Double-counting and the problem of the many

Abstract

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.

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.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.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.