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Identity, variables, and impredicative definitions

Published online by Cambridge University Press:  12 March 2014

K. Jaakko  and
J. Hintikka
K. Jaakko
Affiliation:
University of Helsinki
J. Hintikka
Affiliation:
University of Helsinki
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Extract

Recent discussion serves to bring out, amply and convincingly, the utility of observing the ordinary correct use of words and phrases for the purpose of clearing up philosophical problems. In this paper, I shall endeavour to show, by means of an example, that the reverse method may have its interest, too. An attempt will be made to cultivate a minor deviation from the accepted ways of using certain words and phrases in idiomatic English as well as in the formalized “languages” of the logicians. The words and phrases in question are those for the formalization of which a logician employs (free or bound) variables. Cases in point are the words customarily called quantifiers. The deviation I have in mind affects the relation of these words to the notion of identity. The deviation is illustrated by the following sentences:

(1a) Any two points of a straight line completely determine that line;

(2a) He is John's brother if he has the same parents as John;

(3a) Mazzini did more for the emancipation of his country than any living man of his time.

These examples may be contrasted with the following closely related sentences:

(1b) Any two distinct points of a straight line completely determine that line;

(2b) He is John's brother if he has the same parents as John and if he is not John himself;

(3b) Mazzini did more for the emancipation of his country than any other living man of his time.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 3 , September 1956 , pp. 225 - 245
Copyright
Copyright © Association for Symbolic Logic 1956

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References

1 For a number of uses of quantifiers in ordinary language, see Sapir, Edward, Totality (Linguistic Society of America, Language Monographs no. 6, 1930)Google Scholar.

2 In Nesfield, J. C., Aids to the study and composition of English (London, 1907), p. 176Google Scholar, the sentence (3a) is listed as incorrect, together with the sentences ‘He is more learned than any person now living’ (op. cit., p. 169) and ‘Of all other scholars he is the most accurate‘ (op. cit., p. 169).

3 Hilbert, David, The foundations of geometry, authorized translation by E. J. Townsend, Chicago and London, 1902Google Scholar.

4 Cf. Littlewood, D. E., The skeleton key of mathematics (Hutchinson's University Library, 1949) p. 60Google Scholar.

5 Wittgenstein, Ludwig, Tractatus logico-philosophicus (London, 1922) 5.53–5.352Google Scholar.

6 Carnap, Rudolf, The logical syntax of language (London, 1937) pp. 49–51Google Scholar.

7 The philosophy of Bertrand Russell (ed. by Schilpp, P. A., Evanston, Ill., 1944) p. 688Google Scholar.

8 Jaakko, K.Hintikka, J., Distributive normal forms in the calculus of predicates, Acta Philosophica Fennica, fasc. VI, Helsinki, 1953Google Scholar.

9 Geach, P. T. and von Wright, G. H., On an extended logic of relations, Societas Scientiarum Fennica, Commentationes physico-mathematicae, vol. 16 (1952) no. 1Google Scholar.

10 Zich, O. V., Přispěvek k theorii celých čsel a jednojednoznačného zobrazeni, Rozpravy České Akademie Věd a Umění, Třída II (Matematicko-přírovědecká), vol. LVIII (1948), no. 11Google Scholar. I am indebted to Professor Alonzo Church for calling my attention to this paper, and to Mr. Antti Karppinen for translating the relevant parts of it for me.

11 There remains an ambiguity concerning the interpretation of free variables. Are we to allow the values of two different free variables to coincide? Different answers to this question give rise to a further distinction between different kinds of calculi. We shall not discuss the resulting complications, however; they do not give anything new in principle. One can build a predicate calculus by means of bound variables only.

12 Otherwise, the first half of Tractatus 5.5321 would scarcely make sense.

13 Hilbert, D. and Ackermann, W., Grundzüge der theoretischen Logik, 3rd edition (Berlin, Göttingen, and Heidelberg, 1949), pp. 18–19CrossRefGoogle Scholar.

14 It is not always possible to replace all the free occurences of a variable y in a formula K by free occurences of another given variable x. The use of the notation ‘K(x/y)’ is taken be presuppose that a replacement of this kind is in fact possible. In the case with which we have to do here, the feasibility of the replacement is shown by the above lemma.

15 Wittgenstein was, hence, right in saying that the identity sign is not an essential constituent of logical notation (Tractatus 5.533).

16 An outline of a system of the predicate calculus based upon this relation is given in reference8, § 1.

17 This clause would be redundant if we were presupposing an exclusive interpretation of the variables of our own metalanguage.

18 The fact that (13) is the only assumption we have to make over and above the ordinary predicate calculus without identity in order to obtain the extended calculus with identity has been proved by Sampei, Yemon in Journal of the faculty of science, Hokkaido University, Series I, no. 11 (1950) and by the author in reference8, pp. 63–64Google Scholar.

19 Cf. Gödel, Kurt, Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines mathematischen Kolloquiums, Heft 4 (1931–1932), pp. 34–38Google Scholar, and Kleene, S.C., Introduction to metamathematics (New York, 1952) pp. 492–497Google Scholar.

20 Cf. Łukasiewicz, J., On the intuitionistic theory of deduction, Koninklijke Ne-derlandse Akademie van Wetenschappen, Proceedings of the Section of Sciences, series A, vol. 55 (1952), pp. 202–212Google Scholar.

21 See the surveys by Quine, W. V., New foundations for mathematical logic, American mathematical monthly, vol. 44 (1937), pp. 70–80CrossRefGoogle Scholar (reprinted, with corrections and supplementary remarks, in From the logical point of view, Cambridge, Mass., 1953, pp. 80–101)Google Scholar, and by Wang, Hao and McNaughton, R., Les systèmes axiomatiques de la théorie des ensembles (Paris, 1953)Google Scholar.

22 Cf. Quine, W. V., Mathematical logic (New York, 1940) p. 166Google Scholar, and Gödel, Kurt, Russell's mathematical logic (in The philosophy of Bertrand Russell, edited by Schilpp, P. A., Evanston, Ill., 1944, pp. 123–153) p. 131Google Scholar.

23 Frege, Gottlob, Die Grundgesetze der Arithmetik vol. 2 (Jena, 1903), pp. 253–265Google Scholar.

24 Quine, W. V., On Frege's way out, Mind N.S. vol. 64 (1955) pp. 145–159CrossRefGoogle Scholar. I am indebted to Professor Quine for letting me see a manuscript of this paper prior to its publication.

25 I am indebted to Mr. Peter Geach for valuable information concerning Lésniewski's contradiction. The contradiction is also set forth by Sobociński, Bolesław in L'Analyse de l'antinomie russellienne par Leśniewski, Methodos, vol. 1 (1949), pp. 94–107, 220–228, and 308–316Google Scholar.

26 Previously, Heinrich Behmann has defended a similar opinion (see his paper in Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 40 (1931), pp. 37–48)Google Scholar. He did not blame the paradoxes upon our intuitive ideas about sets or about infinity but upon our ways with variables. His suggestions are rather vague, however, admitting a variety of interpretations. Hence a detailed comparison with our discussion is not possible.

27 Gödel, Kurt, Russell's mathematical logic (cf. reference22), p. 150Google Scholar.

28 For different formulations of the principle, see Poincaré, H., Science and method (New York, Dover Publications, 1952) p. 190Google Scholar; Poincaré, H., Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathematische Physik (Leipzig and Berlin, 1910)p. 47Google Scholar; Russell, Bertrand, Mathematical logic as based on the theory of types, American journal of mathematics, vol. 30 (1908) pp. 222–262CrossRefGoogle Scholar; Scholz, H. reviewing Fraenkel's, A.Zehn Vorlesungen in Deutsche Literaturzeitung, N.F. vol. 4 (1927), columns 2416 to 2426Google Scholar; Gödel, Kurt, Russell's mathematical logic (cf. reference22) pp. 133–137Google Scholar.

29 The oldest theory of this kind is the so-called ramified theory of types, set forth by Whitehead, A. N. and Russell, Bertrand in Principia mathematica (3 vols., Cambridge, England, 1910–1913; 2nd ed. 1925-27) vol. 1, pp. 48–55Google Scholar. Recently, the theory of orders has been given an especially elegant formulation by Quine, W. V. in From the logical point of view (cf. reference21), pp. 123–127Google Scholar.

30 Fitch, F. B., The consistency of the ramified Principia, this Journal, vol. 3 (1938), pp. 140–149Google Scholar.

31 Principia mathematica (cf. reference29) 2nd ed., vol. 1, pp. xliv–xlvGoogle Scholar.

32 Poincaré, H., Sechs Vorträge über ausgewählten Gegenstände aus der reinen Mathematik und mathematische Physik (cf. reference28), p. 47Google Scholar.