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A problem of Rosser and Turquette1

Published online by Cambridge University Press:  12 March 2014

Angelo Margaris
Angelo Margaris*
Affiliation:
The Ohio State University
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Extract

In this paper an affirmative solution is given for the following problem in many-valued logic posed by Rosser and Turquette in [6] p. 110:

For every triple ‹s, t, m› with 1 ≤ s < t < m, is it possible to define a system of m-valued logic which satisfies the following conditions? I. Every statement which always takes values ≦ s is provable. II. No statement which ever takes a value > t is provable. III. Of those statements which always take values ≤ t and sometimes take a value > s, some are provable and some are not.

We state now, temporarily deferring comment, a more demanding alternative to Condition III: III′. For every truth value k such that s < k ≦ t, there are statements P and Q such that each takes the value k at least once, but never takes a value > k, and P is provable and Q is not.

In § 2 a solution is given on the level of the statement calculus. The remainder of the present section is devoted to some negative results which serve to clarify the problem, and one positive result which shall cause us to replace Condition III by III′. We confine our attention to the statement calculus.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 3 , September 1958 , pp. 271 - 279
Copyright
Copyright © Association for Symbolic Logic 1958

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Footnotes

1

From a thesis in partial fulfillment of the requirements for the degree of Ph. D. in the Department of Mathematics of Cornell University. The author thanks Professor Use Novak Gál for her guidance and encouragement, and Professor J. Barkley Rosser for some valuable suggestions.

References

BIBLIOGRAPHY

[1]Church, A., Introduction to mathematical logic, vol. 1, Princeton (Princeton University Press), 1956.Google Scholar
[2]Henkin, L., Fragments of the propositional calculus this Journal, vol. 14 (1949), pp. 42–48.Google Scholar
[3]Kalmar, L., Über die Axiomatisierbarkeit des Aussagenkalküls, Acta scientiarum mathematicarum (Szeged), vol. 7, pp. 222–243.Google Scholar
[4]Łukasiewicz, J., The shortest axiom of the implicational calculus of propositions, Proceedings of the Royal Irish Academy, vol. 52 section A no. 3 (1948), pp. 25–33.Google Scholar
[5]Łukasiewicz, J. and Tarski, A., 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
[6]Rosser, J. B. and Turquette, A. R., Many-valued logics, Amsterdam (North Holland Pub. Co.), 1952.Google Scholar