Vagueness

2. Paradoxes and problems

The oldest puzzle of vagueness, which allegedly derives from Eubulides, is the paradox of the bald man. It goes as follows:

The conclusion is derived from the premises via ten thousand applications of the classical logical rules of modus ponens and universal instantiation, so this argument is valid. Now, premise (2) is certainly true and the conclusion, (4), is certainly false, so it is inferred that premise (3) is false. Therefore, there is an n such that a man with n hairs on his head is bald and a man with n + 1 is not. Therefore, the term ‘bald’ is precise, contrary to appearances. And what is true in this one case is true by parallel reasoning for any general term which is ordinarily classified as vague.

It is worth observing that the paradox of the bald man does not rely essentially on the use of a universal generalization. The same result can be generated by means of a sequence of (ten thousand) conditionals such as in (1a)–(1n), with (2) as a starting point. Since the conclusion, (4), is false, and (2) is true, then at least one of the conditionals must be false. So, given that (2) is true and that each conditional is either true or false, there must be an adjacent pair of conditionals such that the first is true and the second false. This runs directly counter to the idea that ‘bald’ is vague.

If the paradox of the bald man really does demonstrate that the term ‘bald’ is not vague, contrary to what we all ordinarily believe, then ‘bald’ cannot express a vague property. So, the property of being bald cannot itself be vague. Since the paradox of the bald man can be restated so as to apply to any vague general term, one overall conclusion which has been reached is that no general terms or properties are vague. This conclusion, of course, not only threatens the thesis that there are vague abstract objects but also challenges the idea that there is any such thing as vagueness in language at all, however it is understood.

There are also sorites arguments which attack the concrete objects of common sense, on the assumption that these objects have vague boundaries. These arguments, like the paradox of the bald man, can be stated without the use of a universal generalization.

There are other problems for the view that individual concrete objects are vague. One of these is directed against vague objects, conceived of as objects having indeterminate identities (Evans 1978). Suppose that ‘a’ and ‘b’ are (precise) singular terms for vague objects and that ‘a = b’ is indefinite in truth-value. Then, if we let ‘ $\nabla$ ’ symbolize ‘indefinitely’, the following is true:

(5) ascribes to b the property of being indefinitely identical with a. Now surely we have

and hence that a lacks the property of being indefinitely identical with a. By the principle that if object o and object o′ differ in a property, then they are not identical (the contrapositive of Leibniz’s Law; see Identity), it follows that a is not identical with b. Since, on the standard understanding of ‘indefinitely’, it cannot be the case both that it is indefinite whether a is identical with b and that a is not identical with b, there cannot be vague identities.

The deepest problems of vagueness, in my view, are the sorites paradoxes. And these arise whether or not there is vagueness in the nonlinguistic world.