Vagueness

3. Theories

It has been suggested that a proper understanding of vagueness requires the admission that truth and set membership come in degrees, rather than being ‘all or nothing’ (see Zadeh 1965, Goguen 1969). On this approach, real numbers in the interval from 0 to 1 are typically taken to be truth-values, with 1 being fully fledged truth and 0 being fully fledged falsity. The same numbers are assigned to the degrees to which objects belong to sets. So, for example, if Herbert is clearly bald, then the sentence

is assigned the value 1, and Herbert is taken to belong to the set of bald men to degree 1. But if Herbert is a borderline case then (7) is assigned some number less than 1, say, 0.6, and the degree of Herbert’s membership in the set of bald men is now taken to be 0.6 too. In general, a singular sentence ‘Fa’ is treated as having the truth-value n, where 0 ≤n ≤1, if and only if the referent of ‘a’ belongs to the set of Fs to degree n (see Fuzzy logic).

A consequence of this approach is that vague predicates do not sharply divide the world into those things to which they apply and those things to which they do not. Rather, vague predicates apply to objects to varying degrees. As a man loses hair he becomes more and more bald; so the predicate ‘is bald’ applies to him more and more and the assertion that he is bald increases gradually in its degree of truth.

One serious objection to this view is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees-of-truth theorist, comes in precise degrees, as does predicate application and truth. The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to how we think of everyday vagueness (as noted in §1).

It is often supposed by philosophers that vagueness resides in language. One popular view here is that vagueness itself is a matter of semantic indecision (for example, Lewis 1986). On this view, all external objects, properties and relations are precise. Each vague linguistic term has a meaning which can be made precise in a whole range of different permissible ways. For example, ‘over 70 years of age’ is one acceptable way of making precise the meaning of the term ‘old’. But other equally good ways are ‘over 72’ and ‘over 74’. The rules that govern the use of the term ‘old’ are not specific enough for us to be able to choose non-arbitrarily some one precise term as capturing its meaning. When the term ‘old’ was first introduced, no decision was made to link it exclusively with one particular precise property. Instead it was applied widely in a range of cases, thereby acquiring a vague meaning. And what is true here for ‘old’ is true mutatis mutandis for all other vague terms.

Now if it is indeed true that there are no vague properties expressed by vague predicates, but instead ranges of precise properties, then, under some ways of making precise a vague predicate, it will definitely apply to a given object (in virtue of the object having the appropriate precise property) and, under other ways, it will not. Suppose that Alfred is 67. If ‘old’ is sharpened to mean ‘over 70’ then Alfred is not old: he lacks the precise property of being over 70. But if ‘old’ is sharpened to mean ‘over 65’, then Alfred is old – this time he has the right precise property. Is it really true that Alfred is old, then? If these two ways of sharpening the term ‘old’ are equally acceptable – if they both express precise properties falling in the range associated with ‘old’ – then the natural answer is that it is neither true nor false that Alfred is old. He is a borderline case. So, there really is no definite age at which Alfred becomes old: the boundary between being old and not being old is not sharp. By contrast, if Alfred is 97 years of age, then, under any acceptable way of making ‘old’ precise, he will count as old. So here it is unquestionably true that Alfred is old. And if he is 21, then, no matter which acceptable precisification we choose for ‘old’, he will not count as old. The claim that Alfred is old is false.

The thesis that vagueness is semantic indecision thus leads straightforwardly to the following claims. Vague sentences have three possible truth-values: true, false and ‘indefinite’ (neither true nor false). A vague sentence is to be counted as true if it comes out true under all acceptable ways of making precise its component vague terms; as false if it comes out false in every such case; and as neither true nor false if it comes out true under some ways and false under others. These claims entail that the law of excluded middle (LEM) remains true for vague sentences. So, the semantic indecision theory of vagueness provides us with a classical conception of reality (all that there is in the world is precise), a safe haven for vagueness in the linguistic realm and a conservative attitude towards the retention of LEM.

There are grave difficulties, however. In particular, it is far from clear that the semantic indecision approach fares any better than the degrees-of-truth view with respect to sorites paradoxes and robust or resilient vagueness. Another objection concerns LEM. The objection is that

to take one instance of LEM, should not be counted as true if Herbert is a borderline bald man. For if (8) is true then either Herbert is bald or he is not bald. If this is the case then the question, ‘Well, which is he then?’, must surely have an answer; that is, if (8) is true then precisely one of the disjuncts in (8) must be true. So ‘Herbert is bald’ must be either true or false. This runs contrary to the assumption that Herbert is a borderline bald man. So, (8) is not true.

One common feature of both the standard degrees-of-truth approach and the semantic indecision theory is that they employ precise metalanguages in stating the truth-conditions for vague object language sentences (a metalanguage is used to talk about the object language). It is this feature which, in my view, dooms them to failure. A central aspect of the resilient vagueness of ordinary terms is that, in sorites sequences, there simply is no determinate fact of the matter about the transition from true to some other value. Any attempt to state truth-conditions for vague discourse in precise language will inevitably fall foul of this fact.

Another possible approach, then, is to start out using a vague metalanguage, which mirrors in its vagueness the object language. One proposal along these lines is based on the three truth-values, true, false and indefinite. Just as sentences in the object language can be indefinite, so too can sentences in the metalanguage (and indeed in all the higher metalanguages). If this approach is to avoid the sorites paradoxes, it is essential that it not be true that every object language sentence is true or false or indefinite. For this would create sharp dividing lines. But neither can it be false that every such sentence is true or false or indefinite. For this would require further truth-values. Instead, it has been argued that the above generalization about object language sentences must be indefinite. This permits us to hold that the claim that there is a last true sentence, followed by a first indefinite one, in a sorites sequence is indefinite. And if this is the case then sorites arguments cannot be sound (see Tye 1990).

There are many philosophers who would reject all three of the alternative proposals sketched so far. The fourth and final approach I shall mention is the epistemic view (Sorensen 1988, Williamson 1994). According to this position, vagueness is a kind of ignorance. The nonlinguistic world is precise. Standard logic holds even for vague discourse. Every molecule is either inside or outside Everest, for example. We just do not know where the boundaries lie. Likewise, there is always a single hair, the addition of which would turn a bald man into a man who is not bald, even though we cannot say which hair this is. Our sensory and conceptual mechanisms are simply not equipped to make the necessary fine-grained discriminations. Sorites arguments, then, rest upon a false premise.

This view is pleasingly straightforward. Unfortunately, it seems counterintuitive. It denies outright the existence of robust or resilient vagueness. Moreover, it seems to misconceive borderline cases. Our ordinary concept of a borderline case is the concept of a case that is neither one thing nor the other. ‘Definitely’ here does not seem to mean ‘known’ or ‘knowably’.

So, whichever way we turn, we quickly become enmeshed in difficulties. Of all the philosophical mysteries, vagueness is one of the most perplexing.