NOTE-722: Lewis — Many, but almost one

Contribution

A short essay in honour of Armstrong. It takes Unger’s problem of the many and Geach’s paradox of 1001 cats as one problem and solves it twice. One solution uses van Fraassen’s supervaluations. The other is a counting rule: count by almost-identity, which Lewis takes to be extensive mereological overlap. Lewis argues that the two are better combined than chosen between, and shows what each does that the other cannot. Along the way he rejects three rivals: the dualism of things and their constituting matter (Lowe), vague objects, and Geach’s relative identity.

Key insight

Identical against not identical is the wrong opposition. Identity is one end of a spectrum of overlap, and distinctness, meaning no shared part, is the other. Many candidates that differ only at their edges sit near the identity end. So “there is one cat” is almost true, and where the near-identity is not close enough, supervaluation over the undecided boundary still makes it super-true.

Assumptions

• Mereology and perdurance. Things are aggregates of parts, and persist by having temporal parts. Constitution is identity of temporal parts. The problem is set out in space, in time (“there are questionable temporal parts at the beginning of every human life”) and, for those who hold it, across worlds.
• Vagueness is semantic indecision, not a feature of objects: “Semantic indecision will suffice to explain the phenomenon of vagueness.” Footnote 4 adds a condition: there must be many precise candidates to be undecided between.
• Nature is gradual here. Lewis grants that “Some things are by their nature more eligible than others to be referents or objects of thought”, and that the more eligible referent wins by default. That does not help in this case, because “nature is gradual, no handy joint in nature picks out one of the c’s from all the rest” (footnote 6). The essay therefore covers only cases with no natural joint to settle the boundary.
• The goal of a cooperative speaker is super-truth, and the supervaluation rule is a defeasible presumption that pragmatics can suspend.

Key results

• The problem (The problem of the many).
  • “Many surfaces are equally good candidates to be the boundary of the cloud.” All candidates count, so there are many clouds, or none counts, so there are none: “We have many things or we have none, but anyway not the one thing we thought we had.”
  • The 1001 cats is “another instance of Unger’s problem of the many” once the hairs are questionable rather than plucked. Without that, temporal parts already dissolve Geach’s version.
• Two disqualifications rejected.
  • Constituters: unparsimonious, and the constituters are “all too cat-like”, so the result is 1002 cats.
  • Vague objects: Lewis has no correct conception of one. Three pictures (multiplicity, ignorance, fadeaway) each replace the vagueness with precision. The precise candidates are still cat-like, and one vague cat cannot “spoil their cathood”. Footnote 5 adds a point on Evans: a consistent theory of vague objects must take vague identity as sharing some but not all precisifications, not as a truth-value gap.
• Supervaluation (One of the many is a cat).
  • Exactly one candidate is the cat, and which one is arbitrary: “Which one deserves the name ‘cat’ is up to us.”
  • “Call a sentence 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, granted: “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.” This is the horse puzzle.
  • To the objection that the rule forbids even stating the problem: a “fanatical supervaluationism” would. The rule is a defeasible presumption, suspended when it would make nonsense of what is said. “What’s intolerable is to be without any good and natural sense in which there is only one cat.”
  • To the objection that indeterminate reference needs determinate reference first: the concept is fixed on imaginary examples set in sharper worlds.
• Relative identity rejected (The many are not different cats).
  • Geach’s “same cat” is a relation of partial indiscernibility. Lewis accepts counting by relations other than identity, and cites his own roads example from LIT-446. But adding the noun phrase does not settle the cats case as it does “same or different newspaper?” or “same or different issue?”. And Geach never relates “same cat” to identity simpliciter.
• Partial identity (The many are almost one).
  • “the real opposite of identity is distinctness: not distinctness in the sense of non-identity, but rather distinctness in the sense of non-overlap”. Overlap is a spectrum. Siamese twins sharing a finger are near the distinctness end, and any two cat candidates near the identity end. Credit to Armstrong (1978).
  • Number statements are translated with “almost identical” in place of “identical”: “The cats are many, but almost one. By a blameless approximation, we may say simply that there is one cat on the mat.” It is true for most contexts, and stricter standards are possible.
  • Almost-identity is overlap, “not a relation of partial indiscernibility”. No salient respect is guaranteed to be shared.
  • “the relation of almost-identity, in other words extensive overlap, 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.” Whether it is an equivalence among the actual cats “depends on the extent to which the cats differ, and on the threshold for almost-identity (and both of these are matters that we will, very sensibly, leave undecided).”
  • Footnote 9 counts triangles by non-overlap: four small triangles in a square with both diagonals.
• Combination (One solution too many?).
  • There are two kinds of intended interpretation, one-cat and many-cat, and context favours one kind. When attention to the candidates brings in the many-cat kind, almost-identity still gives a sense in which there is one cat.
  • Almost-identity helps supervaluation: “The supervaluation rule comes natural only when the alternative interpretations don’t differ too much.” The ‘bank’ homonym is the contrast.
  • Supervaluation helps almost-identity in two ways. First, Fred’s house with and without the garage overlap substantially but not enough: “we cannot really say they’re almost identical. So likewise we cannot say that the two houses are almost one.” Second, definite descriptions: “The cat on the mat includes hair h17” should be gappy, and relaxed Russellian translations split, one true and one false. “Whichever way we go, supervaluations give us the gappiness we want. It’s hard to see how else to get it.”

Claims

id	claim	strength	support
C1	The problem of the many is everywhere: every thing has questionable parts, spatial and temporal	moderate	examples (cloud, nail, cathode, organism, person’s beginning and end); granted on gradualness of nature
C2	Constituter dualism and vague objects fail to solve it	moderate, argued	parsimony; the candidates stay “all too cat-like”; no correct conception of a vague object
C3	Supervaluation solves it: “there is one cat” is super-true though no candidate is super-truly the cat	strong as a formal move; the cost is granted, not removed	van Fraassen’s method; replies to two objections (Kripke, Strauss)
C4	Counting by almost-identity, meaning extensive overlap, solves it where the candidates overlap almost completely	moderate	the overlap spectrum, credited to Armstrong; true enough “for most contexts”
C5	Almost-identity is not in general transitive, and its threshold is left undecided	strong (a logical point)	sorites by overlap; stated, not formalized
C6	Neither solution suffices alone; combined, each covers the other’s failure	moderate	Fred’s house (overlap too small for almost-identity); definite descriptions (only supervaluation gives the gap); ‘bank’ (supervaluation unnatural when interpretations differ much)
C7	Geach’s relative identity is not the right account of the cats	moderate	the noun phrase does not change hesitancy; no link to identity simpliciter

Concepts

• Problem of the many. Many aggregates with equal claim to be one thing, because its outlying parts are questionable.
• Questionable part. Neither definitely a part nor definitely not.
• Semantic indecision. We never decided exactly which candidate a word applies to. This is Lewis’s whole account of vagueness.
• Super-true / super-false / super-truth-value gap. True under all, false under all, or true under some and false under others of the ways of making the unmade semantic decisions.
• Partial identity. Overlap: having some parts in common and some not.
• Almost-identity. Extensive overlap. Not similarity, and not in general an equivalence relation.
• Distinctness (Lewis’s sense). Non-overlap, not non-identity. The real opposite of identity.
• Eligibility. A natural referent’s default claim to a word when usage leaves it open. Footnote 6 only, and set aside for this problem.

Connections

• Unger (1980; LIT-1020) is the problem’s source. Geach (1980) gives the cats and the relative-identity solution. Lowe (1982) gives the constituter solution. Armstrong (1978) gives the overlap spectrum, and the essay honours him.
• LIT-446 (Survival and Identity; NOTE-373) is cited for counting by identity-along-a-path, the merged roads. That paper already counts by a relation other than identity and handles degrees by supervaluation, so this essay generalizes a move made there for persons. The record reads both from the same page.
• Eligibility points to “Putnam’s Paradox” (Lewis 1984), not held here.

Bearing on the record

The check the LIT set: does “almost one” fairly map onto CLAIM-164’s gap-isolated subspace, or is it only a resemblance? Partly fair, and only for one reading of “the many”. It is unfair for the other. The fit depends on which things are taken to be the many candidates.

• Fair: the many are the perturbed or estimated copies of one subspace. Different noise draws, or partitions that differ in a few marginal variables, give many subspaces, none exactly the true one. By Davis–Kahan they lie within about ‖δT‖/δ of it, and so within about twice that of each other. That is Lewis’s situation: many candidates, differing a little at the edges, almost one. The gap then supplies two things Lewis leaves out. It is a threshold for “almost”, which Lewis “very sensibly” leaves undecided. And because every candidate is anchored to one subspace, there is a bound on how far apart any two can be, so the chain of almost-identities that worries Lewis cannot drift to something entirely distinct. That bounded chain is the record’s addition, not Lewis’s.
• Not fair: the many are the vectors inside a degenerate cluster. CLAIM-164’s degenerate case is that “the k-th vector” names nothing, because any rotation within V is an equally good basis. These candidates are not almost identical. Two of them can be orthogonal, and the re-run’s 86° maximum drift shows they can be far apart. The almost-one route does not apply. Two other parts of the essay fit this case better.
  • Supervaluation. Over the choice of basis, V has a top direction is super-true and v is the top direction is gappy, Lewis’s horse puzzle. On this reading CLAIM-164’s “names nothing” is too strong. Lewis would say the term refers indeterminately, not that it fails to refer. That is a wording point, and it does not touch the mathematics.
  • Naming a different object. The claim’s real move is to make V the referent instead of any vector. That is the disqualification route (“perhaps it is something else altogether, and none of the many are cats”), which Lewis rejects for constituters and vague objects because the candidates stay “all too cat-like”. The objection does not transfer. A single vector is not V-like: it has the wrong dimension. So the record’s route survives Lewis’s objection, though not for any reason he gives.
• Overlap is not angle. Lewis’s almost-identity is mereological overlap. Two nearby subspaces in general position share only the zero vector, yet the angle between them is small. The record’s “almost” is a metric on subspaces, not shared parts. So the mapping keeps Lewis’s logic, counting by a relation weaker than identity, and drops his definition of that relation. Citing Lewis for the metric would be wrong.
• The best point of contact is footnote 6, which the LIT did not name. Lewis says the problem bites because “no handy joint in nature picks out one of the c’s”. In CLAIM-164’s terms a spectral gap is such a joint: a natural boundary that makes one candidate the more eligible referent. So the gap does not illustrate “almost one”. It removes the problem at the level it isolates, and leaves the problem of the many inside the cluster, and at the edge wherever the gap is too narrow for CLAIM-154’s threshold. This bearing is the record’s.

For QUESTION-035 and CLAIM-157.

• Duration. Lewis’s temporal version applies directly to CLAIM-157’s duration clause (“lasts as long as that block persists”). A couple (CASE-046) does not stop being one system at an instant. The block’s clearing of τ fades, so there are many candidate end-times, and the same goes for start-times. Lewis gives the two remedies. Either supervaluate over admissible thresholds, so the song’s middle is definitely one system and its last bars gappy, or count the many candidate durations as almost one. CLAIM-157 does not yet say which, or that it needs either.
• Fred’s house is the nested-block case. Where a block and a block plus a neighbour both clear the threshold, the candidates differ by a whole member, the garage, and are not almost one. For Lewis only supervaluation, or context, decides there. The record’s analogue is CLAIM-152’s target relativity. Lewis’s “Context will favour one sort of interpretation or the other” plays the role a stated target plays in QUESTION-035. That resemblance is the record’s. Lewis’s context is conversational, not predictive.
• Nothing here is evidence for CLAIM-164 or CLAIM-157. Lewis’s argument is about counting and reference in ordinary language. It has no dynamics, no metric and no threshold. It is a philosophical precedent for counting the record’s units by something weaker than identity, and for allowing that units may be many under strict standards.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• No formal semantics. The overlap spectrum is not measured, almost-identity has no threshold, and “context” is not analysed.
• 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.
• The almost-identity solution is admitted to work “for most contexts” only, and not for every instance of the problem.
• The super-truth cost (something is the cat, nothing is super-truly the cat) is accepted, not removed.
• The reply on Geach rests on introspective reports of hesitancy (“it seems to me”).

Open questions

• What fixes the threshold for almost-identity, and whether it can be anything but contextual. For the record, whether a gap-anchored bound is a principled threshold.
• Whether almost-identity among the actual candidates is an equivalence relation. Lewis says it depends on two matters left undecided.
• How the two kinds of intended interpretation are selected by context, and whether a stated target, in the record’s sense, is one such context.