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Simplified axioms for many-valued quantification theory1

Published online by Cambridge University Press:  12 March 2014

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

The basic aim of the present paper is to simplify the axioms for many-valued quantification theory which were developed by Rosser and Turquette in Many-valued logics ([10], pp. 33–34 and pp. 63–64). The simplification which is achieved results not only from a reduction in the number of axiom schemes and rules of inference, but also by obtaining greater formal similarity to some elegant axioms for standard 2-valued quantification theory. This result will not be accomplished by establishing the formal interdeducibility between our simplified axioms and the Rosser-Turquette set. On the contrary, it will be shown that these two sets of axioms are not formally interdeducible.

What will be shown is that whenever the Rosser-Turquette axioms are used to define a plausible set of statements, this set of statements can be defined using our simplified axioms. The exact meaning of “plausible” will become clear from what follows, so for the moment it will be sufficient to remark that a set of statements is plausible if and only if each statement of the set is analytic in the sense that the statements take designated truth-values exclusively. In general, a Rosser-Turquette set of axioms would be used to define a set of plausible statements, and in such a case our simplified axioms could be used to achieve the same result.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 2 , June 1958 , pp. 139 - 148
Copyright
Copyright © Association for Symbolic Logic 1958

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Footnotes

1

The author of the present paper would like to express his appreciation to Professor J. Barkley Rosser and Professor Alonzo Church for reading the manuscript and offering valuable suggestions. Portions of this paper were read at the meeting of the Western Division of the American Philosophical Association, University of Chicago, May 3, 1957.

References

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