Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T21:42:21.298Z Has data issue: false hasContentIssue false
  • English
  • Français

On the logic of quantification

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
U. S. Naval Reserve
Get access Rights & Permissions [Opens in a new window]

Extract

The notation that I shall use here for the logic of quantification is of the familiar kind wherein the letters ‘p’, ‘q’, etc. stand in place of unspecified statements and the letters ‘ƒ’, ‘g’, etc. stand in place of unspecified predicates. The present section will deal with the significance of this notation; the purpose and scope of the paper as a whole can better be indicated afterward, in §2.

By “predicates” I mean, not properties (or classes) and relations, but merely certain notational expressions. As a first approach they may be thought of as expressions like ‘walks,’ ‘is red,’ ‘touches,’ ‘gives to.’ Where ‘ƒ’, ‘g’, ‘h’, and ‘k’ represent these four predicates, we may read ‘ƒx’ ‘gy’, ‘hxy’, and ‘κxyz’ as ‘x walks,’ ‘y is red,’ ‘x touches y,’ ‘x gives y to z.’

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 10 , Issue 1 , March 1945 , pp. 1 - 12
Copyright
Copyright © Association for Symbolic Logic 1945

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Predicates in this sense were introduced in my Elementary logic, p. 119, as an auxiliary to the theory of substitution. But I called them ‘stencils,’ not perceiving how aptly the traditional term ‘predicate’ might be used at this point.

2 See Designation and existence, The journal of philosophy, vol. 36 (1939), pp. 701–709.

3 In higher logic, where quantification over classes and relations takes place, such entities have indeed to be assumed. Even there, however, I prefer not to admit ‘ƒ’, ‘g’, etc. into quantifiers, but to think of classes and relations rather as among the values of the regular variables of quantification ‘x’, ‘y’, etc.; cf. Mathematical logic, §§22–23, and Elementary logic, §56.

The so-called “algebra” or “calculus” of classes and that of relations, as well as much else that is commonly treated in class- and relation-theory, require no quantification over classes and relations. We can, if we like, develop these portions of logic as a “virtual” theory of classes and relations which makes no real assumption of such entities and is wholly translatable into the present schematism of quantification theory; cf. O sentido da nova lógica (Sāo Paulo, 1944), pp. 218–223.

4 Whitehead and Russell's use of ‘p’, ‘q’, etc., and much of their use of their predicate letters ‘φ’, ‘ψ’etc., can be construed as schematic in the above sense, though those authors do not distinguish between this and other interpretations. See my Whitehead and the rise of modern logic, in The philosophy of A. N. Whitehead (Library of Living Philosophers, 1941), pp. 144f.; also Ontological remarks on the propositional calculus, Mind, n.s. vol. 43 (1934), pp. 472–476. The doctrine of schemata is explicit in J. C. Cooley, A primer of formal logic, pp. 11 and 75, and in my Elementary logic, pp. 40, 90f., 116f., though in these two books the respective words ‘form’ and ‘frame’ are used instead of ‘schema.’ The word used in my aforementioned paper on Whitehead and in O sentido da nova lógica is ‘schema’ (‘esquema’).

5 Such is my procedure in Mathematical logic.

6 This refers only to variables ‘x’, ‘y’, etc., not to schematic letters.

7 First shown by Löwenheim, L., Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447–470.CrossRefGoogle Scholar For other decision procedures to the same purpose see Hilbert, and Ackermann, , Grundzüge der theoretischen Logik (2nd edn., 1938), pp. 95–97CrossRefGoogle Scholar; also the references to T. Skolem and H. Behmann there provided; also my O sentido da nova lógica, pp. 126–129. A new one appears in the present paper.

8 Church, A., A note on the Entscheidungsproblem, this Journal, vol. 1 (1936), pp. 40–41, 101–102.Google Scholar

9 Concerning this test there is the following suggestion in O sentido da nova lógica, pp. 65–66: If χ is simpler than ω, work out the table for χ and then extend the table to χ in just those lines where χ came out true. If ω is simpler, work out the table for ω and then extend it to χ in just those lines where ω came out false.

10 The other ten are *103, *110, *118, *137, *157–*162.

11 Namely, *115, *130, *131, *135, *141–*156.

12 This notation is already familiar to readers of Mathematical logic; see pp. 92, 94.

13 Op. cit., pp. 56–57.

14 Op. cit., pp. 74–81. (After Gödel.)