Roy Sorensen

Roy Sorensen is a professor of philosophy at Dartmouth College in Hannover, NH.

An Argument for the Vagueness of Vague (1985)

In his article An Argument for the Vagueness of Vague [Analysis, Vol. 45, No. 3 (Jun., 1985), pp. 134-137 Published by: Blackwell Publishing on behalf of The Analysis Committee] Roy sets out the project to prove his eponymous goal by means of a series of "disjunctive predicates". Ultimately, Sorensen argues that the vagueness of 'vague' creates difficulties for those who claim that (the concept of?) vague predicates are incoherent, and for those who wish to deal with vagueness by a demarkation between vague and non-vague predicates. This latter does not mean replacing a vague with a non-vague predicate, but rather refers to a denial of higher-order vagueness.

He begins by the relatively famous sorites paradox:

(P1) The number zero is small.
(P2) If a number n is small, then n+1 is also small.
(C) 1-billion is small.

This argument, like most sorites, proceeds by a lot of applications of modus ponens. In principle, we could fill in the argument with the 1-billion implicit steps, but Wikipedia would not appreciate the humor. Sorensen notes that most--if not all--of the literature agrees that the defectiveness of the deduction is contained in the vagueness of the predicate 'small', but remains neutral about how to construe this vagueness.

Sorensen then sets out a now-famous further construction to prove that the predicate 'vague' is vague. In order to do this, we need to have a series of predicates, in which we are not sure which ones we should call 'vague' and which we should call 'precise'. To do this, we can produce a series of 'n-small' predicates. For example, '1-small' refers to all numbers which are less than one or small. Generally, an audience will agree that the numbers 1, 2, 3, 4, 5, and more are small. Hence '1-small' refers to 0 because 0<1; and to 1, 2, 3, 4, 5, and more because they are all small. '2-small', to numbers less than two or small. So it refers to 0 and 1 because 0<2 and 1<2; and it refers to 3, 4, 5, and more because they are small. And so forth. Eventually, we reach '1-billion-small'. An audience will again typically agree that this predicate refers to only and every number less than one-billion, because no number greater than one-billion is small.

The more formal definition is:

(Q1) The predicate '1-small' is vague.
(Q2) If the predicate 'n-small' is vague, then the predicate '(n+1)-small' is vague.
(C) The predicate '1-billion-small' is vague.

This is a paradox as well. Certainly the base-case, (Q1), is true. It refers to zero and all small numbers, but the idea of a small number is vague. So '1-small' is vague. Yet just as clearly, '1-billion-small' is not vague. It refers to every number less than one-billion, and every number that is small. But there are no small numbers that are one-billion or greater, so the predicate just refers to every number less than one-billion. So this is a clear predicate. But which predicate do we say is the "last" vague predicate in the series? That would hinge on which number is the "last" small number. Hence it is vague which predicates we should call "vague". QED.

Now suppose a philosopher holds that vague predicates are incoherent. As it turns out, the predicate 'vague' itself is vague, so the predicate 'vague' much also be incoherent. And so it turns out that these philosophers must abandon the use of the term 'vague', unless they are to use incoherent terms with glee. This is also problematic for philosophers like Bertrand Russell, who asserted that logic applies only to nonvague predicates.

References

External Links

* [http://www.dartmouth.edu/~rasoren/rasoren.html Sorensen's personal webpage]