Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-18T06:27:38.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Stratified systems of logic

Published online by Cambridge University Press:  24 October 2008

M. H. A. Newman
M. H. A. Newman
Affiliation:
St John's CollegeCambridge
Get access Rights & Permissions [Opens in a new window]

Extract

The suffixes used in logic to indicate differences of type may be regarded either as belonging to the formalism itself, or as being part of the machinery for deciding which rows of symbols (without suffixes) are to be admitted as significant. The two different attitudes do not necessarily lead to different formalisms, but when types are regarded as only one way of regulating the calculus it is natural to consider other possible ways, in particular the direct characterization of the significant formulae. Direct criteria for stratification were given by Quine, in his ‘New Foundations for Mathematical Logic’ (7). In the corresponding typed form of this theory ordinary integers are adequate as type-suffixes, and the direct description is correspondingly simple, but in other theories, including that recently proposed by Church(4), a partially ordered set of types must be used. In the present paper criteria, equivalent to the existence of a correct typing, are given for a general class of formalisms, which includes Church's system, several systems proposed by Quine, and (with some slight modifications, given in the last paragraph) Principia Mathematica. (The discussion has been given this general form rather with a view to clarity than to comprehensiveness.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 39 , Issue 2 , June 1943 , pp. 69 - 83
Copyright
Copyright © Cambridge Philosophical Society 1943

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

REFERENCES

(1)Bernays, P.Review of Quine(7). J. Symbolic Logic, 2 (1937), 86.CrossRefGoogle Scholar
(2)Carnap, R.Logical syntax of language (London, 1937).Google Scholar
(3)Church, A.Review of Quine (6). Bull. American Math. Soc. 41 (1935), 598–603.CrossRefGoogle Scholar
(4)Church, A.A formulation of the simple theory of types. J. Symbolic Logic, 5 (1940), 56–68.CrossRefGoogle Scholar
(5)Church, A.The calculi of λ-conversion (Princeton, 1941).Google Scholar
(6)Quine, W. V.A system of logistic (Cambridge, Mass., 1934).CrossRefGoogle Scholar
(7)Quine, W. V.New foundations for mathematical logic. Amer. Math. Monthly, 44 (1937), 70.CrossRefGoogle Scholar
(8)Quine, W. V.On the theory of types. J. Symbolic Logic, 3 (1938), 125–39.CrossRefGoogle Scholar
(9)Quine, W. V.Mathematical logic (New York, 1940).Google Scholar
(10)Quine, W. V.Element and number. J. Symbolic Logic, 6 (1941), 136–49.Google Scholar