Abstract
Strong Composition as Identity (SCAI) is the thesis that necessarily, for any xs and any y, those xs compose y iff those xs are non-distributively identical to y. Some have argued against this view as follows: if some many things are non-distributively identical to one thing, then what’s true of the many must be true of the one. But since the many are many in number whereas the one is not, the many cannot be identical to the one. Hence (SCAI) is mistaken. Although I am sympathetic to this objection, in this paper, I present two responses on behalf of the (SCAI) theorist. I also show that once the defender of (SCAI) accepts one of these two responses, that defender will be able to answer The Special Composition Question.
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Notes
Some people might prefer a version of composition as identity that is contingent rather than necessary. That’s fine with me. However, please note that a contingent version of composition as identity can, at best, give a contingent answer to The Special Composition Question. Thanks to an anonymous referee for suggesting that some might prefer the contingent alternative.
(SCAI) is considered by Lewis (1991) before he seemingly rejects it in favor of a view that’s far less radical. (SCAI) has been discussed by Armstrong (1989, 1997), Bohn (2011, 2014a, b), Cameron (2007, 2012), McDaniel (2010), Merricks (1999, 2005), Sider (2007), van Inwagen (1994), Wallace (2011a, b), and Yi (1999a) among others. A variant of this view, going by a similar name, has been defended by Baxter (1988a, b). Moreover, Composition as Identity is the subject of a recent volume of papers edited by Cotnoir and Baxter (2014). Some of these authors have been critical of (SCAI) and others have been sympathetic. Many have stepped forward as mere hypothetical defenders.
See Bohn (2011) for an extensive discussion of Lewis.
A version of this argument is also considered by McKay (2006: 38).
(SCAI) is not an answer to The Special Composition Question. Rather, it is at best an answer to the General Composition Question. Whereas The Special Composition Question asks, for any xs, under what metaphysically necessary and sufficient conditions do those xs compose something or other, The General Composition Question asks, for any xs and any y, under what metaphysically necessary and sufficient conditions do those xs compose that particular y. Both of these questions are formulated by van Inwagen (1990) (though, in the text above, I follow more closely the formulation of Markosian 1998a, b, 2008, 2014). The first of these questions has received quite a bit of attention whereas the second has received very little. One might find this surprising since, given how closely they are related, one might think that answers to the second question should settle answers to the first. There might be a neglected strategy for answering (SCQ) lurking about: first find the best answer to (GCQ) and then see what (SCQ) answer it entails. But Peter van Inwagen (1990: 46–47) has argued that answers to (GCQ) do not in general entail informative answers to (SCQ) even if they do entail answers to (SCQ). Van Inwagen, though, has a particularly restrictive view of what makes an answer to (SCQ) informative and I will be following Markosian’s (1998a) less restrictive notion of informativeness in what follows. In fact, though I won’t argue for this claim here, it is my opinion that no correct answer to The Special Composition Question can be informative in the strict sense. The best we can hope for is an answer that meets Markosian’s notion of informativeness.
The fact that (SCAI) settles The Special Composition Question can be added to a long list of strengths which may be advanced together to make a strong cumulative case for (SCAI). Here is a list of three such strengths: First, Armstrong (1989, 1997), Baxter (1988a, b), Hawley (2014), Lewis (1991), and Sider (2007) have all suggested that (SCAI) captures the intuition that mereology is ontologically innocent and, relatedly, that counting the parts of an object while also counting the whole object is, in some sense, double counting. Second, Wallace (2011a) has pointed out the, in my opinion, underappreciated fact that if (SCAI) is true then we can explain how the material parts of an object collectively occupy the same region as the whole they compose without violating a plausible ban on co-location. And, third, I (2013) have shown that if (SCAI) is correct, then there is a straightforward answer to The Simple Question (under what conditions is a material object a mereological simple): Necessarily, a material object is a simple iff all the things it’s identical to are one in number.
See Markosian (1998a) for a more detailed explanation and defense of this notion of informativeness.
Yi (2014) has argued that English plurals in fact work this way.
Some (SCAI) theorists introduce a notion of partial identity, which is equivalent to mereological overlap. They may take distinctness to be non-partial identity rather than non-identity. Hence they may resist the numerical definitions given above. I will not pursue this kind of objection in this paper. Thanks to an anonymous referee for this suggestion.
The Russell-Style Account suggested above, which uses plural variables, is not exactly Russell’s own account.
If the half orange is an orange, then given the Russell-Style Accounts there are (counterintuitively) two oranges on the table! So, regardless of whether we count the half-orange as an orange, the Russell-Style Accounts give us a counterintuitive answer to the question. I will be ignoring this horn of the dilemma for the remainder of the paper.
I say that the superfluous information might be false rather than that it is false because, as we will see later, the notion of a count might preclude one from legitimately counting both the whole orange and its various undetached halves at the same time.
Or maybe it’s a grapefruit cleverly disguised as an orange. It doesn’t matter. I don’t want to get distracted by math or pomology.
Although, again, the notion of a count might help here.
An anonymous referee has pointed out to me that there are two different questions that might be conflated in this objection: How long is the orange peel? And, how many (non-overlapping) 1 mm segments of orange peel are there? No claim about how to properly answer the first question can be part of a legitimate objection to Russellian-Style Accounts of counting. The second question, moreover, can be answered by the Russellian by saying that there are 314 non-overlapping 1 millimeter segments of orange peel (and, if the conversational context permits, saying that there are additional, smaller, non-overlapping segments as well). However, I think the question asked in the text is different from both of these questions. The question asked in the text is the following: How many millimeters of orange peel are there? This question is asking us to count something rather than measure something. So, it is different from the first question above. Moreover, this question is asking us to count the millimeters of orange peel rather than the 1 millimeter segments of orange peel. So, it is different from the second question above as well. Admittedly, a strong case might be made for the assimilation of my question with one of the two questions distinguished above.
The Russell inspired accounts of numerical claims also have trouble with numerical claims involving negative number and numerical claims involving various orders of infinity. For example, suppose I have overdrawn my bank account by 20 dollars. I might say that there are −20 dollars in my bank account. Or suppose there’s a young philosopher who is just beginning to understand the difference between dense and continuous time. We might tell her that there are a countable infinity of instants in a second if time is merely dense and there are an uncountable infinity of instants in a second if time is continuous. But it’s not even clear that these claims can be analyzed using anything like the Russell inspired methods. Admittedly, the example involving an overdrawn bank account might be dealt with by analyzing the troublesome claim as one about owing the bank 20 dollars rather than a claim about having a negative number of dollars. The example involving different orders of infinity might be dealt with if we analyze those claims using a language that allows for infinite and transinfinite formulas.
The Russell-Style Account of numerical predication introduced in the previous section is irreducibly plural. It introduces a predicate that applies to pluralities which is analyzed in terms of at least one relation that is itself irreducibly plural (the ‘among’ relation). However, more traditional Russell-inspired accounts of counting are not irreducibly plural. On these more traditional accounts, numerical predications are higher order predications.
Thanks to an anonymous referee for this point.
Composition by Identity is equivalent to the following thesis, which we might call Composition by Oneness:
(CBO) Necessarily, for any xs, those xs compose something or other iff those xs are one in number.
One might think that (CBO) has some advantages over (CBI1). First, some people believe that a correct answer to The Special Composition Question should give, for any things, the conditions in virtue of which they compose something or other. If the conditions provided by (CBO) are better in that respect, then it will have an advantage over (CBI1). I take no stand on whether (CBO) or (CBI1) provides more plausible conditions in virtue of which composition occurs. Second, a good answer to The Special Composition Question is supposed to be informative in the sense that one should be able to state it using non-mereological vocabulary. If (SCAI) is true, then perhaps the identity relation and the composition relation are one and the same. And, if those relations are one and the same, then (one might think) any answer to The Special Composition Question that involves identity will violate the informativeness constraint. Hence, (CBO) would be a better answer than (CBI1). I reject this line of reasoning. Even if the composition relation and the identity relation are one and the same, nevertheless, an answer that uses identity vocabulary may be informative. And, moreover, I am skeptical of the claim that (SCAI) entails that the composition relation and the identity relation are one and the same. For a more detailed discussion of these issues see my (2013).
We can also get at the notion of a count by using mereological vocabulary. But be wary! When we introduce the notion of a count by using mereological vocabulary, we are not giving a definition or even a partial definition. We are simply providing a necessary condition, which might help convey the concept of a count. So, then, here is our condition: Necessarily, a domain happens to be a count only if none of the members of the domain overlaps with any of the other members of the domain and anything whatsoever overlaps with at least one member of the domain.
The notion of a count might sound like it has some affinity with Carnapian frames of reference (Carnap 1956). But I intend this view to be a version of ontological realism rather than some kind of Carnapian relativism. It does not follow from anything I’ve said, for example, that if there is a count under which there are twenty four things in the room, then there must also be a count under which there are eight things in the room. There might be a count under which there are oranges in the room and yet no count under which there are tri-oranges in the room. After all, there might not be, in any sense, things that are composed of three oranges. Thanks to the audience at Themes from Baxter II for pushing me to clarify this distinction.
One might even endorse a kind of ways of being view according to which each of the quantifiers that corresponds to a count is a way of being whereas quantifiers that do not correspond to counts are not. For an introduction to contemporary theories of ways of being see McDaniel (2009), my own (2012), and Turner (2010).
The proof that nothing can be both one in number and three in number, under a particular count, is straightforward and follows closely the proof given at the end of Sect. 2.
Plural quantifiers will also be restricted to a count and a similar definition for an unrestricted plural quantifier can be given.
At this point, the view that I am introducing differs from Baxter’s own version of Composition as Identity. Baxter prefers to introduce two notions of identity: intra-count identity and cross-count identity. The first is a relation that obtains between an object, x, and an object, y, that appear in the same count. For example, the orange I plucked from the tree is (intra-count) identical to the orange that grew on the lowest hanging branch of my tree. The second relation, though, is one that obtains between a thing or things that appear under one count and a thing or things that appear under another count. For example, some particular orange molecules are (cross-count) identical to an orange. Only the first identity relation, Baxter claims, obeys an indiscernibility principle. But it is the second relation that Baxter uses in his formulation of Composition as Identity. Baxter, then, can respond to the Lewis-inspired argument by claiming that it is ambiguous. If the arguer, on the one hand, intends to use intra-count identity throughout, then the argument is sound and the conclusion is true. But that spells no trouble for composition as identity. On the other hand, if the arguer intends to use cross-count identity throughout, then the argument is invalid; it employs a false indiscernibility principle. That’s all fine, I suppose, but it isn’t really a defense of (SCAI) and it requires introducing a second identity relation that I’m sure most of us are quite unfamiliar with. I hope to defend (SCAI), retain the indiscernibility principle, and avoid introducing multiple identity relations. For more details on Baxter’s view and his cumulative case against the Indiscernibility of Identicals see his (1988a, b, 1989, 1999, 2001).
An anonymous referee has pointed out to me that this same sort of response might be made by someone who defends the view that there are irreducibly plural numerical predications and, indeed, the sort of suggestion that follows has been made by Wallace (2011b). Instead of relativizing the predication to a count, one simply introduces complex numerical predications that include noun phrases that impose explicit quantifier restrictions. So, instead of saying that there are 24 things in the room under a count, one that happens to include the oranges in the room, we simply say that there are 24 oranges in the room. This suggestion also allows one to respond to the objectionable brute necessities by adopting a Russell-Style Account of irreducibly plural complex numerical predications.
Thanks to Ross Cameron, Aaron Cotnoir, Hud Hudson, Shieva Kleinschmidt, Kris McDaniel, and Chris Tillman for discussing the ideas of this paper with me. Thanks also to an audience at the Themes from Baxter II conference in Ligerz, Switzerland, for helpful comments on an early draft of this paper. Finally, special thanks to two anonymous referees for this journal who provided extensive and very helpful comments on two earlier drafts.
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Spencer, J. Counting on Strong Composition as Identity to Settle the Special Composition Question. Erkenn 82, 857–872 (2017). https://doi.org/10.1007/s10670-016-9847-1
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DOI: https://doi.org/10.1007/s10670-016-9847-1