Die Übermodalität der Notwendigkeit in Logischer Betrachtung. by Walter BruggerReview by: A. R. TurquetteThe Journal of Symbolic Logic, Vol. 18, No. 4 (Dec., 1953), p. 329Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266562 .

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REVIEWS 329

'There is a possible x which is A' means 'Possibly there is an actual x which is A.' In Chapter 4, the ontology of modalities of N. Hartmann, (already discussed in

XIV 140(3),) and other conceptions, are clarified verbally and by figures. Leibniz's notion of compossibility is analyzed. 'q is compossible,' symbolized 'Cp(q)', is defined by 'M(q . E)', where 1 is the totality of assumed conditions. The dual notion of "conecessity" is introduced, defined: Cn(q) = vCp(mq) = 1' < q. Becker marvels that "conecessity has never been introduced into logic or into ontology"; however, it is the same as the notion of "relative necessity" defined in Lewis and Langford (p. 161). Heidegger's non-classical conception of modalities is discussed without formal analysis. Appendices have to do with independence proofs, distinction of S2 red. from a finite truth-value system, the three-valued systems of Lukasiewicz and Moisil, and other details. WILLIAM T. PARRY

OSKAR ANDERSON. Zur Axiomnatik der Wahrscheinlichkeitslehre. Kontrolliertes Denken, Untersuchungen zum Logikkalkul und zur Logik der Einzelwissen- schaften (Festschrift fur Wilhelm Britzelmayr), edited by Albert Menne, Alexander Wilhelmy, and Helmut Angstl, rotaprint, Kommissions-Verlag Karl Alber, Munich 1951, pp. 4-7.

It is argued that only certain set-theoretical and logical treatments of probability theory satisfy the rigorous demands of modern pure mathematics to which Anderson believes probability theory should conform. Cram6r's set-theoretical probability axioms (See Harold Cramer, Mathematical methods of statistics, Princeton 1946) and Reichenbach's logical axioms (XVI 48) are given particular attention. It is claimed that both are equally rigorous, and a preference between them depends upon the thinking habits of the investigator. For example, there are sound reasons for a social scientist to prefer Cramer's axioms to Reichenbach's, while in subjects such as physics one might well prefer Reichenbach's axioms to Cramer's. A. R. TURQUETTE

HELMUT ANGSTL. Ciber Gleichungen im Aussagenkalkil. Ibid., pp. 8-9. Let A (p1, . . ., Pn , x) denote an expression built up out Of P1,. ... p , and the

variable x using the basic set of operators of the propositional calculus. Let

a(P1 > . . . , AP) denote a statement function Of P1 , Pn . Considered as truth- value functions, there are 22" different functions a(p1, ... pn). Using conjunctive and disjunctive normal forms, Angstl gives a rather straightforward method for deciding which (if any) of the 22" different a(p1, ... , pn)'s can be substituted for x in A (P1, . . ., pn , x) to give a tautology. The method which Angstl uses in solving this problem is reminiscent of what Schrbder calls the method of McColl. For example, see 427, p. 591; 394, p. 185; and 395, p. 26. A. R. TURQUETTE

WALTER BRUGGER. Die Ubermodalitdt der Notwendigkeit in logischer Betrachtung. Ibid., pp. 18-21.

The author argues intuitively, and not too convincingly, that in addition to the common modal distinctions of the possible, impossible, necessary, and contingent, one must distinguish a kind of basic modality called "Tbermodalitat der Not- wendigkeit" which results from the absolute necessity of pure identity ("des Iden- tischsein an sich mit sich selbst"). This basic modality plays a role for Brugger which is not unlike a Kantian synthetic a priori principle and is subject to similar criticisms.

A. R. TURQUETTE

EBERHARD FELS. Ordtnungsrelationen in einer subjektiven Werttheorie. Ibid.,

pp. 38-50. The author advocates the use of symbolic logic as a helpful tool in establishing a

rigorous and complete theory of economics. To this end, the present essay formalizes some of the basic concepts and definitions in the economic theory of John R. Hicks

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