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A system of formal logic without an analogue to the curry W operator1

Published online by Cambridge University Press:  12 March 2014

Frederic Brenton Fitch
Frederic Brenton Fitch*
Affiliation:
Yale University
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Extract

The system of formal logic to be presented in this paper has the following main properties:

(1). The Mengenlehre paradoxes do not arise in it.

(2). It seems to be free from other contradictions.

(3). It does not resort to a theory of types.

(4). It uses negation and material implication and does not weaken the principle of excluded middle.

(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.

(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….

(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.

(8). A theory of integers and real numbers cannot, apparently, be deduced from it.

(9). A form of the axiom of choice is provable in it.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 1 , Issue 3 , September 1936 , pp. 92 - 100
Copyright
Copyright © Association for Symbolic Logic 1936

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Footnotes

1

The material of this paper is in part an adaptation from the author's dissertation, A system of symbolic logic that avoids the paradoxes without a theory of types, presented for the degree of Doctor of Philosophy in Yale University, 1934. In part it also consists of work done while the author held a Du Pont Research Fellowship at The University of Virginia, 1934–35, and a Sterling Fellowship at Yale University, 1935–36. The author is indebted to Dr. J. B. Rosser, Professor Alonzo Church, and Professor F. S. C. Northrop for much helpful criticism in connection with the writing of this paper. To Dr. Rosser is due the suggestion of including some such axiom as 1.15′ (see Appendix) as an alternative to 1.15.

References

2 For a defense of the principle of excluded middle on philosophical grounds, see Baylis, C. A., Are some propositions neither true nor false?, Philosophy of science, vol. 3 (1936), pp. 156–166CrossRefGoogle Scholar.

3 Cf. Curry, H. B., Grundlagen der kombinatorischen Logik, American journal of mathematics, vol. 52 (1930), pp. 509–536, 789–834Google Scholar; Some additions to the theory of combinators, ibid., vol. 54 (1932), pp. 551–558; The universal quantifier in combinatory logic, Annals of mathematics, vol. 32 (1931), pp. 154–180Google Scholar; also Rosser, J. B., A mathematical logic without variables, Annals of mathematics, vol. 36 (1935), pp. 127–150Google Scholar, and Duke mathematical journal, vol. 1 (1935), pp. 328–355Google Scholar.

4 Schönfinkel, M., Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305–316CrossRefGoogle Scholar.

5 Bernays, P., Axiomatische Untersuchung des Aussagenkalküls der “Principia Mathtmaiica,” Mathematische Zeitschrift, vol. 25 (1926),pp. 305–320CrossRefGoogle Scholar.

6 Loc. cit.

7 Apparent variables from the standpoint of combinatory logic, Annals of mathematics, vol. 34 (1933), pp. 381–401CrossRefGoogle Scholar.