THEORY-452: A thing with many equally good candidate boundaries can be counted as one in two ways that cover each other's failures: supervaluation makes 'there is one cat' true on every precisification though no candidate is definitely the cat, and counting by almost-identity, meaning extensive mereological overlap, makes it true enough where the candidates overlap almost completely

Source

David Lewis (1993), “Many, but almost one”, LIT-1015, in Ontology, Causality and Mind; read in full in NOTE-722 from the 1999 reprint (Papers in Metaphysics and Epistemology, pp. 164–182), with all ten footnotes.

What was actually shown

The problem. “Many surfaces are equally good candidates to be the boundary of the cloud.” Either all candidates count, and there are many clouds, or none does, and there are none: “We have many things or we have none, but anyway not the one thing we thought we had.” Geach’s 1001 cats are the same problem once the hairs are questionable rather than plucked.

Two rivals set aside. Constituter dualism is unparsimonious, and the constituters stay “all too cat-like”, which gives 1002 cats. Vague objects get no correct conception: each of three pictures replaces the vagueness with precision, and the precise candidates are still cat-like.

Supervaluation. Vagueness is semantic indecision. A sentence is super-true “if and only if it is true under all ways of making the unmade semantic decisions”. Each intended interpretation puts exactly one candidate in the extension of “cat”, so “there is just one cat” is super-true. The cost is granted, not removed: “Although it is super-true that something is a cat on the mat, there is nothing such that it is super-true of it that it is a cat on the mat.” The rule is a defeasible presumption that pragmatics can suspend.

Almost-identity. “The real opposite of identity is distinctness” in the sense of non-overlap. Overlap is a spectrum, credited to Armstrong. The candidates sit near the identity end, so “The cats are many, but almost one”, which is true for most contexts. Almost-identity is overlap, “not a relation of partial indiscernibility”. It “is not in general an equivalence relation. Many steps of almost-identity can take us from one thing to another thing that is entirely distinct from the first.” Its threshold is left undecided.

Each needs the other. Fred’s house with and without the garage overlap substantially but are not almost one, so supervaluation must decide. Definite descriptions (“The cat on the mat includes hair h17”) get their truth-value gap only from supervaluation: “It’s hard to see how else to get it.” In the other direction, “The supervaluation rule comes natural only when the alternative interpretations don’t differ too much”, which almost-identity secures.

What could have come out otherwise. One remedy could have handled every case alone. The garage case and the gappy description show that neither does, and that is the essay’s argument for combining them.

What this does not say

• That the problem arises everywhere. Footnote 6 grants that a more eligible referent would win by default. The problem bites only where “nature is gradual, no handy joint in nature picks out one of the c’s from all the rest”. Where there is a natural joint, the account is not needed.
• Where the threshold for “almost” lies. Lewis leaves it, and whether almost-identity is an equivalence among the actual candidates, “very sensibly” undecided. The account cannot count anything until a context supplies both.
• That almost-identity is closeness in some metric. It is shared parts. Two nearby subspaces in general position share only the zero vector, yet the angle between them is small. Borrowing the counting move does not borrow the relation (NOTE-722).
• That the cost of supervaluation is removed. Something is the cat and nothing is super-truly the cat. Lewis accepts this, he does not explain it away.
• That it holds without its assumptions. It assumes perdurance and an unrestricted supply of precise candidates (footnote 4). Those who deny either, such as van Inwagen, are pointed to, not answered.
• That it is evidence for the record’s units. The essay is about counting and reference in ordinary language. It has no metric, no dynamics and no threshold.

Connections

• The record’s units. NOTE-722 finds the map onto CLAIM-164 partly fair. For perturbed copies of one gap-isolated subspace, “almost one” fits, and the gap supplies the threshold Lewis leaves open. For vectors inside a degenerate cluster it does not, because they can be orthogonal; there supervaluation fits, and “names nothing” would be “refers indeterminately”. Both bearings are the record’s.
• QUESTION-035. The temporal version of the problem applies to the duration of a combined form, whose clearing of a threshold has no sharp end. Fred’s house is the nested-block case. NOTE-722 reads Lewis’s context as the role a stated target plays there, and says that resemblance is the record’s. No answers edge is claimed, because the essay supplies remedies for counting, not a criterion for the question.
• LIT-446 (Survival and Identity, NOTE-373) already counts by a relation other than identity and handles degrees by supervaluation, so this essay generalizes a move made there for persons.
• Fiat boundaries. Smith and Varzi (LIT-1024, THEORY-622) also hold that vagueness is a matter of demarcation, not of the material, and that every candidate aggregate is precise.