Vagueness

How many grains must you take away before a heap stops being a heap?

Here is a heap of sand. Take one grain away; still a heap, obviously. Take another; still a heap. Nobody thinks a single grain is what stands between a heap and a non-heap. Yet repeat the move enough times and you are looking at one grain, and one grain is not a heap.

So two things you are certain of cannot both be true. That is the sorites paradox, named from the Greek for heap and credited to Eubulides in the fourth century BC. It is not a trick of wording. It runs on tall, bald, rich, child, reasonable force, and on almost every word you used today.

Lay the argument out so we can see which part to attack. The first premise says ten thousand grains make a heap. The second — call it the tolerance principle — says that if n grains make a heap, so do n minus one. Apply the second ten thousand times and you conclude that one grain is a heap, which is false. A valid argument with a false conclusion has a false premise, so one of the two has to go, or the logic does.

The first premise is not in doubt. So attention falls on tolerance: is there really no single grain whose removal makes the difference?

Try denying it. Then there is some exact number n such that n grains are a heap and n minus one is not. This sounds absurd — what could make grain 3,417 the one that matters? — but absurdity is not refutation. This is epistemicism, defended most fully by Timothy Williamson. On this view vague predicates have sharp boundaries and classical logic is untouched; vagueness amounts to irremediable ignorance of where the boundary lies. Why the ignorance? Because a word's meaning is fixed by the whole vast pattern of how a community uses it, which nobody can survey; and because knowledge requires a margin for error, so any belief about a case that close to the line would be true only by luck. The cost is obvious: it asks you to accept a fact — the exact grain — that nothing could ever determine or reveal.

Try instead keeping the intuition that borderline cases are genuinely unsettled. Supervaluationism, developed by Kit Fine, says a vague predicate is compatible with many admissible ways of making it precise — draw the line at 3,000, at 5,000, at 8,417, any of which respects the clear cases. A statement is true if it comes out true on every admissible precisification, false if false on every one, and neither if they disagree. So "this borderline pile is a heap" is neither true nor false. Bivalence goes; but the elegant part is that classical tautologies survive. "Either it is a heap or it is not" is true, because it holds on each precisification, even though neither disjunct is. And tolerance is simply false — on each precisification there is a boundary — while no particular number is the boundary.

Or attack the sharpness of truth itself. Degree theories treat truth as a matter of degree, so "this is a heap" might be true to degree 0.6, and tolerance is very nearly true rather than flatly true. The paradox dissolves as a long chain of almost-valid steps whose small errors accumulate. The standard objection is that this smuggles precision back in through the window — what could make the degree exactly 0.6 rather than 0.61? — and that it strains on connected pairs: if you are a borderline case of tall, "you are tall or you are not tall" should be plainly true, not half-true.

There is a further twist all three must handle. Suppose we admit borderline cases. Is there a sharp boundary between the clear heaps and the borderline ones? Evidently not — "borderline" is itself vague, and the whole problem regenerates one level up. That is higher-order vagueness, and it is a standing tax on any theory that tries to solve the paradox by adding a middle category.

The three responses disagree about different things. Epistemicism keeps classical logic and bivalence and locates the problem in us. Supervaluationism keeps classical tautologies but drops bivalence, locating the problem in the language's failure to decide. Degree theories drop both, locating it in truth itself. Reject tolerance and you owe an account of the boundary; keep it and you owe an account of the logic.

It is also worth resisting the tempting thought that vagueness is a defect to be legislated away. Precise substitutes exist — we could stipulate 1,000 grains — but a stipulated boundary buys sharpness at the cost of arbitrariness, and pushes the disputes to the edge rather than removing them. That is why law so often prefers a standard ("reasonable care") to a rule, and then assigns someone to draw the line case by case.

Which is the payoff. Vague predicates work. Communication needs agreement about the clear cases and tolerance in the middle, not a shared boundary — and since almost every use falls in the clear region, nobody ever discovers that no boundary was agreed. The paradox is a puzzle about logic and meaning, not a malfunction in ordinary talk.

• How courts and regulators handle sorites-shaped disputes, where somebody must draw the line the language never did.
• Whether some vagueness is in the world itself rather than the language — indeterminacy about the boundary of a cloud or a mountain.

Every term the collection defines is gathered in the glossary.