Zeitschrift für mathematische Logik und Grundlagen der Mathematik by Alan RoseReview by: Gene F. RoseThe Journal of Symbolic Logic, Vol. 29, No. 4 (Dec., 1964), p. 213Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270402 .

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REVIEWS 213

ALAN ROSE. Some formalisations of No-valued propositional calculi. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 204-209.

Let C, N, H, and 1 be the truth-functions defined over the rational numbers from 0 to I so that Cxy = min(l, 1-x + y), Nx = 1 -x, Hx = x/2, and lx = 1. Finite axioms systems, with modus ponens and substitution as the only rules of inference, are given for the following sets of primitive functors (1 being the designated element): (1) C, N, H, 1; (2) C, N, H; (3) C, H; and (4) C, H, 1. (For (1), an earlier formalization was reported in the author's XXIX 216(1).

GENE F. ROSE

ALAN ROSE. Sur un ensemble independent de foncteurs primitits pour le calcul propositionnel, sequel constitue son propre dual. Comptes rendus hebdotmadaires des seances de l'Acaddmie des Sciences (Paris), vol. 250 (1960), pp. 4089-4091.

ALAN ROSE. Nouvelle formalisation du calcul propositionnel bivalent don't les foncteurs primitils Torment un ensemble qui constitue son propre dual. Ibid., pp. 4246-4248.

Let G, H, t, and f be the two-valued truth-functions such that G(p, q, r) = (p & q) v

(p & Nr) v (q & NY), H(p, q, r) = (p & Nq) v (p & r) v (Nq & r), t(p) = the desig- nated value, and 1(P) = the undesignated value, where &, v, and N are the classical truth-functions for conjunction, disjunction, and negation. In the first paper, it is noted that G, t, and /, as well as H, t, and f form a functionally complete set of independent primitive functors (i.e., every truth-table can be defined in terms of the primitives, but with any primitive lacking, not every truth-table can be defined). The duals of G, H, t, and f are G, H with arguments in reverse order, f, and t re- spectively (the dual of an n-ary function P being the function Q such that Q(xl. xn) = NP(Nx, . Nxn)).

In the second paper, the primitive functors H, t, and f are formalized by four axioms and a rule of detachment. GENE F. ROSE

A. A. ZINOV'EV. Philosophical problems of many-valued logic. Revised edition, edited and translated by Guido Kfing and David Dinsmore Comey. Synthese library. D. Reidel Publishing Company, Dordrecht, Holland, 1963, XIV + 155 pp.

GUIDO KtNG and DAVID DINSMORE COMEY. Foreword from the editors. Therein, pp. VII-IX.

Anonymous. Bibliography of the publications of Aleksandr Aleksandrovi6 Zinov'ev. Ibid., pp. X-XI. A. A. ZINOV'EV. Author's preface. Ibid., pp. XII-XIV. GuIDo KONG and DAVID DINSMORE COMEY. Translators' notes. Ibid., pp. 149-150. This is a translation of a revised edition of Zinov'ev's XXVIII 255. The revision

was prepared by the author especially for this English-language edition. The trans- lators indicate that their translation is "a fairly literal one," but in general it reads better than might have been expected in view of technical difficulties associated with some of the subject matter of the book. Zinov'ev indicates that he is more concerned with "the philosophical problems raised by many-valued logic" than with its "mathe- matical and technical" nature. However, one of the interesting aspects of Zinov'ev's treatment is the manner in which these "philosophical problems" are almost always in- separably connected with technical issues. For example, practically no attention is paid to "dialectics" and in the small amount of space devoted to the subject, it is only indicated that there is no connection between "dialectics" and logic.

Zinov'ev is a young Russian logician at the Institute of Philosophy in Moscow and according to the translators, is the only Russian philosopher who has a first-hand technical knowledge of modern mathematical logic. In the past, most of the work on

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