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The combinatory foundations of mathematical logic

Published online by Cambridge University Press:  12 March 2014

Haskell B. Curry*
Affiliation:
The Pennsylvania State College

Extract

In investigations of the foundations of mathematics we can distinguish two separate tendencies. On the one hand, one may seek to define his subject with greatest possible explicitness: to obtain a formulation which satisfies the most exacting demands for precision, and which is at the same time free from paradoxes and adequate for the purpose. On the other hand, besides the problem of formulation, there is that of simplification; one can seek to find systems based upon processes of greater and greater primitiveness. The reduction of a piece of mathematics to a formal system, and still further to a completely formalized system (as explained, for example, in my New York address a year ago), is a step toward the first of these objectives. But it is evident that one can formalize in various ways, and that some of these ways constitute a more profound analysis than others. Although from some points of view one way of formalization is as good as any other, yet a certain interest attaches to the problem of simplification, as is shown by the attention which some of the greatest mathematicians have devoted to it.

The researches about which I am reporting today are directed toward the second of these objectives. In fact we are concerned with constructing systems of an extremely rudimentary character, which analyze processes ordinarily taken for granted. This is properly part of the business of mathematical logic. Of course there are those, even among logicians, who doubt the utility of this sort of thing—who profess to have no interest in improvements which do not lead to increases in deductive power or what not. However that may be, this second objective certainly has some interest in its own right; and it is this interest which has formed the primary motivation for these researches.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1942

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References

1 VI 100.—Bibliographical references are made in the manner customary in reviews in this Journal, as explained at the beginning of the review section, except that references are not always to reviews, but sometimes to leading articles. Certain expressions defined in the paper cited here have been used below without further explanation. Except for this no technical acquaintance with mathematical logic is assumed here.

2 In connection with the following discussion cf. 3965; also Schönfinkel 3041.

3 The notation originally appeared in 3594, p. 352. For later explanations see II 39 and Church's recent monograph, The calculus of lambda conversion, Princeton University Press, 1941.

4 In the discussion below, is not required actually to contain x. However, we can think of as depending on x fictitiously; just as in elementary calculus we regard constants as functions.

5 Strictly speaking we are concerned not with a specific system , but with a type of system . There may be various systems satisfying further postulates in addition to those mentioned below.

6 3041.

7 Church expounds his system (see footnote 3) not as a formal system, but as a generalized calculus (in the sense of VI 100, p. 233). For a more detailed statement of it as a formal system see Example 6 of my paper Outlines of a formalist philosophy of mathematics (not yet published).

8 I 74 (2).

9 3962, p. 511. It was intended there to give the name ‘combinatory logic’ to the subject of the investigation rather than to any particular method of studying it.

10 3041. Schönfinkel used the letters ‘T’, ‘Z’, ‘C’, in place of ‘B’, ‘C’, ‘K’ respectively. The reason for this divergence in notation is that I had worked for some time with ‘B’, ‘C’, ‘W’, and ‘I’ before I knew of Schönfinkel's paper.

11 3962, 3964, VI 41–61.

12 VI 47–51, 54–61.

13 5461.

14 A partial consistency proof was contained in 3962 (Kapitel II, Abschnitt C) but the proof using the Church-Rosser theorem is more elegant. For the latest form of consistency proof see VI 54–61.

14a The presence of these paradoxical terms is an advantage, because it enables the paradoxes to be represented in the system where it is possible to analyze them. Cf. 3967, p. 588 f.

15 See, for example, 3598, I 73, II 39, III 178, and Church's recent monograph, cited in footnote 3, which contains a bibliography.

16 3596, p. 863. The development of the theory is due largely to Rosser (5461) and Kleene (4972, II 38(1) and (2)).

17 II 38 (2). For a more exact exposition of the results than can be given here, see Church's monograph (cited in footnote 3) pp. 40 ff.

18 As explained by Church (I 73, p. 351, footnote 9) the notion was proposed by Gödel in his Princeton lectures of 1934 (cf. 41814) and was credited by him in part to an unpublished suggestion of Herbrand; the first systematic exposition is due to Kleene (II 38(1)).

19 For Turing's computability see II 42; for the equivalence mentioned, III 89.

20 I 73.

21 I 40–41.

22 I.e., ordinarily we think of our systems as embracing (in intuitive interpretation) a category of entities much wider than those which are explicitly represented, so that our systems are capable of indefinite extension without altering the concept of generality.

23 Cf. Quine's definition of implication in 4585 p. 45.

24 3963 and 3965.

25 Cf. 3967, p. 585.

26 See footnote 24. The general theorems mentioned below are developed in the second paper there cited and also in 3966. For the properties of F on this basis see I 65, §5.

27 Cf. 3967 and I 65. (The latter paper contains numerous serious misprints; a mimeographed sheet of corrections will be sent on request as long as the supply lasts.) In these papers, however, the theory of the universal quantifier is presupposed.

28 Note the similarity of the postulates for F and those for P. If in any of the former postulates we change F to P and drop the combinator we have the corresponding postulate for P.

29 Cf. Theorem 3.9 of my paper I 65. For Quine's criterion see II 86 or his book, V 163.

30 A form of the deduction theorem (cf. 5071, p. 155) is probably deducible in this system, although the details have not been worked out yet.

31 5451.

32 See my paper The paradox of Kleene, and Rosser, , Transactions of the American Mathematical Society, vol. 50(1941), pp. 454516.Google Scholar

33 This is shown in my paper The inconsistency of certain formal logics (not yet published).

33a A canonical term, however, need not have a normal form in the sense of Church, and consequently can be meaningless from his point of view.

34 In particular the Church-Rosser Theorem mentioned in §3.

35 4422.

36 I 75 and IV 31.

37 Note that the present system, in so far as it contains a propositional algebra at all, contains only an intuitionistic propositional algebra. However, it was shown by Gödel (41811) that an arithmetic based on an intuitionistic propositional algebra is not essentially more restricted than that based on a classical one. It is probable that something analogous holds here.

24
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