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Further results on infinite valued predicate logic

Published online by Cambridge University Press:  12 March 2014

L. P. Belluce
L. P. Belluce*
Affiliation:
University of California, Riverside
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Extract

In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

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Articles
Information
The Journal of Symbolic Logic , Volume 29 , Issue 2 , June 1964 , pp. 69 - 78
Copyright
Copyright © Association for Symbolic Logic 1964

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References

[1]Belluce, L. P. and Chang, C. C., A weak completeness theorem for infinite valued first order logic, this Journal, vol. 28 (1963), pp. 43–50.Google Scholar
[2]Chang, C. C., Theory of models of infinite valued logic. I, II, III. (abstracts), Notices of the American Mathematical Society, vol. 8 (1961), pp. 68–69.Google Scholar
[3]Chang, C. C., Logic with positive and negative truth values, in Acta Philosophica Fennica, fasc. 16 (1963), pp. 19–39.Google Scholar
[4]Chang, C. C., Algebraic analysis of many valued logics, Transactions of the American Mathematical Society, vol. 88 (1958), pp. 467–490.CrossRefGoogle Scholar
[5]Chang, C. C., A new proof of the completeness of the Łukasiewicz axioms, Transactions of the American Mathematical Society, vol. 93 (1959), pp. 74–80.Google Scholar
[6]McNaughton, Robert, A theorem about infinite valued sentential logic, this Journal, vol. 16 (1951), pp. 1–13.Google Scholar
[7]Mostowski, A., Axiomatizability of some many valued predicate calculi, Fundamenta mathematicae, vol. 50 (1961), pp. 165–190.CrossRefGoogle Scholar
[8]Scarpellini, B., Die Nicht-Axiomatisierbarkeit des unendlichwertigen Prädikatenkalkuls von Łukasiewicz, this Journal, vol. 27 (1962), pp. 159–170.Google Scholar
[9]Rose, Alan and Rosser, J. B., Fragments of many valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 1–53.CrossRefGoogle Scholar
[10]Rosser, J. B., Axiomatization of infinite valued logics, Logique et analyse, n.s. vol. 3 (1960), pp. 137–153.Google Scholar
[11]Hay, Louise S., An axiomatization of the infinitely many valued predicate calculus, M. S. Thesis, Cornell University (1958).Google Scholar
[12]Tarski, A., Sur les ensembles definissables de nombres reels. I, Fundamenta mathernaticae, vol. 17 (1931), pp. 210–239.CrossRefGoogle Scholar
[13]Robinson, A., Complete Theories, North Holland Publishing Company, Amsterdam, 1956.Google Scholar
[14]Rutledge, J. D., A preliminary investigation of the infinitely many-valued predicate calculus, Ph.D. Thesis, Cornell University, 1959.Google Scholar