On Definitions by Hugues LeblancReview by: Leon HenkinThe Journal of Symbolic Logic, Vol. 16, No. 3 (Sep., 1951), pp. 213-214Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266404 .

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REVIEWS 213

The author's recommendation for dealing with the so-called modal paradoxes (cf. Quine VIII 45) is that a distinction be made between sameness 'Gleichheit) and identity (Identi- tat), the latter holding between a and b only when 0 a = b, and that identity be required for substitution in modal contexts. Thus we may not infer that the number of planets is necessarily greater than 7 unless we can show that this number is identical with (and not merely the same as) 9. Since the author follows Carnap (XIII 218) in explicating rOSi as 'S is L-true,' the previous remark requiring explicit specification of L is again applicable.

In a concluding section the author, disturbed by the "ambiguity" of one-place predi- cates, suggests that properties and classes be identified whenever the property in question can be proved to satisfy the principle of abstraction. Once again the tacit reference to a system must be eliminated. Further it might be remarked that the blurring of the distinc- tion between properties in intention and properties in extension (classes), though a matter of general mathematical practice, is not universally a desideratum.

Erratum: p. 14, replace the first occurrence of 'a = b' by 'a = a'. NICHOLAS RESCHER and J. F. THOMSON

JAMEs K. FEIBLEMAN. Class-membership and the ontological problem. Philosophy of science, vol. 17 (1950), pp. 254-259.

Quine and Goodman (XIII 48(2), XIII 49) have begun an analysis of the adequacy of the nominalistic thesis within the framework of symbolic logic. In the present paper Feibleman comments on the "realism-idealism-nominalism controversy," including strong criticism of the Quine-Goodman position, and then goes on to express some opinions about the de- velopment and future of symbolic logic. The paper has the form of a series of opinions with very little effort at substantiation. It begins with several principles which are called postu- lates although having the form of definitions (e.g. " 'reality' =Df 'having independent being' "), but there is no later reference to the postulates in the body of the article. Among the author's opinions is the belief that extensional logic is basically nominalistic. The attack on Quine is based in part, at least, on a lack of understanding of Quine's ideas, since Feibleman believes that the consistent nominalist must eschew all words except names of particular concrete individuals. As to symbolic logic, the author believes that "for a long while (it) has tried to be nominalistic," and that "a completely satisfactory logic cannot be established on the basis of existence postulates alone," but must be supple- mented by an intentional logic. LEON HENKIN

HUGUEs LEBLANC. The semiotic function of predicates. The journal of philosophy, vol. 46 (1949), pp. 838-844.

The author describes three interpretations of the r6le of predicates. The Platonic theory regards predicates as denoting abstract classes. The nominalistic position is that predicates are not names at all, but combine with names of concrete objects to form sentences which express meaningful assertions about those objects. The Aristotelian viewpoint holds that predicates denote "components" of concrete entities, a physical object being the sum of all the components which it possesses-although a component must not be confused with a part. The author attempts no justification for the term "Aristotelianism," and describes this theory as a "milder form of Platonism"; he does not indicate whether components are to be extensionally identified, and says nothing to contravene the suspicion that components are simply classes of which one says that they belong to their members, rather than that the members belong to them. As long as quantified predicate variables are not considered, the author maintains that we are equally free to choose among these three theories, although he agrees with Quine that quantified predicate variables are incompatible with the nomi- nalistic interpretation. In the latter portion of the paper the nominalistic attitude toward predicates is examined in finer detail, and the author illustrates the interesting thesis that predicates act either to support or undermine those aspects of individual variables and names (considered as physical objects) which favor or detract from their ability to function as signs. LEON HENKIN

HUGUES LEBLANC. On definitions. Philosophy of science, vol. 17 (1950), pp. 302-309. The author is principally concerned with the definition (within some logical system S)

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214 REVIEWS

of terms having a prior meaning. He seeks to avoid a discussion of the notion of synonymy by introducing, in syntactical terms, the concept of an absolute definition. Such a definition occurs when there is a finite axiom system Aa which fully characterizes the term a to be defined, and when the formula b = (7x)Ax is provable in S, where b is the definiens to be employed for a in the system S. (However, the proof that (3 !x)Ax may require a stronger logic than that of S.) The suggestion is made that the concept "a is synonymous with b" be replaced by the relative notion "the definition 'a stands for b' is absolute in S."

The author recognizes certain shortcomings of his proposal, such as the fact that for many notions a no complete finite axiom system can be constructed. But the most funda- mental objection is overlooked, namely, that even the most complete axiom system deter- mines its models only to within isomorphism, so that no significant concept can ever be given an absolute definition!

To this reviewer it seems that the author's admission that "Meanings . . . are obscure, and synonymy of meanings has never been analyzed . . . in a satisfactory way" is a chal- lenge to carry out an analysis of these concepts (perhaps along the lines of Church, in this JOURNAL, vol. 11 (1946), p. 31), rather than an excuse to avoid them as is done in the pres- ent work. LEON HENKIN

LUCIANO ALLENDE LEZAMA and ARMANDO ASTI VERA. El concept de inversion en el pensamiento. Episteme (Buenos Aires), no. 1 (1947), pp. 11-23.

This is a rambling essay in analogy and contrast. Logical negation is brought into analogy with inversion of functions. Truth-table logic is brought into analogy with its arithmetical counterpart, which is then contrasted with the rest of arithmetic; and the "important conclusion" is drawn that the "logistical schemas" (meaning perhaps the truth tables) do not suffice for arithmetic. 2 is said to be the first natural number, because it is the be- ginning of aggregation; but 1 and 0 are said to be epistemologically prior to the natural numbers, and an analogy is found in the proton and electron as prior to the atom. 1, 0, and the concept of number are found also to correspond to the Hegelian thesis, antithesis, and synthesis. Further observations include an analogy between the Mobius strip and the uni- fication of space and time. W. V. QUINE

Definici6n. Ibid., pp. 24-26. It is claimed that the concept-mapping scheme noted in XVI 139(1-5) permits us to

define definition as "the classifying of concepts, ordered and systematized according to proximum genus and specific difference." W. V. QUINE

MANLEY H. THOMPSON, Jr. The logical paradoxes and Peirce's semiotic. The journal of philosophy, vol. 46 (1949), pp. 513-536.

Using Peirce's analysis of the paradox of the liar as a guide, the present paper defends the kind of "solution" to a logical paradox which results from exploring the implications, and exposing the confusion of meanings, of the propositions which generate the given para- dox. It is argued that such an analysis is "fundamentally different" from and superior to that proposed by formal logic, which merely constructs formal systems the rules of which enable one to avoid a paradox, but which, unlike Peirce's analysis, fails to "explain" the paradox.

To this reviewer, such a thesis reduces to the claim that rules for avoiding contradiction do not "solve" the problems entailed by contradiction, since a "solution" should explain the contradiction. This is like saying that angels do not solve the problem of evil since they never encounter sin. However, even if such a strong meaning is attached to the term "solu- tion," one must not forget that there is usually more than one way to "solve" a given prob- lem. In particular, systems of formal logic not only enable one to avoid paradox, but can "explain" a paradox in the sense that paradoxical consequences can be shown to follow from specific violations of the rules of a given formal system. It is difficult to see how this mode of "explanation" is "fundamentally different" from Peirce's method of analysis. Actually, it is likely that Peirce's treatment can be formalized. In the end, then, the issue would reduce to the question of a choice of formal systems, and surely the present author would not wish to claim divine rights for a formalization of Peirce's mode of analysis.

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