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Independent Axioms for Infinite-Valued Logic1

Published online by Cambridge University Press:  12 March 2014

Atwell R. Turquette
Atwell R. Turquette*
Affiliation:
University of Illinois
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Extract

Recent interest in Łukasiewicz' Lℵ0, raises the question whether this axiomatic system can be simplified [2]. It is known that Łukasiewicz' fourth axiom CCCPQCQPCQP is dependent [4] p. 51. The axiomatic system resulting from deleting the fourth axiom from Lℵ0 will be shown to be “minimal” in the sense that the axioms and rules of inference are mutually independent; consequently, no further simplification of Lℵ0, is possible. By switching basic operators, however, simplified modifications of Lℵ0, can be constructed.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 3 , September 1963 , pp. 217 - 221
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

Portions of this paper were presented at the meeting of the International Congress of Mathematicians, August 1962, Stockholm, Sweden.

References

[1]Łukasiewicz, Jan, A system of modal logic, The journal of computing systems, vol. 1 no. 3 (1953), pp. 111–149.Google Scholar
[2] Alan Rose and Rosser, J. B., Fragments of many-valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 1–53.Google Scholar
[3]Rosser, J. B. and Turquette, A. R., Many-valued logics, Amsterdam (North Holland), 1952.Google Scholar
[4]Tarski, Alfred, Logic, semantics, metamathematics, translated by Woodger, J. H., Oxford (Clarendon), 1956.Google Scholar