It has been proposed that the law of nonsontradiction be revised to permit the simultaneous truth and falsity of the key sentences of the logical paradoxes, e.g., "This sentence is false". In an attempt to show to what extent this bizarre suggestion of inconsistent models or truth-value "gluts" is a coherent suggestion it is proved that a firs&order language for number theory can be semantically closed by having its own global truth predicate under some non-standard interpretation and thus that it actually can contain the Liar sentence. It is proved that in this interpretation the Lii sentence is both true and false, although not every sentence is.
To get a better solution to the logical paradoxes it has been proposed by Priest (1979) that we learn to live with logical inconsistency. The suggestion is that it is time to revise the law of non-contradiction and permit a few key sentences, such as "This sentence is false", to be simultaneously true and false. Priest attempts to provide a precise theory that would carry out his proposal as it applies to number theory, but he does not prove his claim that there are appropriate models of this theory. The present paper will prove that his language actually can be semantically closed by having its own global truth predicate under some non-standard interpretation and thus that it actually does contain the Liar sentence. In addition it will be proved that in this interpretation the Liar sentence is both true and false, although not every sentence is.
The following development of Priest's technical project is useful because it reveals to how great an extent the bizarre suggestion of inconsistent models can be shown to be a coherent suggestion. It is also interesting to see the similarity in structure between this semantics of so-called truth-value "gluts" and the semantics of truth-value "gaps" proposed by Kripke (1975). However, at the more practical level of choosing among competing solutions to the paradoxes, the cost of Priest's proposal is at least as great as Kripke's. Neither theory treats the Strengthened Liar sentence ("This sentence is not true") the way it treats the ordinary Liar. And, I would conjecture, the proof theory associated with Priest's semantics will necessarily turn out to be rather byzantine.