Deducibility and many-valuedness

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Abstract
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.

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