialectics as belonging to formal logic, in it he sees a form and means of cooperation between people, etc.

On the Syllogistic of G. Boole

Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.

The first Russian textbook on logic: Historical, logical, and textual analysis

The article is devoted to one of the first Russian textbooks on logic written by Makariy Petrovich (1733–1765), a professor of the Moscow Slavic-Greek-Latin Academy. The analysis of this work demonstrates that it was not a simple translation of some Latin textbook (such translations were published later, after the foundation of the Moscow University). The textbook is a result of Makariy’s (and his predecessors) years of teaching logic in Russian religious schools. It is, to some extent, an original work. The article explores the peculiarities of Makariy’s rendition of logic. For example, the chapters on syllogistics illustrate that Makariy was somewhat innovative: he replaced Latin names for the modes of the categorical syllogism with made-up Cyrillic words (while keeping to the rules defining the choice of Latin letters). It seems that he strived to make logic more practical and as a result, wrote the textbook entirely in Cyrillic. The article considers the historical context of Makariy’s work and its place in the history of logic. The authors make several assumptions as to why Mikhil Lomonosov did not publish the first Russian textbook on logic and also why Makariy’s book was not printed. The authors’ comparative study of XVIII century textbooks on logic makes it possible to specify some probable sources for “Logic”, both Latin and Greek. In addition, Makariy’s approach to the translation of logical terms (some of them he translated into Russian while others were transliterated) is considered. After thorough examination of all three surviving manuscripts of Makariy’s “Logic”, the authors conclude that this textbook is worthy of proper publication.

CHRISTOPH SIGWART (1830–1904) AND HIS LOGIC IN THE WORKS OF ISIDOR PRODAN (1854–1919/1920)

In this article, a little-studied problem of the critical analysis of the philosophical and logical position of the representative of German philosophical tradition Christoph Sigwart (1830–1904) in the university philosophy, especially in the work of a Kharkiv private-docent Isidor Prodan (1854–1919/1920) is presented. At first, the main periods of the scientific and creative career of Isidor Prodan, including his studying at the Czernowitz (Chernivtsi) Gymnasium (1864–1872) and the philosophical faculty at the University of Vienna (1872–1875) are considered. His teacher in Vienna was a very famous German and Austrian professor Franz Brentano (1838–1917), the author of the work “Psychology from an Empirical Standpoint” (1874) and the founder of “descriptive psychology” and intentionalism. Then his teaching of logic and philosophy at the Gymnasiums of Kishinev (Moldova), Izmail (Ukraine), Riga (Latvia), Tartu (Estonia), and Moscow (Russia) from 1876 till 1900 is emphasized. Then the features of the teaching and the publications of Isidor Prodan in his “Kharkiv period” (1906–1916) are pointed out, during which he was a private-docent at the department of philosophy. Isidor Prodan’s works at this time comprise three areas: 1) History of logic (Aristotle, Leibniz, Spencer, Sigwart), 2) philosophy of common sense (Thomas Reid and the Scottish School of Common Sense), 3) critique of Kant and Neo-Kantianism (Hermann Cohen, Wilhelm Windelband, Hans Vaihinger, Heinrich Rickert, Ernst Cassirer e. a.). In the last group, his work “The Truth about Kant (A Secret of his Success)” (1914) was of great importance. His very important work was the monograph “The Cognition and its Object (Justification of Common Sense)” (Kharkiv, 1913). The positions of well-known philosophers (Plato, Descartes, Berkley, Leibniz, and Hume) and less-known authors (Lodge, Preyer, and Schneider) were here analyzed. Isidor Prodan’s critical interpretation of the logical viewpoint of Christoph Sigwart in his two-volume work “Logic” (1873, 3rd ed., 1904) occupies an important place in this analysis. In turn, Isidor Prodan’s important achievement was the popularization of the ideas of this German logician and philosopher, in particular, because of his translation of extracts from the work “Logic”.