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The thesis that ordinary material objects are mereological sums in the standard sense has been remarkably popular among three-dimensionalists and four-dimensionalists alike. This chapter considers two prominent representatives: Judith Jarvis Thomson, for the three-dimensionalist camp; and David Lewis, for the four-dimensionalist camp. The question of why Thomson's temporalized and modalized version of standard mereology still does not adequately capture the characteristics of ordinary material objects is postponed until Chapter 4. The remainder of the chapter considers Lewis' argument in favor of Unrestricted Composition, in particular in the recently expanded version offered in Theodore Sider's, Four-Dimensionalism: An Ontology of Persistence and Time; as well as Lewis' defence of the Composition-as-Identity Thesis in his Parts of Classes. It is argued that the three-dimensionalist need not be swayed by either component of Lewis' view, since they are ultimately founded on question-begging reasoning.

The doctrine of the Trinity is a conjunction of these three claims: There are three distinct Divine Persons: the Father, the Son, and the Holy Spirit; each Divine Person is God; and there is exactly one God. However, if there are three distinct Persons that are each God, we should get the result that there are three Gods. It seems Trinitarian Christians are having trouble counting: they need 3 to equal 1. There has been a flurry of discussion about the claim that ‘composition is identity’, i.e., that pluralities are identical to the things that they compose (if there is something they compose). Many can literally be one. This chapter argues that this claim is not helpful to Trinitarians. Section I begins by discussing the many-one identity claim. Section II applies the claim to the Doctrine of the Trinity, presenting how it might seem to help the Trinitarian. Section III argues that upon closer inspection it proves not to help at all, but instead leaves us with the same options that we began with. The chapter also discusses the difficulties appeals to Composition as Identity generate for the intelligibility of monotheism. It concludes that, while it is an interesting metaphysical thesis, the Composition as Identity claim is not useful in helping reconcile the claims in the Doctrine of the Trinity.

Let’s provisionally understand compositional pluralism as the doctrine that there is more than one basic parthood relation. This chapter investigates to what extent compositional pluralism and composition as identity can form a coherent package of views. Since there are good arguments for compositional pluralism and the author feels the intuition that gives some support to composition as identity, he is motivated to determine this. But regardless of whether you feel attraction to either of these two views, it is worth thinking about how versions of them might interact with each other, since doing so is likely to lead to interesting thoughts about parthood and identity. Accordingly the chapter distinguishes between several versions of compositional pluralism and composition as identity. It argues that many versions of compositional pluralism are compatible with even very strong versions of composition as identity.

Composition as identity, as understood here, is the thesis that a whole and all its parts collectively is the same thing under two different modes of presentation. Unrestricted composition is the thesis that any plurality composes something. The chapter first articulates a version of composition as identity. Then it argues that this entails universalism, contra some objections to that effect. Finally, this result throws some new light on the more general debate over composite objects and quantification.

Composition is the relation between a whole and its parts—the parts are said to compose the whole; the whole comprises the parts. But is a whole anything over and above its parts taken collectively? Are the many parts identical to the one whole? This chapter traces the motivations, varieties, and problems with the view that has come to be known as composition as identity. It also provides an introduction and background in formal mereology and plural logic that is necessary for understanding the contemporary debate.

The connection between whole and part is intimate: not only can we share the same space, but I’m incapable of leaving my parts behind; settle the non-mereological facts and you thereby settle what is a part of what; wholes don’t seem to be an additional ontological commitment over their parts. Composition as identity promises to explain this intimacy. But it threatens to make the connection too intimate, for surely the parts could have made a different whole and the whole have had different parts. This chapter attempts to offer an account of parthood that is intimate enough but not too intimate: the parts generate the whole—the whole exists in virtue of the parts—but they are not themselves the whole.

A plurality or plural object is a single object that is also many, and pluralitism is the thesis that there is such an object. This chapter argues that pluralitism and closely related theses (e.g. the many–one identity thesis and the composition as identity thesis) violate logic. To do so, it formulates an approach to the logic and semantics of plural constructions that results in plural logic, and relates treatments of plural constructions to accounts of natural number. It gives a critical examination of Frege’s views of numbers and plural constructions, and sketches the view of plural constructions as devices for talking about the many (as such), and an account of natural numbers as properties of a special kind, plural properties.

‘Composition as identity’ is the radical claim that the whole is identical to the parts—radical because it implies that a single object can be identical to many objects. Composition as identity, together with auxiliary assumptions, implies the principle of ‘collapse’: an object is one of some things if and only it is part of the fusion of those things. Collapse has important implications: the comprehension principle of plural logic must be restricted, plural definite descriptions such as ‘the Cheerios in the bowl’ are empty, composition as identity does not preclude emergent properties (contrary to what McDaniel has argued), and drastic simplifications of mereological and logical ideology are extensionally available.

This chapter gives a regimented, formalized presentation of Donald Baxter’s theory of Composition as Identity. It highlights how Baxter’s theory differs from the more commonly discussed variant that David Lewis flirted with. Baxter’s theory depends heavily on his Theory of Aspects, which restricts Leibniz’ Law, and of Count-Relative Identity. The chapter interprets the latter as a form of Ontological Pluralism and generates a novel notation for the former. It proposes certain principles as axioms of Baxter’s theory, regiments them, and gives them an informal gloss. It also highlights other potential axioms as choice points for Baxterians. It ends discussing objections and remaining questions.

This article is a critical discussion of Aaron Cotnoir's 'Composition as General Identity' (chapter 7 in this volume). Discussion focuses on the relation being the same portion of reality as, asking what grounds we have to regard this as an identity relation rather than, for example, a counterpart relation. Two different metaphysical pictures are briefly explored: either portions of reality are ordinary objects, or they are something distinct from ordinary objects.

Mereological Essentialism (ME) claims that an object is identical to its parts in every world in which it exists. Composition as Identity (CAI) claims that a composite object is (collectively) identical to its parts. Many think that CAI is false because CAI entails ME, and ME is absurd. This chapter shows how a CAI theorist can defend herself against this kind of objection—as well as other (similar) modal objections—by adopting Modal Parts, a view where ordinary objects are trans-world mereological sums of world parts.

This chapter gives a detailed presentation of the permissivist, eliminativist, and conservative theses under discussion. It distinguishes among a variety of different permissive and eliminative theses (e.g., universalism, plenitude, nihilism, and organicism) and explains how the arguments from the previous chapter are meant to motivate the different revisionary theses. It address a variety of attempts to show that there is a trivial answer to whether universalism and/or eliminativism is true. Finally, it includes some remarks about methodology and other background assumptions.