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Published online by Cambridge University Press: 12 March 2014
If we think of denotation in a broad sense, we may say not only that, e.g., ‘Marni’ denotes the actual dog Marni but also that ‘dog’ denotes severally the dogs Marni, Fido, etc. What are ordinarily called class names thus come to denote severally the members of the class but not the class itself. This kind of a multiple denotation relation is a very simple as well as a very powerful semantical relation. In fact it can be taken as the fundamental relation of formal (extensional) semantics.
Conceptually this denotation relation is akin to the relation of satisfaction studied by Tarski, although the two must be carefully distinguished. Satisfaction (in its simplest form) is a relation between objects and sentential functions of one variable. Denotation is taken as a relation holding, on the one hand, between what is ordinarily called a proper name and the object which it names, and, on the other, between what may be called a virtual class denotator and the objects which are members of the virtual class in question. (This relation will be characterized more precisely below.) Clearly the two relations, satisfaction in its simplest form and multiple denotation, are interdefinable in an appropriate formalism.
1 Cf. the relation D studied in Martin, R. M. and Woodger, J. H., Toward an inscriptional semantics, this Journal, vol. 16 (1951), pp. 191–203Google Scholar. (See the abstracts, this Journal, vol. 14 (1949), pp. 74–75, and vol. 15 (1950), p. 79.) This paper will be referred to as T.I.S. The relation D is closely related to the relation for which Carnap suggests the term ‘applies’. See Carnap, R., Meaning and necessity, 1947, p. 97, footnote 1Google Scholar.
2 See Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1936), pp. 261–405Google Scholar.
3 See Skolem, Th., Über einige Grundlagenfragen in der Mathematik, Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo 1929, I, Math.-Naturv. Klasse, 1929, no. 4, esp. pp. 3–9Google Scholar.
4 See Church, A., Introduction to mathematical logic, 1944, esp. p. 37Google Scholar, The rules concerning quantification are to be presupposed formulated in such a way that expressions containing vacuous quantifiers are meaningless.
5 Tarski, loc. cit., esp. pp. 284–303.
6 Church, loc. cit., p. 37.
7 Tarski, loc. cit., p. 289. Tarski's other axioms are already provided for.
8 See Quine, W. V., Mathematical logic, 1940, pp. 291–305Google Scholar.
9 Cf. D4.1 of the author's A homogeneous system for formal logic, this Journal, vol. 8 (1943), pp. 1–23Google Scholar.
10 Cf. Tarski, loc. cit., pp. 285–286.
11 Cf. Tarski, loc. cit., p. 287.
12 The leading idea here is similar to that of the corresponding definition in T.I.S.
13 Cf. Tarski, loc. cit., pp. 311–313.
14 See Quine, loc. cit., p. 296.
15 See, e.g., Hilbert, D. and Ackermann, W., Principles of mathematical logic, 1950, p. 12Google Scholar.
16 Thus, e.g., where ‘p1’, … ‘pq’ are atomic universal formulae or negates of such, each ‘pi’ is a minimal component of a disjunction ‘p1 ∨ p2 ∨ … ∨ pq’, and where ‘A1’, …, ‘Am’ are any such disjunctions, each ‘Ai’ is a minimal component of a conjunction ‘A1·A2· … ·Am’.
17 Cf. Tarski, loc. cit., pp. 305–306.
18 The kind of formulation presupposed is similar to that of Tarski, loc. cit., pp. 365–366.
19 See Hilbert–Ackermann, loc. cit., p. 83, or A. Church, loc. cit., pp. 59–61.
20 Cf. Tarski, loc. cit., pp. 327–363.
21 See Tarski, loc. cit., pp. 363–369. See also the abstract. Type theory vs. set theory, by Kemeny, J., this Journal, vol. 15 (1950), p. 78Google Scholar.
22 See Tarski, , On undecidable statements in enlarged systems of logic and the concept of truth, this Journal, vol. 4 (1939), pp. 105–112, esp. p. 110Google Scholar.
