It is, of course, when our words describe the world that they are true. So for example, ‘Telly is bald’ was a true description of Aristotelis Savalas when he was a baby. (As he himself said, “We’re all born bald, baby.”) Now, Telly did not go from being a bald baby to not being bald by growing just a few hairs, because ‘bald’ has not got so precise a definition. So if, as seems possible, Telly did not suddenly grow a lot of hair, then he will only gradually have stopped being bald. There could, possibly, have been times when ‘Telly is bald’ was true, later times when ‘Telly is bald’ was not true, and times in between when something else was the case – or could there? If ‘Telly is bald’ was neither true nor untrue at those intermediate times, then ‘Telly is bald’ was not true and was true, which is ruled out by the meaning of ‘not’.
So, such intermediate times seem to be logically impossible. And yet, we can hardly know a priori that Telly suddenly grew a lot of hair. And while we can introduce new terms that are less vague than ‘bald’ – e.g. 100 hairs or less and you are bald101, otherwise you are not – that would hardly solve our problem with ‘bald’. So let us assume, for the sake of argument, that Telly stopped being bald gradually: What was going on at the intermediate times? Well, some of those around Telly may have been thinking of him as bald, while others thought of him as not bald. And the vagueness of ‘bald’ gives us no reason to think that any of them were wrong. But, Telly was certainly not very bald at such times, and nor was he clearly not bald, so why not think of him as having been about as bald as not? Were ‘is bald’ about as true as not of Telly, ‘Telly is bald’ would not so much not be true as be only about as true as not, and it would not so much not be untrue as be about as true as not. So, that would solve our problem.
We reason best with descriptions that are either true or else not true, but the words of natural languages are a little vague,1 so the two classical truth-values, ‘true’ and ‘false’,2 meet at a place – in logical space – where descriptions are described as well by ‘not true’ as by ‘true’. For another example, imagine a rough table-top being gently sanded flatter and flatter. Eventually it becomes flat enough to count as flat, in the usual contexts. But sanding away just a few scratches would hardly have flattened it, so the borderline between flat and not flat is more like a pencil line than a mathematical line. Our table-top will, briefly, be only vaguely flat, or about as flat as not. ‘Flat’ is not, in that sense, well defined: It is a vague predicate, not a definite predicate. But, there is a sense in which it is defined perfectly well: There are such things as tables, which are flat by design; and there are, similarly, bald men. Precisely redefining ‘flat’ and ‘bald’ – and ‘man’ and ‘table’ – in order to avoid the problem of vagueness would lose us some of our ability to refer to reality. Indeed, we would lose rather a lot of that basic function of language, because most of our words are to some extent vague.
This also solves such puzzles as the Sorites: We might suppose, for example, that the truth-value of ‘the table-top is flat’ could not change with the sanding away of a single scratch. If so, then gently sanding a rough table-top for even a very long time could not make true ‘the table-top is flat’. But, while ‘not flat’ is contradicted by ‘flat’, it is not necessarily contradicted by ‘about as flat as not’. So as the table-top begins to be about as flat as not, we would not be wrong to call it ‘not flat’. Our calls could change from ‘not flat’ to ‘about as flat as not’ in the blink of an eye, with no sanding at all. (Our original supposition is less plausible when there is a borderline truth-value.)
There seems to be a ubiquitous vagueness in natural language, but it is not really a problem. It is surprising, but only because it is so unproblematic that it usually goes unnoticed. Our words are defined as precisely as our purposes have required them to be, and the slight vagueness means that we can always make them more precise. When ‘Telly is bald’ becomes problematic, for example, ‘Telly is getting hairy’ will be more straightforwardly true. ‘Getting hairy’ is hardly less vague than ‘bald’, but its borderlines are in different places. We can usually move the borderlines out of the way, even though we cannot remove all the vagueness. And since we do not have to do much with descriptions that are about as true as not – other than identify them as needing to be replaced with truer descriptions – hence we need only adjoin ‘about as true as not’ to the classical truth-values. Indeed, we should only do that: The precision of formal logic is inapposite when we have left behind the sharp division between something being the case and it not being the case. A more formal definition could only be an inaccurate – if deceptively precise – mathematical model of the most natural definition.
Following Aristotle, the classical definitions are as follows. To say of what is the case that it is the case, or of what is not the case that it is not the case, that is to speak truly. And to say of what is the case that it is not the case, or of what is not the case that it is the case, that is to speak falsely. So an adequate adjunct could be: To say of what is about as much the case as not that it is the case, or that it is not the case, that is to say something that is about as true as not. A description that is much truer than not will be true enough to count as true, by definition of ‘much’, while one that is not much truer than not will be about as true as not by definition of ‘about’. And if we need to make sharper distinctions, then we need to avoid borderline cases and use classical logic.3
Now, descriptions are normally of other things, but self-description is allowed – e.g. ‘this is in English’ is a true self-description – so consider this example: This description is true. Let us call that self-description ‘T’ (for Truth-teller). T says only that T is true, so it is certainly possible for T to be true; but another possibility is that T is false, because if T was false then it would follow from the meaning of T (that T is true) only that T was not true. And since there is no more to T than that – since T does nothing but describe itself (as true) – hence there is no reason why T should be true rather than not true, or false rather than not false. So it would make sense were T about as true as not.
Furthermore, some self-descriptions are paradoxical if they are not about as true as not (the post below concerns the Liar Paradox).
1. Bertrand Russell, ‘Vagueness’, The Australasian Journal of Psychology and Philosophy 1 (1923), 84–92, reprinted in Rosanna Keefe and Peter Smith (eds.), Vagueness: A Reader (Cambridge, MA: MIT Press, 1997), 61–68. For the state of the art, see Richard Dietz and Sebastiano Moruzzi (eds.), Cuts and Clouds: Vagueness, Its Nature and its Logic (New York and Oxford: Oxford University Press, 2010).
2. ‘“X is Y” is false’ just means that X is not Y, so in classical logic, where either X is Y or else X is not Y, ‘false’ and ‘not true’ are interchangeable.
3. A good introduction to the mathematics of classical logic is Stewart Shapiro, Classical Logic.