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Logic based on inclusion and abstraction

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Harvard University
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Extract

All notions of mathematical logic, as measured e.g. by Principia mathematica, can be constructed from a basis which embraces variables and just two further primitives: the familiar notions of inclusion and abstraction. Inclusion will be expressed by the usual form of notation “(x⊂y)”, which may be read “x is included in y.” Abstraction will be expressed by the form of notation “x3 …” as with Peano, the blank being filled by any formula (statement or statement form); the whole may be read “the class of all objects x such that ….”

A system of logic based on these primitives will be presented in this paper. The system involves the familiar theory of class types, which, for metamathematical convenience, will be recorded in the system by use of distinctive styles of variables; thus the variables “x”, “y”, and “z”, also with subscripts when further variety is needed, will take individuals as values; the variables “x′”, “y′”, and “z′”, also with subscripts, will take classes of individuals as values; the variables “x″”, “y″”, and “z″”, also with subscripts, will take classes of classes of individuals as values; and so on. The metamathematical notion of term, covering all variables and also all expressions of abstraction, is now describable recursively thus: a variable is a term, and the number of accents which it bears is called its rank; and if terms of equal rank are put for “ζ” and “η” and a variable of rank n is put for “α” in “α3(ζ ⊂ η)”, the result is a term of rank n+1. A formula finally, is any result of putting terms of equal rank for “ζ” and “η” in “(ζ ⊂ η)”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 4 , December 1937 , pp. 145 - 152
Copyright
Copyright © Association for Symbolic Logic 1937

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References

1 In the notation of Principia mathematica the sign “3” is dropped, and instead a circumflex accent is placed over the initial variable; but this presents typographical difficulties.

2 However, see Leonard, and Goodman, , A calculus of individuals, in this volume of the Journal, pp. 63–64 Google Scholar.

3 See my System of logistic , pp. 37–38, 45.

4 See Wiener, , A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 387–390 Google Scholar; Kuratowski, , Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1920), p. 171 Google Scholar; or my New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 75–76 Google Scholar.

5 [Note added December 2, 1937.] R1 and R2 are suppressive in favor of:

R1′· ⊢⌈α n ⊂ β n · ⊃: β ⊂ γ n · ⊃ · α ⊂ γ⌉,

R2′· ⊢⌈α n 3 (α∈γ n+1 · ⊃ ϕ) ⊂ α3(α∈β n+1 · ⊃ · α∈γ) · ⊃ · β ⊂ γ⌉.

For, by the reasoning used in proving M27 below, we see that:

(i) If⊢ϕ, and ψ results from substituting ζ n for α n in ζ, then ⊢ ψ.

From R1′ (with α, β, and γ chosen as distinct variables free in neither ζ, η, nor θ) we can now derive R1 by three applications of (i); and from R2′ (with β and γ chosen as distinct variables free in neither ϕ, ζ, nor η) we can derive R2 by two applications of (i).

6 R3 and R5 answer to the rules of subsumption and concretion used in my System of logistic (pp. 43, 48). The rule of concretion is a specialization of a rule introduced still earlier by Church in A set of postulates for the foundation of logic, Annals of mathematics, vol. 33 (1932), see p. 355 (II)Google Scholar.

7 See Tarski, , Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit, Monatshefte für Mathematik und Physik, vol. 40(1933), pp. 97–103 CrossRefGoogle Scholar, especially Definitions 7 and 9. His notation will be modified in the present account.

8 See Łukasiewicz, and Tarski, , Untersuckungtn über den Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), p.34 (reprint p. 6), Satz 6Google Scholar; also my essay Truth by convention, in Philosophical essays for A. N. Whitehead (New York 1936), pp. 107–112, esp. note 18Google Scholar.