1.3.7. 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 presents similar problems to indexicality, the fact that sentences containing such terms may not have definite truth values even when the context is specified means that we will have to handle such sentences in a different way than we do sentences exhibiting ordinary forms of indexicality. We can understand entailments involving indexical terms—such as

—to hold because the propositions expressed by the two sentences are related in a certain way in every context of use. But we cannot understand the entailment

to hold for the same reason because the sentences involved may not express definite propositions in any context of use.

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 classes of things that the terms small and large are 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 entailment no matter how these delineations were specified. So, 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 are vague. This approach to vagueness is also somewhat analogous to our approach to presupposition, and it can also be thought of as a use of super-valuations (which were first used in this sort of way by David Lewis).

Ignoring vagueness of course means that we will ignore yet another important feature of language. The logical forms we will consider do not employ vague terms, so ignoring vagueness will not limit our ability to study them. But, because the relation of entailment does not recognize vagueness, it cannot be used to study certain ways of deriving information from things that are said. The accommodation of vague language can be analogous to accommodation of indexicality and can support analogous ways of deriving information from what is said. While This is hot will often be intended to provide information about whatever this refers to, it can also serve to calibrate judgments of hotness. That is, it can help someone to specify the significance of hot in a given context since it insures that the thing pointed to falls within (and, indeed, some distance within) the range of hot things on any delineation of that range that is allowed by the context.

And non-deductive inferences can reflect these ways of deriving information. The fact that we derive information in this way can explain a traditional logical puzzle known as the sorites paradox (or paradox of the heap, from a particular example trading on the vagueness of heap). The argument

is not deductively valid because the things refered to by this and that could well fall on opposite sides of a delineation. But it seems like a reasonable argument; and, if we suppose that we accommodate vague language by considering only delineations on which what has been said is true and not just barely true, the conclusion will be true on any delineation allowed by a context that results from accommodating the premise. The paradox comes by imagining a series of things, with each successive thing asserted to be only a little cooler than the one before with the last clearly not hot. Each step in the series could be justified by an argument like the one above, but the final result seems unacceptable.

This result would not be surprising if we understand the displayed argument to be the result of accomodation. Suppose first that we attempted to collect all the steps in the series into a single argument.*

This would not be reasonable because accommodating the first premise need not place the temperature assigned to A far enough from allowable delineations to support the truth of the conclusion.

On the other hand suppose we were faced with a series of arguments

one for each successive pair of terms in the series. If we really were willing to accommodate the premise at each stage, we would end up accepting the final conclusion; but the allowed delineations of hot would have shifted also at each stage and the final conclusion would end up acceptable.

Someone who really refused to accept the final conclusion would refuse to accommodate the premise of one of the arguments along the way. The paradoxical inference can seem to be supported if this way of extracting information from an assertion is thought to behave just like deductive inference by allowing us to link together inferences that are good individually. That is, the sorites paradox shows us that the non-deductive relation associated with this way of deriving information is not transitive.