Double-counting and the problem of the many


There is a defeasible constraint against double counting. When I count colours, for instance, I can’t freely count both a color and its shades. Once we properly grasp this constraint, we can solve the problem of the many. Unlike other solutions, this solution requires us to reject neither our counting judgments, nor the metaphysical principles that seemingly conflict with them. The key is recognizing that the judgments and principles are compatible due to the targeted effects of the defeasible constraint.

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  1. 1.

    This presentation is adapted from Weatherson (2016).

  2. 2.

    Unger (1980) denies (i), and rejects the existence of clouds. López de Sa (2014) and Williams and Robert (2006) reject (i) because they think there are a multitude of clouds. McGee and McLaughlin (2000) reject (ii) because they think that it is false under all precisifications, though each candidate constitutes a cloud relative to some precisification. Korman (2015) rejects (ii) because he thinks that just one of the candidates constitutes a cloud, though it is metaphysically indeterminate which. Woods (forthcoming) rejects (ii) because he thinks that there is one maximal candidate. Jones (2015) rejects (iii) because he thinks that a single cloud can be constituted by multiple different collections. Chisolm (1976) and Noonan (1993) reject (v) because they think we count by a relation weaker than identity.

  3. 3.

    Sattig (2010) also develops a view that takes (i)–(vi) to be consistent. Space precludes a comprehensive discussion of his view, but, as he recognizes, his view is based on a semantic theory that lacks empirical support. (Though he doesn’t take this to be a problem.) Given that I take there to be substantial empirical support for my view, it has at least that advantage over Sattig’s.

  4. 4.

    I’m simplifying the view and omitting the functional variable that Stanley and Szabó posit to explain binding readings.

  5. 5.

    One worry is that these cases exemplify ambiguity in the restricting noun, rather than restriction on its domain. To see that an ambiguity view is misguided note two things: (1) such a view would lead to an infinite number of senses for terms like ‘colour’, given that there are an infinite number of levels of colour-individuation, (2) there are occurrences of ‘colour’ in which it must have both red and maroon in its extension (e.g. ‘Red and maroon are both colours’.) Cf. Liebesman and Magidor (2017).

  6. 6.

    This technical notion of salience differs from our intuitive notion. As Kratzer (2005) stresses, there are intuitively salient domains that are nonetheless not eligible to be the values of domain variables.

  7. 7.

    I’ll speak of domains as themselves double-counting. This is shorthand for the claim that, a domain d, relative to a context c and noun n is such that if n’s domain variable were saturated with d in c, that would suffice for double-counting.

  8. 8.

    Other languages contain other colour words and, given that, other domains will be salient. Russian, for instance, contains distinct lexical items for light and dark blue. Some have argued that this gives rise to processing differences: see Winawer et al. (2007).

  9. 9.

    Baxter (1988) and Cotnoir (2013), by contrast, build a double-counting constraint directly into their semantics.

  10. 10.

    A rejoinder: when the predicate is copied, it is the free variable that is copied, not its saturation. A response: on this view, the occurrence of ‘as is’ in (9) would be very odd, as we’d be ascribing different properties to red and maroon.

  11. 11.

    Kratzer contrasts the lunatic with the pedant. The latter responds to an restricted quantificational claim by denying it and asserting a less restricted counterpart. The contrast is that the pedant seems annoying, but within his rights as a speaker, while the lunatic is not.

  12. 12.

    This example comes from Liebesman and Magidor (2017). Korman (2015) uses an almost identical example.

  13. 13.

    There is a tempting metaphysical rejoinder to this examples: to argue that in the wooden thing case the arbitrary parts are not wooden things because they don’t exist, perhaps because arbitrary undetached parts do not exist more generally. (See van Inwagen (1981) for an early influential defense.) Tempting as it is, this rejoinder is unconvincing. Imagine that each of the table’s five parts were constructed from two pieces of wood glued together, though the gluing was seamless. In that case, it is uncontroversial that the ten parts exist, though there is still not a true reading of (19), at least without additional contextual supplementation. Even if we cannot arbitrarily decompose objects/events, some decompositions will still exist. We can then generate examples using these decompositions.

    1. (19)

      Ten wooden things are in the room.

  14. 14.

    This case, and the two-volume book case, are adapted from cases discussed extensively by Krifka (2009).

  15. 15.

    There’s an important feature of this case that doesn’t apply to the others: there isn’t a unique set of shirt/pants pairs that witnesses the truth of (22). Rather, it seems that any set of non-overlapping pairs will work just as well as any other. We can make sense of this linguistically by taking the domain variable associated with ‘outfits’ to be indeterminate between any domain that doesn’t double-count. Any acceptable resolution of the indeterminacy ensures the truth of (22).

  16. 16.

    Korman (2015: 221) makes this point forcefully.

  17. 17.

    In fact, Korman (2015: 202) claims that relevantly similar counterpossibles are true.

  18. 18.

    This is not uncontroversial, see Weatherson (2016) and López de Sa (2014) for competing interpretations of Lewis (1993).

  19. 19.

    Another contrast: Lewis and I motivate our counting claims with different cases and, as I argue in Liebesman (2015), I don’t think his cases work.

  20. 20.

    The force of this prima facie advantage will depend on a variety of issues involving indeterminacy and interpretation.

  21. 21.

    In general, one may reasonably think that different versions of the problem of the many can be solved in different manners. See Simon (2018) for a defense of the view that the version of the problem of the many that arises for experiences is particularly difficult, and that the best solution to it may be to adopt property dualism.

  22. 22.

    More carefully, for the purposes of understanding the notion of functional overlap, we can understand the function of an object as given by a subset of its set of dispositions. This account may be unsuitable for analyzing the notion of a function more generally, or for other purposes.

  23. 23.

    The condition that M(d-d′) > 0 guarantees that there is genine overlap, not just irrelevance. After all, if we lift this requirement then we could guarantee that a domain overlaps relative to an M solely by adding some entities that M maps to 0, e.g. a set containing five shirts and one dog would functionally overlap relative to PCC because dogs can’t clothe people.


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Thanks to Bernhard Nickel, Jonathan Payton, Matti Eklund, Jeremy Fantl, Nicholas Jones, Ofra Magidor, Alexis Wellwood, Barry Schein, Jeremy Goodman, Mark Richard, and Dan Korman, two anonymous referees as well as audiences at USC and The Society of Exact Philosophy. Support for this paper was provided by an Insight Grant from The Social Sciences and Humanities Research Council of Canada.

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Appendix: Functional overlap

Appendix: Functional overlap

What is functional overlap? There are several challenges in understanding the notion. First, there’s a worry that we don’t know what it is for functions to overlap. After all, overlap is standardly understood in terms of part-sharing, and it is not clear that functions have parts, let alone share them. Second, there’s a worry that many objects aren’t associated with functions. I’ll address these in turn.

Consider our outfits for charity from Sect. 2.4. An arbitrary shirt/pants pair has the power to clothe a single person. We can understand this power dispositionally. If we think of clothing somebody as protecting them from the environment, then a shirt/pair pants has the disposition to clothe somebody when worn. The idea, then, is that the function of a shirt is given by some of its dispositional properties.

We can think of the function of an object as given by a subset of its total dispositions.Footnote 22 Just which subset? This is partly determined by what we are counting, and partly determined by context. If we are counting houses, then the relevant function (in many contexts) is that of sheltering, which is what we associate with the kind house. When we are counting artworks, there will likely be a different associated function. What if something is both a house and an artwork? Well, then the function that is relevant to double-counting will depend on whether we are restricting our quantifier with ‘house’ or ‘artwork’.

Taking the function of a kind of object to be a contextually-selected set of dispositions, we can now consider our other question: what is it for functions of two objects to overlap? I won’t give a fully general answer here—one that applies to any functions whatsoever—but I will give a sufficient condition that covers some functions. Return to our outfits: in the charity scenario we associate the kind outfit with the dispositional property of clothing (understood as the dispositional property of protecting from the environment when worn). Particular outfits have this general property, but they also have a more specific variant: a single pants/shirt pair has the disposition to clothe one person. A wholly distinct pair (one that differs in both components) also has the disposition to clothe one person. Those pairs, taken together, have the disposition to clothe two people. This, however, is not true of overlapping pairs. Consider the pairs p1/s1 and p1/s2 that differ in their shirt-components but not in their pants-components. Taken separately, each has the disposition to clothe one person, but taken together they do not have the disposition to clothe two—after all two people can’t wear a single pair of paints at single time. These pairs functionally overlap. We can make this a bit more precise by noting that some dispositional properties correspond to measure functions.

Measure functions are functions from entities to values on a conventional scale. For instance, the measure function volume-in-liters is a function from entities to numbers, where the numbers measure the volume (in liters) of those entities. Some measure functions are dispositional, in the sense that they measure powers/abilities. For instance there is a measure function people-can-clothe (call it PCC) that maps objects to the number of people they can clothe, such that, e.g. p1/s1 is mapped to one. For dispositional properties like the property of clothing one person there will be a corresponding measure function/measure pair, in this case (PCC, 1). Necessarily, if something instantiates the property of clothing one person when worn, PCC maps it to 1. This is the sense in which the measure function/measure pair corresponds to the dispositional property. More general properties like being able to clothe people correspond to the measure functions themselves, in the following sense: necessarily, if an object has the property of being able to clothe people, then it maps PCC to some positive number. Measure functions like PCC may abstract away from some irrelevant physical properties of the objects they are measuring. Even if s1 has a few more molecules than s2, PCC may map both s1/p1 and s2/p1 to 1: after all those extra molecules on s2 do not affect the number of people it can clothe.

For our purposes, measure functions are functions from both individual objects, and pluralities of objects. A measure function M is non-additive relative to some objects o1 and o2 just in case, M(o1)+M(o2) does not equal M(o1,o2), where the latter signifies that the plurality of o1 and o2 is the argument of M. We’ve already seen how this can happen. If o1 and o2 share a component, then the total measure may reflect this. Finally, we can understand the relevant notion of functional overlap: function f overlaps for objects o1 and o2 just in case f (understood as a disposition) corresponds to a measure function M and M is non-additive relative to o1 and o2. Again, the outfit case illustrates this. The function of clothing people that is shared by p1/s1 and p1/s2 overlaps for them because that function corresponds to PCC and PCC is non-additive for p1/s1 and p1/s2.

We can now articulate a sufficient condition for a domain to double-count relative to a context c, based on this understanding of functional overlap.

A candidate domain d for a noun N double-counts relative to c if the function F associated with N in c corresponds to a measure function M, and M(d) is identical to M(d′), where d′ is some proper subset of d such that M(d-d′) > 0.

In articulating this condition, I slid from a measure function applying to a plurality, to that function applying to a domain, this was merely shorthand: when I wrote of a measure-function applying to a domain I intend that to be understood as the measure function applying to the plurality of the members of the domain. The idea behind this sufficient condition is that some object in the domain must add nothing whatsoever to the measure, and that suffices for double-counting. Again, this constraint makes perfect sense of the outfit case. When we go beyond fifty shirt/pants pairs, we will add nothing whatsoever to the measure given by PCC.Footnote 23

Consider, for contrast, Sutton’s two houses that share a massive wall. The relevant F is sheltering, and the measure function M is a function from amount of sheltered space they provide. Despite their shared wall, M maps the two houses to twice the number that it maps each to individually, so the sufficient condition for functional overlap is not met, despite the massive mereological overlap of the houses.

Another worry about invoking functional overlap is that in for many kinds K and contexts c, there just is no function associated with K in c. One response is that this simply doesn’t undermine our sufficient condition for double-counting: when there’s no associated function then we can’t double-count via functional overlap. There needn’t always be the possibility of functional overlap for it to sometimes be actual. Another response is that our notion of functional overlap is a proprietary one, and is not directly connected with independent investigations into teleology. Rather, we should think of it as whatever notion allows us to make sense of the outfit and house cases. These cases show us that some notion of overlap is relevant to NDC, but it is not simple part-sharing. Our notion of functional overlap is merely a first attempt to capture the notion of overlap that makes sense of our judgments in these cases. We have no reason to expect that it directly connects to indpendent discussions of the nature of functions.

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Liebesman, D. Double-counting and the problem of the many. Philos Stud 178, 209–234 (2021).

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  • Problem of the many
  • Quantifier domain restriction
  • Nominal restriction
  • Double-counting
  • Semantics of counting
  • Pragmatics of counting