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Vagueness: an introduction (sort of)

Fred Ablondi tells you Everything You Always Wanted To Know About Vagueness. But not quite.

The ‘Problem of Vagueness’ is, fittingly, not one precise problem, but several related ones. A vague predicate can be defined in terms of boundarylessness (i.e., lacking sharp boundaries): F is vague if there are cases in which it is not determinate whether some particular x is F or not. (Vague predicates are also often defined as having borderline cases.) For example, when does the successive removal of grains of salt make a heap of salt a nonheap? How short can a person be for an increase of 1/1000 of an inch in height to make her no longer short? Over the last 25 years or so, several issues relating to vagueness have received considerable attention from philosophers in the analytic tradition. In this brief introduction to the topic, I wish to discuss a few of them.

One feature of vague predicates which has been the subject of philosophers’ attentions is the fact that they fall prey to the sorites paradox. Like so much in contemporary philosophy, the sorites paradox can be traced back to the ancient Greeks; Eubulides is usually credited with what has been called ‘the problem of the heap’. Consider the following argument involving the vague predicate ‘is rich’:

1. A person with £5,000,000,000 is rich.
2. Someone does not go from being rich to being not rich by losing £1.
3. Therefore, someone with £4,999,999,999 is rich.
By repeated applications of 2, we get:
(C) Therefore, someone with £6 is rich.

Of course, we can reverse the process and generate a similar paradox involving the predicate ‘is poor’. (Other predicates frequently used in sorites examples are ‘is bald’ and ‘is tall’.) On the face of it, the premises are true, and imply the conclusion (C), yet (C) is clearly false. There have been several attempts to solve, or at least explain, the paradox, but none have met with universal approval.

Another problem which vagueness generates has to do with its origins: what is the source of vagueness? Here too there are many views, and, corresponding to these, we encounter many ‘solutions’ to the problems to which vagueness gives rise. Some would say vagueness is nothing more than a result of imprecision in our language. On this view, a precisification of language would eliminate vagueness. Critics of this position point out that such a precisification, were it even possible, would eliminate many of the thing we value most in our world. For example, ‘goodness’, ‘justice’, and ‘beauty’ are all vague terms, as are terms denoting concrete objects, such as ‘pebble’ and ‘conscious being’. Still, some philosophers are comfortable taking this to ‘nihilistic’ conclusions, denying, for example, that there are even such things as ‘heaps’.¹ Others,while still maintaining that vagueness is a purely semantic phenomena, ‘solve’ the problem by appealing to the concept of supervenience, claiming that vague expressions supervene on elements of a precise language.

Other philosophers, known as epistemists, agree that there is no ‘real’ vagueness in the world, but rather than attributing vagueness to our language, they hold it to be a result of a limitation of our knowing powers.² Consequently, there is no ‘solution’ to the problem of vagueness – it’s just something we have to learn to live with (and why should we humans think we can know everything, anyway?). For example, according to the epistemist, there is a pound amount such that if x has £n, then x is rich, but if x has £(n-1), then x is not rich. The selling point of this view is that it retains for classical logic the principle of bivalence (i.e., it is either true that x is F, or it is false that x is F). But aside from being quite counter-intuitive, inasmuch as it declares there is, for example, a one hair difference between a bald and a non-bald person, this view has the curious feature of being unable to say just what it is we are unable to know.³ Note too that neither the epistemic nor the semantic account deny the phenomena of vagueness; they are (rival) attempts to explain from where it comes.

Some philosophers, on the other hand, ‘bite the bullet’ and admit not only that vague sets and vague properties are real, but that there exist vague objects in the world. Frequently used examples of such objects include clouds, deserts, and Mt. Everest. For example, it is indeterminate exactly where a desert stops and starts. There are grains of sand which are definitely in the Sahara Desert, and grains of sand which are definitely not. But there are some grains of which it could neither be said of them that they are nor that they are not part of the Sahara Desert. Imagine a field on the edge of the Sahara Desert on which there are no grains of sand. If a strong wind blows several million grains onto the field, where now does the Desert begin? The difficulty answering this and similar questions with precision or exactness makes a strong case for the existence of vague objects.⁴

The late philosopher Gareth Evans argued against the possibility of vague objects on logical grounds (though there is, it should be noted, some debate about exactly what Evans was seeking to demonstrate). Imagine an object a about which it is indefinite whether a = b not due to any lack of knowledge on our parts, but because (we're assuming as our premise) the identity of a and b really is vauge. It is surely not the case that it is indefinite whether a = a. In fact, it is both determinate and true that a = a. Now by Leibniz’s law, if a = b, there can be no property which a possesses which b does not. So if a has the property of definitely being identical with a, and b does not – in fact, it has the property of being indefinitely identical with a – then we know that it is definite that a ≠ b. Therefore, there can be no vague objects. Evans’ argument has been both attacked and defended, though most philosophers today do not accept it.

Another issue receiving attention involves higher-order vagueness, that is, the question of whether ‘vague’ is itself vague. Of the borderline cases between the class of definitely tall men and the class of definitely non-tall men, aren’t some more borderline than others? In other words, of all those men who are neither clearly tall men nor clearly non-tall men, aren’t there those who are less clearly neither tall nor non-tall than others? Aren’t some regions of the ‘gray area’ grayer than others? If, as it seems, ‘vague’ is indeed vague, what are the implications for an account of vagueness? It would surely seem that any acceptable theory of vagueness would have to accommodate the existence of higher-order vagueness. Yet what does the existence of higher-order vagueness say for the hopes of producing a theory of vagueness which is able to say anything precise?

There is also the matter of the possible implications vagueness has for classical logic. Work in the latter has included the development of fuzzy logics, in which truth values range in real numbers from 1 (determinately, or absolutely true) to 0 (determinately, or absolutely false). For instance, the statement “This is a heap” said of a collection of eleven grains of salt might be assigned a truth value of 0.2 – though note: even this assignment is vague! Another approach is supervaluation, according to which a statement involving a vague term is true if and only if it is true on every precisification of that term, false if and only if it is false on every precisification, and neither true nor false if it is true on some and false on others. For example, it would be truly said of a collection of 1,000,000,000 grains of salt that it is a heap no matter where we might reasonably draw the line separating heaps from non-heaps, and it would likewise be false if said of two grains of salt that it was a heap, no matter where we might reasonably draw the line. With regard to a collection of 20 grains of salt, however, it would be neither true nor false to say of it “This is a heap,” since on some precisifications it would be true (e.g., if the line separating heaps from non-heaps were set at 15 grains), and on some it would be false (e.g., if the line were set at 25).

Vagueness may seem at this point to be a problem only a philosopher could love. But I would, in closing, ask the reader to consider the implications for ethics if it is granted that ‘good’ is vague, for biology, religion and ethics (and legislation) if ‘child’ is vague, and for philosophy of mind and our view of the person if ‘consciousness’ is vague.

(I would like to thank Emily Austin, John Churchill, and Roy Sorensen for their helpful comments on earlier drafts of this article.)

Fred Ablondi is an assistant professor at Hendrix College, Arkansas