¹ That it is in the values of the variables, and not in the supposed designata of constant terms, that the ontology of a theory is to be sought, has been urged by Quine, W. V. in Noies on existence and necessity, The journal of philosophy, vol. 40 (1943), pp. 113–127 CrossRefGoogle Scholar; also in Designation and existence, ibid., vol. 36 (1939), pp. 701–709.

² As for example in Goodman's, Nelson A study of qualities (1941, typescript, Harvard University Library).Google Scholar Qualitative (“abstract”) particles of experience and spatio-temporally bounded (“concrete”) particles are there regarded as equally acceptable basic elements for a system. Devices described in the present paper will probably make it possible so to revise that study that no construction will depend upon the existence of classes.

³ The simple principle of class abstraction, which leads to Russell's paradox and others, is this: Given any formula containing the variable ‘x’, there is a class whose members are all and only the objects x for which that formula holds. See Quine, W. V., Mathematical logic, pp. 128–130.Google Scholar For a brief survey of systems designed to exclude the paradoxes, see pp. 163–166, op. cit.; also Element and number, this Journal, vol. 6 (1941), pp. 135–149.

⁴ According to quantum physics, each physical object consists of a finite number of spatio-temporally scattered quanta of action. For there to be infinitely many physical objects, then, the world would have to have infinite extent along at least one of its spatio-temporal dimensions. Whether it has is a question upon which the current speculation of physicists seems to be divided.

⁵ A nominalistic syntax language may, of course, still contain shape-predicates, enabling us to say that a given inscription is, for example, dot-shaped, dotted-line-shaped, Odyssey-shaped. See §5 and §10.

⁶ The usual definition, which was first set forth by Frege in 1879 (Begriffschriƒt, p. 60), has become well-known through Whitehead and Russell and other writers. It is presented once more in the next section.

⁷ It might be supposed that the nominalist must regard as unclear any predicate of individuals for which there is no explanation that does not involve commitment to abstract entities. But unless “explanation” as here intended depends upon standards of clarity, which do not concern the nominalist as nominalist, a suitable explanation can always be supplied trivially by equating the predicate in question with any arbitrarily concocted single word.

⁸ The nominalist need not necessarily regard such a sentence as ‘There are 10¹⁰⁰⁰ objects in the universe’ as meaningless, even though there be no translation along these lines. For, this sentence can be translated as ‘The universe (as an individual) has 10¹⁰⁰⁰ objects as parts’ where ‘has 10¹⁰⁰⁰ objects as parts’ is taken as a primitive predicate of individuals. But while this translation satisfies purely nominalistic demands, there may be extra-nominalistic reasons of economy or clarity for wanting a translation that contains no such predicate. And wherever and for whatever reasons a translation of an expression is wanted in terms of certain predicates or a certain kind of predicates, the search for such a translation is a problem for the nominalist—though of course neither he nor any one else claims that every predicate can be defined in terms of every possible set of others.

⁹ A systematic treatment of ‘part’ and kindred terms will be found in The calculus of individuals and its uses by Leonard, Henry S. and Goodman, Nelson in this Journal, vol. 5 (1940), pp. 45–55.Google Scholar Earlier versions were published by Tarski and Leśniewski. Although all of these would have to undergo revision to meet the demands of nominalism, such revision is for the most part easily accomplished and does not affect any of the uses to which the terms in question are put here.

¹⁰ We use ‘platonistic’ as the antithesis of ‘nominalistic.’ Thus any language or theory that involves commitment to any abstract entity is platonistic.

¹¹ We might, equally consistently with nominalism, construe marks phenomenally, as events in the visual (or in the auditory or tactual) field. Moreover, although we shall regard an appropriate object during its entire existence as a single mark, we could equally well—and even advantageously if we want to increase the supply of marks—construe a mark as comprising the object in question during only a single moment of time.

¹² The idea of dealing with the language of classical mathematics in terms of a nuclear syntax language that would meet nominalistic demands was suggested in 1940 by Tarski. In the course of that year the project was discussed among Tarski, Carnap, and the present writers, but solutions were not found at that time for the technical problems involved.

¹³ The sign ‘=’, when it occurs as the main connective in definitions in this paper, is not to be thought of as expressing identity. It is to be regarded rather as constituting, in combination with the ‘D’ which precedes each definition-number, a mark of definitional abbreviation; and it may occur between name-matrices and statement-matrices indifferently. The definition D1 is to be understood as a convention to this effect: ‘Cxyzw’ is to be understood as an abbreviation of ‘(∃ t) (Cxyt · Ctzw’ and a similar understanding is to obtain when any other variables are used in place of ‘x’, ‘y’, ‘z’, and ‘w’, provided that a variable distinct from them is used in place of ‘t’. Other definitions are to be construed analogously.

¹⁴ Using essentially the method of Frege's definition of the ancestral of a relation, we might say that x is a formula if it belongs to every class which contains all atomic formulas and all quantifications and alternative denials of its members. But this definition is unallowable because of its use of quantification over classes; cf. §4.—There is indeed a completely general method, in syntax, of deriving ancestrale and kindred constructions without appeal to classes of expressions. This is the method of “framed ingredients” which appears in Quine, Mathematical logic , §56. The method consists essentially of these two steps: (1) the Frege form of definition is so revised that the classes to which it appeals can be limited to finite classes without impairing the result; (2) finite classes of expressions are then identified with individual expressions wherein the “member”-expressions occur merely as parts marked off in certain recognizable ways. However, when as nominalists we conceive of expressions strictly as concrete inscriptions, we find the method of framed ingredients unsatisfactory, because its success depends too much on what inscriptions happen to exist in the world. Actually, though, the nominalistic definition of proof in the present paper will be simpler than that in terms of framed ingredients; for it will not require the lines of a proof to be concatenated, nor to be marked off by intervening signs.

¹⁵ This is Łukasiewicz's simplification of Nicod's axiom schema. See Łukasiewicz, Jan, Uwagi o aksyomacie Nicod'a i o “dedukcyi uogólniającej”, Księga pamiqtkowa Polskiego Towarzystwa Filozoficznego we Lwowie, 1931, pp. 2–7 Google Scholar; also Nicod, Jean, A reduction in the number of primitive propositions of logic, Proceedings of the Cambridge Philosophical Society, vol. 19 (1917–1920), pp. 32–41.Google Scholar

¹⁶ They answer to 4.4.4, 4.4.5, and 4.4.6 of Fitch, F. B., The consistency of the ramified Principia, this Journal, vol. 3 (1938), pp. 140–149 Google Scholar; also to *102-*104 of W. V. Quine, Mathematical logic , p. 88.

¹⁷ Hailperin, Theodore, A set of axioms for logic, this Journal, vol. 9 (1944), pp. 1–19.Google Scholar

¹⁸ This is Nicod's generalization of modus ponens; see footnote 15.

¹⁹ According to the classical principles of syntax, any two expressions x and y have concatenate x⌢y and moreover x⌢y is always distinct from z⌢w, unless the characters occurring in x and in y are successively the same as those in z and in w. This combination of principles is as untenable from the point of view of a platonistic syntax of shape-classes as from the point of view of nominalism.