Review: S. V. Ablonskij, Functional Constructions in k-Valued Logic

symbolic logic in Russia has been done by mathematicians. This unique background of the author is reflected throughout the present work and is, no doubt, a major reason why his conception of the philosophical problems of many-valued logic is so closely connected with the technical and mathematical side of the subject. English readers with very little knowledge of recent logical developments in Russia should find it a rewarding experience to follow Zinov'ev's philosophical approach, even if only a few really new concepts are found. In this connection, the reader will discover a more extensive and better than usual treatment of the interesting work of such logicians as BoCvar, Sestakov, and Jablonskij. I t should not be inferred, however, that Zinov'ev focuses attention on Russian logic alone. On the contrary, many-valued logic is surveyed within a broad context of modern logical developments both in and out of Russia. Lukasiewicz and Post are acknowledged to be the chief originators of many-valued logic. Emphasis is placed on the philosophical motivation of Lukasiewicz and the complete absence of such motivation in Post. I t is claimed that analysis of the modal functor "possible" led Lukasiewicz to his three-valued logic, while Post was interested merely in a formal generalization of two-valued logic. "The current of ideas of intuitionistic logic," with its rejection of the law of excluded middle, is interpreted as further stimulating the development of many-valued logic. Brouwer, Heyting, Kolmogorov, Glivenko, and Jaskowski are taken as important representatives of this current. Other recognized sources of stimulation were such systems as BoSvar's three-valued logic for solving the classical paradoxes, the quantum logics (Birkhoff, von Neumann, Destouches-Fevrier, Reichenbach), the circuit logics of Sestakov, and various systems of strict implication (Lewis, Ackermann, von Wright, Rosser, and this reviewer). No attention is given in this connection to such early forerunners of many-valued logic as Peirce and MacColl, but there is agreement with Lukasiewicz that Aristotle was a many-valued logician and that many-valued logics should be called non-Chrysippean rather than non-Aristotelian. On the formal side, emphasis is placed on the truth-table development of manyvalued logic, although a chapter is devoted to an exposition of the quantification theory of Rosser and the present reviewer. Zinov'ev feels that it is the more-than-twovaluedness which is the essence of many-valued logic. On the side of interpretation, this leads to the problem of defining many-valued truth-values and it is pointed out that this should not be confused with the problem of finding applications for manyvalued logical systems. This emphasis on more-than-two-valuedness makes it a bit surprising to find that Zinov'ev leans over backwards to show that there is no conflict between many-valued logic and traditional two-valued logic, especially the "laws of thought" of the latter. Of course, there is a sense in which this is true, but a deep desire for unification should not blind one to significant differences. The reader should be alert to some rather obvious typographical errors.