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A weak completeness theorem for infinite valued first-order logic

Published online by Cambridge University Press:  12 March 2014

L. P. Belluce  and
C. C. Chang
L. P. Belluce
Affiliation:
University of California, Los Angeles
C. C. Chang
Affiliation:
University of California, Los Angeles
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Extract

This paper contains some results concerning the completeness of a first-order system of infinite valued logic

There are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 1 , March 1963 , pp. 43 - 50
Copyright
Copyright © Association for Symbolic Logic 1964

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References

[1]Belluce, L. P., Some remarks on the completeness of infinite valued predicate logic (abstract). Notices of the American Mathematical Society, vol. 7 ( 1960), p. 633.Google Scholar
[2]Belluce, L. P. and Chang, C. C., A weak completeness theorem for infinite valued predicate logic (abstract). Notices of the American Mathematical Society, vol. 7 (1960), pp. 632–633.Google Scholar
[3]Chang, C. C., Algebraic analysis of many valued logics, Transactions of the American Mathematical Society, vol. 88 (1958), pp. 467–490.CrossRefGoogle Scholar
[4]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
[5]Hay, Louise S., An axiomatization of the infinitely many-valued predicate calculus, M. A. Thesis, Cornell University, 1959.Google Scholar
[6]Mostowski, A., Axiomatizability of some many valued predicate calculi, Fundamenta mathematicae, vol. 50 (1961), pp. 165–190.CrossRefGoogle Scholar
[7]Rasiowa, H. and Sikorski, R., A proof of the completeness theorem of Godei, Fundamenta mathematicae, vol. 37 (1950), pp. 193–200.CrossRefGoogle Scholar
[8]Rosser, J. B., Axiomatization of infinite valued logics, Logique et analyse, n.s., vol. 3 (1960), pp. 137–153.Google Scholar
[9]Rutledge, Joseph D., A preliminary investigation of the infinitely many-valued predicate calculus, Ph. D. Thesis, Cornell University, 1959.Google Scholar
[10]Scarpellini, B., Die Nicht-Axiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz, this Journal,Google Scholar