Enthymemes by Alan Ross Anderson; Nuel D. Belnap, Jr.Review by: Nicholas RescherThe Journal of Symbolic Logic, Vol. 27, No. 1 (Mar., 1962), pp. 115-116Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2963753 .

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REVIEWS 115

In the last section of the paper the author considers the view that some propositions about the future may have a third "neuter" truth-value. This view affects the logic of the operators F and G; it may, for instance, justify a rejection of (6b). S. KANGER

HAKAN TbRNEBOHM. On two logical systems proposed in the philosophy of quantum- mechanics. Theoria (Lund), vol. 23 (1957), pp. 84-101.

This is a useful and quite elementary paper on some logical properties of two systems of three-valued logic: a system (called LDF by Tdrnebohm and called L3c by its author) proposed by Mme. Paulette Fevrier, and a system (called Ln by Tbrnebohm) proposed by Reichenbach.

LDF contains two different "negations," two "conjunctions," three "disjunctions," one "conditional," and two "biconditionals"; Lu has three "negations," one "con- junction," one "disjunction," three "conditionals," and two "biconditionals." Tbrne- bohm establishes the formulas (a -b) -(b1 a), a * b 3 b * a, and a v b bv a as valid for all biconditionals, conjunctions and disjunctions in both systems. The formula a a = a holds in Lu (for both biconditionals), but fails for one of the "con- junctions" of LDF (the "always false" one); likewise a v a -a fails for one of the disjunctions of LDF . a _ a holds for only one negation in both LDF and LR. a D a holds in LDF but fails for one "conditional" in Lu.

The paper also includes certain statistics on the number of three-valued systems with certain properties. These should be taken with a grain of salt, since T6rnebohm counts two systems as different even if they have the same truth-tables, if, for example, there is a connective which is called a "disjunction" in one system and a "conjunction" in the other. As long as no restrictions have been imposed upon the use of these terms, I doubt if this is a useful method of classification. (The heuristic value of the terms "disjunction," "negation," etc., in discussing particular systems, such as LR and LDF, is evident.)

The reviewer wishes to point out the unsuitability of Mme. F6vrier's suggestion that an always false "conjunction" is of help in understanding complementarity in quantum mechanics. Certain conjunctions may well be always false in quantum mechanics - e.g., "The position of this particle is x and its momentum is y" - but this is not clarified by symbolizing the "and" as an always false truth-function, any more than the logical falsity of p * P would be clarified by interpreting the "*" as an always false truth- function.

The bottom formula on page 90 should read (a = f) = (--M) = 1. HILARY PUTNAM

ALAN Ross ANDERSON and NUEL D. BELNAP, Jr. Enthymemes. The journal of philosophy, vol. 58 (1961), pp. 713-723.

Given any implication-relation -*, let us construct a corresponding implication- relation => by the definition: A => B =df (3y)[Y & ([Y & A] -+ B)]. The principal finding of the present paper may be stated thus: If -* is taken to be the implication relation of the system E of Anderson and Belnap XXIII 457, then => is the intuition- istic implication of Heyting XXI 367. This complements Myhill's finding (XX 178) that if -* is the strict implication of Lewis's system S4, then => is material implication; and Heyting's remark (loc. cit.) that if -* is intuitionistic implication, then so is a.

(Obviously if -* is material implication itself, then => is also.) Consider further the following variant of the above definition: A -< B =df

(3r)[Nr & ([r & A] -* B)], where N represents the modality of necessity. The authors establish that when -* is taken as their E-implication, then = < becomes the strict implication of the positive fragment of Lewis's system S4 (in the formulation where the primitives are conjunction, alternation, and strict implication). Speaking generally,

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116 REVIEWS

and thus loosely, it is clear that whenever -* is some type of strict implication, so is =<. (And again, it is obvious that when --- is material implication, then =< is also.)

The authors offer considerable heuristic discussion to the effect that if -+ is taken to be their E-implication relation, then the corresponding 4. (viz. intuitionistic implication) offers a satisfactory means for capturing the traditional conception of "enthymematic implication." The reviewer cannot fully agree. For consider any implication relation -- meeting the inevitable requirements that for any A we have A --. A; and that whenever A -- B, then for any C we have (A & C) -+ B. It then follows that /or any true statement B we have A *. B for any statement A whatsoever (since B itself serves as the required r). This unpalatable consequence that a true statement is "enthymematically implied" by any and every statement plainly does not square with the traditional notion of enthymematic argument. It would seem that the minimal modification of the definition of *. in the direction of an acceptable construction of the "enthymematic implication" of traditional discussions is embodied in the definition: A -< B =df (3r)[r & .-(r -S B) & ([r & A] -R B)]. Here -* could appropriately be construed as a strict-implication of the Lewis type, or as the E- implication of Anderson-Belnap. (Clearly if -> were material implication and B were true, A-< B could never be true; and if were strict implication and B necessary, then A-< B could never be true.) An investigation of this more demanding type of "enthymematic implication" would seem of interest. NICHOLAs RESCHER

P. BANKS (pseudonym). On the philosophical interpretation o/logic: An Aristotelian dialogue. Dominican studies, vol. 3 (1950), pp. 139-153.

In this lively conversation between Messrs. Paleo, Neo, and The Aristotelian,

N. and A. team up against P. in favour of the outlook and methods of modem logic, which, however, A. claims, against N. and P. to be Aristotelian in spirit. The historical arguments by which this claim is supported may seem rather beside the point when we finally read that Aristotelianism is just an unfortunate name for common sense and scientific spirit. However the first intent is certainly to sustain the thesis on a stricter interpretation of the term. Thus A. variously asserts or argues against N.'s attack, that the basic sentences of Principia mathematics attribute a property to an individual, so that the viewpoint is substantialist; that the metalogical rules underlying logistic are formulations either of the dictum de omni or of the principle of identity; "every single law of logistic is constructed according to the Aristotelian scheme of subject and attribute" (later modified in effect to "is equivalent to some sentence constructed etc."); the Organon contains more than the syllogistic, e.g., some propositional and relational laws; an evident and bivalent metalogic governs all so- called logics, and in this the principle of excluded middle holds (though polyvalent logics are not to be held un-Aristotelian for suspending assent to it); operationalism is a philosophical theory having nothing to do with logic itself and "even opposed

to its spirit"; the pretended nominalism of modern logic is often only anti-Platonism, and "Platonism is a nuisance." Occasion is taken along the way to demolish the obscurantism of P.: "I do not understand all your meta-things, and doubt very much if they have any use"; "what is more than the old logic, is wrong." For A. holds with

progress in logic, on traditional foundations. The chief adverse criticism the reviewer would pass on A. is for his suggestion that

it matters to logic what philosophy logicians adopt. That philosophical views have

influenced the directions in which logicians have worked is undeniable. But in the

final count their work is not judged by philosophical standards. Indeed one may think it advantageous to the progress of logic that such different stimuli should be

operative. Ivo THOMAS

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