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An extension of a theorem of margaris

Published online by Cambridge University Press:  12 March 2014

Alan Rose
Alan Rose*
Affiliation:
The University, Nottingham, England
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Extract

Margaris has shown1 that for every triple 〈s,t,m〉 of integers such that 1 ≦ s < t < m it is possible to construct a formalisation of an m-valued propositional calculus satisfying the following conditions: I. Every statement which takes only truth-values belonging to the set {1, …,s} is provable. II. Every provable statement takes only truth-values belonging to the set {1, …, t}. III. There exist statements Pk, Qk which take only truth-values belonging to the set {1, …, k) and neither of which takes only truth-values belonging to the set {1, …, k—1} such that Pk is provable and Qk is unprovable (k = s+1, …, t). The systems of Margaris are all functionally incomplete and he appears to suggest2 that it is impossible to construct functionally complete systems having the required properties.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 3 , September 1960 , pp. 209 - 211
Copyright
Copyright © Association for Symbolic Logic 1960

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References

1 Margaris, Angelo, A problem of Rosser and Turquette, this Journal, vol. 23 (1958), pp. 271–279.Google Scholar

2 Op. cit., last paragraph of §1.

3 Rose, Alan and Rosser, J. Barkley, Fragments of many-valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 1–53.CrossRefGoogle Scholar

4 Łukasiewicz, Jan and Tarski, Alfred, Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 30–50.Google Scholar

5 Rosser, J. B. and Turquette, A. R., Many-valued Logics, Amsterdam, 1952, p. 17.Google Scholar

6 Słupecki, Jerzy, Der volle dreiwertige Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 29 (1936), pp. 9–11.Google Scholar

7 Op. cit., p. 25.