Vagueness

Vague expressions, such as "tall," "red," "bald," "heap," "tadpole," and "child," possess borderline cases where it is unclear whether or not the predicate applies. Some people are borderline tall: not clearly tall and not clearly not tall. It seems that the unclarity here is not merely epistemic. There need be no fact of the matter about which we are ignorant: the borderline predications are indeterminate, neither true nor false, and are exceptions to the principle of bivalence (which is a key feature of classical LOGIC). Relatedly, vague predicates lack well-defined extensions. On a scale of heights, there is no sharp division between the tall people and the rest. If candidates for satisfying some vague F are arranged with spatial closeness indicating similarity, no sharp line can be drawn round the cases to which F applies. Vague predicates are thus naturally described as having fuzzy boundaries: but according to classical SEMANTICS, all predicates have well-defined extensions, again suggesting that a departure from classical conceptions is needed to accommodate vagueness.

Vague predicates are also susceptible to sorites paradoxes. Intuitively, a hundredth of an inch cannot make a difference as to whether a person counts as tall: such tiny variations, which cannot be discriminated by the naked eye (or even by everyday measurements), are just too small to matter. This insensitivity to imperceptible differences gives the term its everyday utility and seems part of what it is for "tall" to be a vague height term lacking sharp boundaries -- which suggests [S1] if X is tall, and Y is only a hundredth of an inch shorter than X, then Y is also tall. But imagine a line of men, starting with someone seven-foot tall, and each of the rest a hundredth of an inch shorter than the man in front of him. Repeated applications of [S1] as we move down the line imply that each man we encounter is tall, however far we continue. And this yields conclusions that are clearly false, for instance, that a man five feet high, reached after three thousand steps, still counts as tall. Similarly, there is the ancient example of the heap (Greek soros, hence the paradox's name). Plausibly, [S2] if X is a heap of sand, then the result Y of removing one grain will still be a heap. So take a heap and remove grains one by one; repeated applications of [S2] imply that even a solitary last grain must still count as a heap. Familiar ethical "slippery slope" arguments share the same sorites structure (see, e.g., Walton 1992 and Williams 1995).

Borderline-case vagueness must be distinguished from mere underspecificity (as in "X is an integer greater than thirty," which may be unhelpfully vague but has sharp boundaries). Nor is vagueness simple AMBIGUITY: "tadpole" has a univocal sense, though that sense does not determine a well-defined extension. Context-dependence is different again. Who counts as tall may vary with the intended comparison class; but fix on a definite context and "tall" will remain vague, with borderline cases and fuzzy boundaries.

A theory of vagueness must elucidate the logic and semantics of vagueness. The simplest approach argues that appearances are deceptive; we can retain classical logic and semantics for vague terms after all. On this epistemic view borderline case predications are either true or false, but we do not and cannot know which. The sorites paradox is avoided by denying principles like [S1]: there is a sharp divide between the tall men and the others in our series, though we are ignorant of where this boundary lies and so wrongly assume that it does not exist. For an ingenious defense of this initially surprising view, see Williamson (1994).

Competing theories reject classical logic and semantics. One option is to countenance nonclassical degrees of truth, introducing a whole spectrum of truth values from 0 to 1, with complete falsity as degree 0 and complete truth as degree 1 (see, e.g., Machina 1976). Borderline cases then each take some intermediate truth value, with "X is tall" gradually decreasing in truth value as we move down the sorites series. When Y is only a hundredth of an inch shorter than X, the claim "if X is tall, Y is also tall," which appears true, may actually be slightly less than completely true. Repeated application of the sorites principle [S1] then introduces an additional departure from the truth at each step, eventually reaching falsity.

The degree theory is typically associated with an infinite-valued logic or FUZZY LOGIC, and various versions have been proposed. It is normally assumed at least that, if V(P) is the value of P, then

V(not-P) = 1 - V(P)
V(P & Q) = minimum of V(P), V(Q).

But suppose Tek is taller than Tom, and V(Tek is tall) is 0.5, V(Tom is tall) = 0.4. Then, by the standard rules, V(Tom is tall and Tek is not) = 0.4. Such a result is arguably implausible. If Tek is taller than Tom, then is it not entirely ruled out that Tom should be tall and Tek not -- that is, should not V(Tom is tall and Tek is not) = 0? (see Chierchia and McConnell-Ginet 1990: Chapter 8, for more examples). However, not all degree theorists accept that the propositional connectives obey truth-functional rules (see Edgington 1997).

The other popular option is supervaluationism. The basic idea is to treat vagueness as a matter of semantic indecision, as if we have not settled which precise range of heights is to count as tall. A proposition involving "tall" is true (false) if it comes out true (false) on all the ways in which "tall" could be made precise (ways, that is, which preserve the truth-values of uncontentious cases of "X is tall"). A borderline case, "Tek is tall," will thus be neither true nor false, for it is true on some ways of making "tall" precise and false on others. But a classical tautology like "either Tek is tall or he is not tall" will still come out true because it remains true wherever a sharp boundary for "tall" is drawn. In this way, the supervaluationist adopts a nonclassical semantics while aiming to minimize divergence from classical logic (see Fine 1975). On any way of making "tall" totally precise, there will be some X who counts as tall when Y a hundredth of an inch shorter does not, making [S1] false. So because [S1] is false on each precisification, [S1] counts as false simpliciter. But note, nobody counts as the last tall man on every precisification of "tall," so supervaluationism avoids commitment to a sharp boundary.

Both the supervaluationist and the degree theorist can naturally accommodate a range of linguistic phenomena. For example, both can give semantic treatments of comparatives. The degree theorist may say that "X is taller than Y " counts as true just if "X is tall" is true to a greater degree than "Y is tall." The supervaluationist may say that the comparative claim holds if the ways of making "tall" precise which make Y count as tall are a proper subset of those that make X count as tall. Likewise, both theorists can deal with various modifiers: for example "X is quite tall" is true if "X is tall" has a sufficiently (but not too) high degree of truth, or alternatively if "X is tall" is true on sufficiently many (but not too many) precisifications of "tall." (See Kamp 1975; Zadeh 1975.)

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-- Peter Smith and Rosanna Keefe

References

Chierchia, G., and S. McConnell-Ginet. (1990). Meaning and Grammar. Cambridge, MA: MIT Press.

Edgington, D. (1997). Vagueness by degrees. In Keefe and Smith (1997a), pp. 294-316.

Fine, K. (1975). Vagueness, truth and logic. Synthese 30:265-300. Reprinted in Keefe and Smith (1997a).

Kamp, J. A. W. (1975). Two theories about adjectives. In E. Keenan, Ed., Formal Semantics of Natural Languages. Cambridge: Cambridge University Press, pp. 123-155.

Keefe, R., and P. Smith. (1997a). Vagueness: A Reader. Cambridge, MA: MIT Press.

Machina, K. F. (1976). Truth, belief and vagueness. Journal of Philosophical Logic 5:47-78. Reprinted in Keefe and Smith (1997a).

Walton, D. N. (1992). Slippery Slope Arguments. Oxford: Clarendon Press.

Williams, B. (1995). Which slopes are slippery? In Making Sense of Humanity. Cambridge: Cambridge University Press, pp. 213-223.

Williamson, T. (1994). Vagueness. London: Routledge.

Zadeh, L. A. (1975). Fuzzy logic and approximate reasoning. Synthese 30:407-428.

Vagueness

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