Abstract
In this paper, I argue that the debate on Composition as Identity—the thesis that any composite object is identical to its parts—is deadlocked because both the defenders and the detractors of the claim have so far failed to take its philosophical core at face value and have, as a result, defended and criticized respectively something that is not Composition as Identity. After establishing how Composition as Identity should properly be understood and proposing for it a new interpretation centered around the novel notion of metaphysical information, I set forth a strategy to defend it that crucially rests on the indefinite extensibility of the domain of quantification. I eventually suggest that “Composition as Analysis” is a name that better reflects the content and theoretical proposal of the thesis that Composition as Identity is supposed to be.
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Notes
The most common line of objection against CAI is based on its alleged incompatibility with the Indiscernibility of Identicals, which is a constitutive principle of the logic of identity. If, in fact, CAI is the thesis that the one object is identical to its many parts, then Indiscernibility of Identicals fails and composition fails to meet a fundamental requirement to be considered the relation of identity. I will say more about this in due course.
There are many possible formulations of mereological fusion. The one I have adopted is the one originally used by Lewis (1991: p. 73). For a discussion of other, and less standard, formulations, see Varzi (2016). Nothing of what I will say, however, depends on a preferred formulation of mereological fusion.
From now on, unless otherwise specified, I will refer to strong CAI simply as “CAI”.
Here I use “thing” rather than “object” so as to leave the possibility open that objects are taken to be things satisfying certain further requirements.
Lewis uses it for the first time in Parts of Classes.
To be sure, Cotnoir (2013) offers a tentative account of CAI that rests on a set-theoretical understanding of the notion of “portion of reality”. Since, however, he ends up defending the idea that parts and whole correspond to the same portion, rather than that they are the same portion, I include him among those who understood CAI as saying that parts and whole are identical insofar as they are different ways of speaking/counting/carving up the same thing.
I am not here making the claim that Lewis’s CAI implies unrestricted composition or the other way around. I am simply saying that Lewis works under the independent assumption that classical mereology is true and hence that unrestricted composition—that is part of the axiomatic core of the theory—is itself true.
Recently, Bennett (2017) argued for the existence of a unified family of relations—that she calls “building relations”—that are characterized, among other features, by being asymmetric and by exhibiting a relative fundamentality among the related objects, of which one is always prior to the other. Although when I use “building” in this context I do not have Bennett’s specific usage in mind, I take the absence of any sort of priority/privileged direction between an object and its parts to be a crucial component of the claim that composition is identity. I am therefore happy to grant that, if CAI is true, surely composition is not a member of the family of building relations individuated by Bennett. For analogous reasons, and as yet another consequence of what I have just said, I take any attempt to account for CAI in terms of grounding, that is to say, any attempt to make sense of CAI by saying that the existence of a composite object is grounded in the parts it is made up—see in particular Cameron (2014)—to be untenable insofar as based on a substantial misunderstanding of CAI. Thinking of the composite object as grounded the parts would, in fact, account for the intuition that the parts make the object up without this constituting a further ontological commitment; but this is exactly, as we have argued so far, the wrong intuition to hold when it comes to CAI.
The idea being, of course, that we abandon a fully Quinean approach to ontological commitment and hence the claim that we are committed to all and only the objects we include in our domain of quantification.
That, let us recall, consists in locating the double-counting at the level of ontological commitment and hence in embedding in a theory of ontological commitment the idea that an inventory of the things in the world that includes both the objects and their parts is redundant.
For a plural logician, pluralities are, in fact, nothing more that convenient syntactical abbreviations to speak of many objects at the same time.
The formulation, notation, and name of the principle are owed to Sider (2007).
Notice that this also comes with the consequence that there are overall fewer pluralities of objects, since what would otherwise count as extensionally distinct pluralities are made here collapse into one extensionally enriched plurality.
To be sure, also Lists would not survive our revision of Plural Comprehension and the novel understanding of pluralities that results from it. But, given what we said earlier, this is an expected and welcomed result.
The sense in which a plurality can be extensionally definable will be clarified in the paragraphs that follow. The general idea, however, is that we are dealing with pluralities that crucially can time by time be extensionally defined but are never actually so, since they can always acquire new members.
Linnebo speaks simply of “Collapse”; I opted for a different notation in order to distinguish this collapse from the Collapse of \(\prec \) into parthood. It is to be stressed, however, that Collapse\(^{\star }\) can be considered a form of collapse of \(\prec \) into \(\in \).
From \(\forall u (u \prec tt \leftrightarrow u \notin u)\), in fact, we get, via Collapse\(^{\star }\), \(\exists t \forall u (u \in t \leftrightarrow u \prec tt)\), which leads to a set defined by the following condition: \(\forall u (u \in t \leftrightarrow u \notin u)\), of which \(t \in t \leftrightarrow t \notin t\) is an instance. This proof and the related discussion can be found in Linnebo (2010, p. 147).
I will say more about the way in which the modality at issue should be understood in the following section.
The offered readings of the modalized version of Collapse\(^\star \) and Plural Comprehension are based on the ones offered by Linnebo (2010, p. 157).
To be sure, the concerns raised by Linnebo about the possibility of restricting Collapse\(^\star \) all rested on the intuition that any restriction of Collapse\(^\star \) would betray the core of an iterative conception of sets: that nothing is required for some objects to form a set. The restriction we suggest, however, does not seem to be affected by this problem. Integral part of an iterative conception of sets seems to be, in fact, also the idea that all the elements of a set must be present at the moment of the set-formation process—i.e. that the same set cannot gain new elements. The restriction of Collapse\(^\star \) we suggest seems to account for exactly this intuition.
The state of maximal information, as already stressed, can never be reached, and this fits well with the idea that the quantification domain is infinitely extensible and an absolutely unrestricted quantification is never possible.
That the same element can be the bottom in different orders is not a problem: think for instance of the empty set in the partial ordering generated by the relation of set inclusion.
One might want to claim that each domain expansion should start with the individuation of at least two parts, the part we explicitly quantify over and its complement. Accepting this amounts to accepting a mereology in which at least the principle of weak supplementation holds. Since, however, we want to be as neutral as we can with respect to the mereology we presuppose, I am not building that assumption within my framework. On the relationship between my account and the general mereological debate I shall say more in the last section.
Linnebo (2018), however, seems to renounce to a purely set-theoretical characterization of the relevant modality in favor of an interpretational characterization that is explicitly connected to Fine’s proposal.
See, in particular, Bøhn (2009), who explicitly suggests that many-one identity statements could be thought of as informative identity statements and cashes out the information at a semantical level.
I cannot discuss here the details of the sort of monism that my account is likely to come with. However, it is important to notice that the resulting picture is monistic insofar as the maximal portion of reality is what exists at any level of expansion including the non-expanded level, at which we have a full fledged existential monism, since the world is the only thing that exists. Does this mean that, in our picture, we can never have parts that exist alone, independently of the whole they are parts of? Not necessarily. What the parts cannot do without is the world, that is the maximal portion of reality and hence, in some sense, the maximal whole. But we can easily think of an expansion of the quantifier domain that picks out all and only the (physical) atoms the world is made up of without quantifying over the molecules. What we cannot do is to pick out the molecules at a further level of expansion within the same string of information in which we have already quantified over the atoms. But we can always start a new string of information and pick out the molecules.
Since there are just atoms within each domain expansion, the principle according to which for any two objects there is their mereological fusion (which is the principle of binary summation, at the basis of universalism) does not hold. Adding, by the same stipulation that allowed us to add a bottom to the discrete order, the top is not enough: the top in fact would be the fusion of all the atoms, but there still would not be objects which are the fusion of exactly two atoms, given any two atoms. Still, by adding a top and the bottom to the partial order most of the other axioms can be recovered, although in most cases in a trivial form.
Thanks to the audiences at a Logic & Metaphysics Workshop at CUNY (2019), a Graduate Workshop at Columbia University (2018), a Logos seminar at the University of Barcelona (2017), the First Eidos Graduate Winter School in Metaphysics at the University of Neuchatel (2017), an Eidos seminar at the University of Geneva (2017), and the WFAP Graduate Conference at the University of Wien, (2016), where I presented earlier versions of this paper. Thanks also to Claudio Calosi, Alex Skiles, Maria Scarpati, and Giorgio Lando for helpful discussions at the early stages of this project. Thanks to the referees of Synthese for their useful comments. And thanks to Achille Varzi for months of discussions, mentoring, and encouragement.
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Botti, M. Composition as analysis: the meta-ontological origins (and future) of composition as identity. Synthese 198 (Suppl 18), 4545–4570 (2021). https://doi.org/10.1007/s11229-019-02396-2
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DOI: https://doi.org/10.1007/s11229-019-02396-2