Notes / Vagueness
The grain that never matters
Working note — a draft kept in the open. Tell me where it’s wrong.
Here is the oldest puzzle I know that is also a true report. One grain of wheat is not a heap. If some collection of grains is not a heap, adding one more grain does not make it a heap — what could a single grain do? By those two premises there are no heaps, however many grains you pour. And yet I am looking at one.
The reasoning is valid and the conclusion is absurd, so a premise is false; the whole literature is an argument about which one, and at what price. I find it useful to lay the options out as a ledger, because each saves something and is billed for it.
Three ways to pay
Deny the inductive premise — epistemicism. Williamson holds that “heap” has a perfectly sharp boundary: there is a precise number n such that n grains are not a heap and n+1 grains are. We cannot know where it falls, and our ignorance is principled, grounded in the way knowledge requires a margin for error. The reward is enormous: classical logic and bivalence survive untouched. The bill is a sharp line that nothing in our use of the word put there and nothing in the world could reveal. I admire the nerve of it and cannot make myself believe it.
There is a sharpest boundary; we just cannot locate it. the epistemicist’s wager
Deny a hidden assumption — supervaluationism. Kit Fine’s move is to admit there is no fact that settles the borderline cases. “Heap” can be made precise in many admissible ways; a sentence is true if it comes out true on every precisification, false if false on every one, and neither when the precisifications disagree. We keep the comforting result that “either it is a heap or it is not” is true — it holds on every sharpening — without having to say which disjunct holds. This is the view I am most tempted by, and its trouble is honest: higher-order vagueness. The line between the clear cases and the borderline cases is itself blurred, and the line between that blur and its surroundings, and so up. The penumbra has a penumbra.
Revise the logic — degree theories. Let truth come in degrees between 0 and 1, so that “this is a heap” can be half true. Elegant, and it captures the felt gradualness; but it tends to replace one sharp line with infinitely many (exactly where does 0.5 sit?), and assigning a real number to the heap-hood of a pile feels like answering a question nobody asked.
What the paradox is really reporting
I think the sorites is not a malfunction to be repaired but a description of how the predicate was made. We learn “heap,” “red,” “bald,” “reasonable” on clear cases, and we are then sent out to go on. Nothing in that training installed a boundary, because boundaries were never the point; the point was to be able to act, sort, and speak in the ordinary run of cases, which are not borderline. The tolerance premise — one grain never matters — is not a mistake we make about the word. It is something close to the meaning of the word, an expression of the fact that the practice was built to be insensitive to single grains.
This is where my pragmatism shows its hand. If meaning were a hidden core, the borderline case would be a failure to see the core clearly, and epistemicism would be the natural diagnosis: the line is there, our eyes are weak. But if meaning is use — a mastery of how to go on, answerable to others — then the penumbra is not ignorance of a line. It is the absence of one, and competence with a vague word includes knowing how to carry on across that absence: how to decide a borderline case when something turns on it, with reasons, knowing the reasons could have gone the other way.
Notice what that last clause smuggles in. “When something turns on it.” In the kitchen, nothing turns on whether the pile is a heap, so we never decide. In a courtroom, everything can turn on whether a structure is a “building” or a delay is “reasonable,” and a decision must be made anyway, today, with consequences. Same predicate, same penumbra; only now there is a person whose job is to extend the practice under pressure and to answer for the extension. That is the subject of the next note.
Touching: Fine (1975), Williamson Vagueness (1994), Sorensen (2001), Keefe (2000). See the canon §A. Continues in The skateboard at the gate.