1.3.6. Vagueness

One way of understanding vague terms is to suppose that their significance varies with the context of use but is not completely determined by it. The meaning of a word like small depends on the line to be drawn between what is and what is not small. This line is settled to some degree by features of the context of its use—whether the word appears in a discussion of molecules or of galaxies, for example—and some contexts will pin it down more precisely than others. But there is usually, and perhaps always, some indeterminacy remaining, and the class of things that count as small in a given context will have fuzzy edges.

Although the context dependence of vague terms means that vagueness is somewhat analogous to indexicality, the fact that sentences containing vague terms may not have definite truth values even when the context is specified means that we cannot handle such sentences in quite the same way as we do sentences exhibiting ordinary forms of indexicality. We can understand logical relations among sentences involving indexical terms—such as

—to hold even without specifying the context in which they are used because the propositions expressed by the two sentences are related in a certain way in every context of use. But we cannot understand the relation

to hold for the same reason because the sentences involved may not express fully definite propositions in any context of use. In most contexts small is not perfectly precise, and there will be some possibilities for the population of Crawfordsville that will put it in the gray area. That means that for such a context, we will not be able to say that Crawfordsville is small and Crawfordsville is not small have opposite truth values in every possible world because there will be some possibilities under which neither has a definite truth value.

Still, there is a way of extending our approach to indexicality to provide an approach to vagueness. In both cases we can understand deductive properties and relations to hold for sentences because of the propositions that would be expressed by the sentences if certain factors were specified. In the case of the first example above, the relevant factor, the time of utterance, is specified by any actual context of use. In the second example, the relevant factors are precise delineations of the class of things that the term small is true of. These delineations are not fully determined by an actual context of use, but we can still say that the propositions expressed by the sentences in the second example would represent a case of contradictory sentences no matter how these delineations were specified. If we limit consideration to logical properties and relations that hold no matter what delineation is chosen, we will be able to take the same attitude to delineations as we do to contexts of use. That is, just as we will always take for granted an unspecified context of use, we will take for granted but leave unspecified precise delineations of all vague terms. And that means that we will speak of sentences as if no terms were vague.

Of course, some terms are vague, so perfectly precise delineations of their application are not real features of at least some contexts of use. This means that our approach to vagueness has at least a somewhat different character than our approach to indexicality. How far different it is depends on aspects of vagueness that are controversial. If the second pair of sentences is taken to be equally as contradictory as the first in spite of their vagueness, then delineations are simply an artificial device used to capture real relations (just as the artificial assignment of coordinates can be used to capture real geographic relations). On the other hand, if the second pair of sentences is taken to be less than fully contradictory due to their vagueness, our use of delineations will count as an idealization (comparable to assuming the absence of friction) that might help simplify the study of logic but that will yield results that hold only with some qualification.