Issues of vagueness

Some people, like 6′ 7″ Gina Biggerly, are just plain tall. Other people, like 4′ 7″ Tina Littleton, are just as plainly not tall. But now consider Mary Middleford, who is 5′ 7″. Is she tall? Well, kind of, but not really – certainly not as clearly as Gina is tall. If Mary Middleford is kind of but not really tall, is the sentence Mary Middleford is tall true? No. Nor is the sentence false. The sentence Mary Middleford is tall is neither true nor false. This is a counterexample to the Principle of Bivalence, which states that every declarative sentence is either true, like the sentence Gina Biggerly is tall, or false, like the sentence Tina Littleton is tall (bivalence means having two values). The counterexample arises because the predicate tall is vague: in addition to the people to whom the predicate (clearly) applies or (clearly) fails to apply, there are people like Mary Middleford to whom the predicate neither clearly applies nor clearly fails to apply. Thus the predicate is true of some people, false of some other people, and neither true nor false of yet others. We call the latter people (or, perhaps more strictly, their heights) borderline or fringe cases of tallness.

Vague predicates contrast with precise ones, which admit of no borderline cases in their domain of application. The predicates that mathematicians typically use to classify numbers are precise.