Field's Paradox and Its Medieval Solution

History and Philosophy of Logic 31 (2):161-176 (2010)
Abstract
Hartry Field's revised logic for the theory of truth in his new book, Saving Truth from Paradox , seeking to preserve Tarski's T-scheme, does not admit a full theory of negation. In response, Crispin Wright proposed that the negation of a proposition is the proposition saying that some proposition inconsistent with the first is true. For this to work, we have to show that this proposition is entailed by any proposition incompatible with the first, that is, that it is the weakest proposition incompatible with the proposition whose negation it should be. To show that his proposal gave a full intuitionist theory of negation, Wright appealed to two principles, about incompatibility and entailment, and using them Field formulated a paradox of validity (or more precisely, of inconsistency). The medieval mathematician, theologian and logician, Thomas Bradwardine, writing in the fourteenth century, proposed a solution to the paradoxes of truth which does not require any revision of logic. The key principle behind Bradwardine's solution is a pluralist doctrine of meaning, or signification, that propositions can mean more than they explicitly say. In particular, he proposed that signification is closed under entailment. In light of this, Bradwardine revised the truth-rules, in particular, refining the T-scheme, so that a proposition is true only if everything that it signifies obtains. Thereby, he was able to show that any proposition which signifies that it itself is false, also signifies that it is true, and consequently is false and not true. I show that Bradwardine's solution is also able to deal with Field's paradox and others of a similar nature. Hence Field's logical revisions are unnecessary to save truth from paradox
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References found in this work BETA
M. H. Lob (1955). Solution of a Problem of Leon Henkin. Journal of Symbolic Logic 20 (2):115-118.

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Citations of this work BETA
Miroslav Hanke (2013). Implied-Meaning Analysis of the Currian Conditional. History and Philosophy of Logic 34 (4):367 - 380.
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