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Toward a calculus of concepts

Published online by Cambridge University Press:  12 March 2014

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

By concepts will be meant propositions (or truth-values), attributes (or classes), and relations of all degrees. The degree of a concept will be said to be 0, 1, or n (> 1), and the concept will be said to be medadic, monadic, or n-adic, according as the concept is a proposition, an attribute, or an n-adic relation. The common procedure in systematizing logistic is to treat these successive degrees as ultimately separate categories. The partition is not rested upon properties of the thus classified elements within the formal system, but is imposed rather at the metamathematical level, through stipulations as to what combinations of signs are to be accorded or denied meaning. Each function of the formal system is restricted, thus metamathematically, to one degree for its values and to one for each of its arguments. The theory of types imports a further scheme of infinite partition, imposed by metamathematical stipulations as to the relative types of admissible arguments of the several functions and stipulations as to the types of the values of the functions relative to the types of the arguments.

The elaborateness of the metamathematical grillwork which thus underlies formal logistic accounts in part for the tendency of those interested in logistic less for the matters treated than for the structures exemplified to limit their attention to the propositional calculus and the Boolean calculus of attributes (or classes), which, taken separately, are independent of the partitioning. A second reason for the algebraic appeal of these departments is their freedom from bound (apparent) variables: for use of bound variables fuses systematic considerations with notational or metamathematical ones in a way which resists ordinary formulation in terms of fixed functions and their arguments. Freedom from bound variables may be regarded, indeed, as the feature distinguishing algebra from analysis.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 1 , Issue 1 , June 1936 , pp. 2 - 25
Copyright
Copyright © Association for Symbolic Logic 1936

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References

1 Society of Fellows, Harvard University.

2 0-adic. See Collected papers of C. S. Peirce, vol. 3, p. 294Google Scholar.

3 While this gain in generality includes that described in Logistic [Quine, , A system of logistic], pp. 3, 191–193Google Scholar, the synthesis here undertaken is more basic. Whereas in Logistic the cleavage between degrees reduces to difference of type, under the present scheme even this residuum of extra-formal cleavage disappears. Furthermore, though in Logistic most functions are released from confinement to arguments of single degrees, they are not extended to arguments of all degrees, like and unlike, positive and zero, as under the present scheme.

4 This operation so contrasts with the related operation of ordination used in Logistic (q.v., pp. 10–16). The present procedure involves no consideration of sequences of sequences, with the attendant internal groupings.

5 This use of ‘з’, readable as ‘such that,’ is due to Peano.

6 Whitehead, and Russell's, Principia mathematica, 2nd. ed., vol. 1Google Scholar, will be thus referred to. For all references to symbols thereof use the index to definitions, pp. 667–668.

7 This is not the same generalization as occurs in Logistic (pp. 161–162), for the latter depends upon groupings within sequences. (V. supra, note 4.) Analogous divergence of generalizations results throughout relational theory. The present generalizations differ also quantitatively from those of Logistic (v. supra, note 3); thus the function described in (2) admits medadic arguments, while the analogue in Logistic does not. With other functions the difference of scope is greater.

8 Alike in degree; Sheffer's term.

9 The term disjunction is used here for Sheffer's stroke-function, the denial of conjunction; never for alternation.

10 See Quine, , Ontological remarks on the propositional calculus, Mind, n.s. vol. 43 (1934), pp. 472–176CrossRefGoogle Scholar.

11 See Frege, , Grundgesetze der Arithmetik, vol. 1, p. 50Google Scholar; Carnap, , Logische Syntax der Sprache, pp. 38, 192–202Google Scholar.

12 Still, those inclined may think of ‘T’ as ‘Tα’ and incorporate the vacuous argument similarly into derivative constants such as ‘F’.

13 C. S. Peirce's terms ‘composition’ and ‘aggregation’ are clearly preferable to ‘logical multiplication’ and ‘logical addition.’

14 See Logistic, pp. 91–92.

15 See Logistic, pp. 125, 127.

16 Not idemgradual; Sheffer's term.

17 In this direction one interesting point might be mentioned, viz. that Cnvδα and α are idem-gradual regardless of the degree of δ.

18 I am indebted to Rosser for suggestions leading to a reduction in the length of this proof; also for the correction of a theoretical error and several mechanical ones elsewhere in the paper.

19 Logische Syntax der Sprache, p. 76.

20 In this respect the present procedure agrees with that of PM, q.v., pp. 66, 71–72, 81, 187–188, 200.

21 In the more general case where may represent concepts other than propositions, answers rather to Church's . Cf. his A set of postulates for the foundation of logic, An of mathematics, 2 s. vol. 33 (1932), pp. 351–354Google Scholar.

22 Pp. 19, 39, 85.