Meaning of the word "the"

My memory of this, which I picked up studying linguistics (though I don't know where it originated), is that statements have a set of presuppositions, which the speaker asserts to be true and noncontroversial by using them, and then additionally have a truth value for the main proposition only if the presuppositions all hold. There's a presupposition for every noun phrase introduced with "the" that an appropriate referent exists; and presuppositions may also be introduced in a variety of other ways, such as by embedding statements in certain ways ("she knew that X" presupposes X and asserts her knowledge of X), with type compatibility (using a pronoun presupposes that the context has one most salient person of the appropriate gender), with "too", and in a number of other ways.

What you call a sentence for which presuppositions fail, such as "John's wife has red hair" when John is unmarried, is a matter of definition. The presupposition failed and so the supposition cannot be evaluated, and there is no further fact of the matter. It could be false, or meaningless, or even ungrammatical depending on how you define terms, and arguing over those definitions is quite unilluminating.

Hmm. If "the wife of John" is the null set, it seems false, rather than meaningless, to predicate "red hair" on the null set.