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Deducibility and many-valuedness

Published online by Cambridge University Press:  12 March 2014

D. J. Shoesmith  and
T. J. Smiley
D. J. Shoesmith
Affiliation:
Cambridge University, Cambridge, England
T. J. Smiley
Affiliation:
Cambridge University, Cambridge, England
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Extract

Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 4 , December 1971 , pp. 610 - 622
Copyright
Copyright © Association for Symbolic Logic 1972

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