Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T02:37:34.954Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of Russell's resolution of the semantical antinomies with that of Tarski

Published online by Cambridge University Press:  12 March 2014

Alonzo Church
Alonzo Church*
Affiliation:
University of California, Los Angeles, California 90024
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we treat the ramified type theory of Russell [6], afterwards adopted by Whitehead and Russell in Principia mathematica [12], so that we may compare Russell's resolution of the semantical antinomies by ramified type theory with the now widely accepted resolution of them by the method of Tarski in [7], [8], [9].

To avoid impredicativity the essential restriction is that quantification over any domain (type) must not be allowed to add new members to the domain, as it is held that adding new members changes the meaning of quantification over the domain in such a way that a vicious circle results. As Whitehead and Russell point out, there is no one particular form of the doctrine of types that is indispensable to accomplishing this restriction, and they have themselves offered two different versions of the ramified hierarchy in the first edition of Principia (see Preface, p. vii). The version in §§58–59 of the writer's [1], which will be followed in this paper, is still slightly different.

To distinguish Russellian types or types in the sense of the ramified hierarchy from types in the sense of the simple theory of types, let us call the former r-types.

There is an r-type i to which the individual variables belong. If β1, β2, …, βm are any given r-types, m ≧ 0, there is an r-type (β1, β2, …, βm)/n to which there belong m-ary functional variables of level n, n ≧ 1. The r-type (α1, α2, …, αm)/k is said to be directly lower than the r-type (β1, β2, …, βm)/n if α1 = β1, α2 = β2, …, αm = βm, k < n.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 4 , December 1976 , pp. 747 - 760
Copyright
Copyright © Association for Symbolic Logic 1976

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]Church, Alonzo, Introduction to mathematical logic, Volume I, Princeton, 1956.Google Scholar
[2]Copi, Irving M., The theory of logical types, London, 1971.Google Scholar
[3]Fraenkel, Abraham A. and Bar-Hillel, Yehoshua, Foundations of set theory, Amsterdam, 1958; second edition 1973, by Fraenkel, Bar-Hillel, and Azriel Levy with collaboration of Dirk van Dalen.Google Scholar
[4]Grelling, Kurt and Nelson, Leonard, Bemerkungen zu den Paradoxieen von Russell und Burali-Forti, Abhandlungen der Fries'schen Schule, n.s. vol. 2 (1907–1908), pp. 301–324.Google Scholar
[5]Henkin, Leon, Banishing the rule of substitution for functional variables, this Journal, vol. 18 (1953), pp. 201–208.Google Scholar
[6]Russell, Bertrand, Mathematical logic as based on the theory of types, American journal of mathematics, vol. 30 (1908), pp. 222–262.CrossRefGoogle Scholar
[7]Tarski, Alfred, Pojecie prawdy w jezykach nauk dedukcyjnych, Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III, no. 34, Warsaw, 1933.Google Scholar
[8]Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen (German translation of [7] with added Nachwort), Studia philosophica, vol. 1 (1936), pp. 261–405.Google Scholar
[9]Tarski, Alfred, The concept of truth in formalized languages (English translation of [8]), Logic, semantics, metamathematics, Papers from 1923 to 1938, by Alfred Tarski, London, 1956, pp. 152–278.Google Scholar
[10]Weyl, Hermann, Das Kontinuum, Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig, 1918.CrossRefGoogle Scholar
[11]Weyl, Hermann, Der circulus vitiosus in der heutigen Begründung der Analysis, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 28 (1919), pp. 85–92.Google Scholar
[12]Whitehead, A. N. and Russell, Bertrand, Principia mathematica (three volumes), Cambridge, 1910–1913; second edition, Cambridge, 1925–1927.Google Scholar