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Natural deduction based set theories: a new resolution of the old paradoxes

Published online by Cambridge University Press:  12 March 2014

Paul C. Gilmore
Paul C. Gilmore*
Affiliation:
Department of Computer Science, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada
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Abstract

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 393 - 411
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

The research reported on in this paper was supported by grants of the Natural Science and Engineering Research Council of Canada.

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