Abstract
The mereological predicate ‘is part of’ can be used to define the predicate ‘is identical with’. I argue that this entails that mereological theories can be ideologically simpler than nihilistic theories that do not use the notion of parthood—contrary to what has been argued by Ted Sider. Moreover, if one accepts an extensional mereology, there are good philosophical reasons apart from ideological simplicity to give a mereological definition of identity.
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Notes
Comparing the complexity of different primitive terms is hard, but not impossible as is shown by (Goodman 1951, 46ff).
Note that ‘is part of’ sometimes goes by the name ‘is an improper part of’ and should not be confused with ‘is a proper part of’ [defined in (D1)]. One could instead take ‘is a proper part of’ as the primitive term and define ‘is part of’ as follows: \(\hbox {P}{xy} =_\mathrm{df} \hbox {PP}{xy} \vee x=y\). However, if one wishes to define ‘=’ and only accept Core Mereology, then one should not take ‘is a proper part of’ as primitive. See for an explanation footnote 9 below. Thanks here to an anonymous referee for this journal for asking me to clarify this.
Note that (D1) is ‘strict proper part’, an alternative definition of proper part is: \(\hbox {PP}{xy}=_{\mathrm{df}}\hbox {P}{xy}\wedge {x}\ne {y}\). This difference does not matter for our discussion.
Thanks to an anonymous referee for Erkenntnis for pointing this out to me.
The term ‘pure nihilism’ is in a way unfortunate, since Sider’s version of nihilism is restricted to fundamental reality (Sider 2013, pp. 252–253). Since he distinguishes between what exists fundamentally and what exists derivatively Sider’s nihilism may be, in a sense, impure: fundamentally, there only exist atoms; but non-fundamentally there (possibly) exist objects composed of atoms. The term ‘pure fundamental nihilism’ might have been more apt, but it’s rather cumbersome.
Thanks to an anonymous referee of this journal for asking me to clarify this.
Of course, the objects that the nihilist things that exist need not all be F; maybe some are F and others are G. In that case she would need both the terms ‘is an F’ and ‘is a G’ (and so on for any number of terms). I am assuming, for simplicity, that all the objects that according to the nihilist exist have something in common and thus all satisfy one predicate.
This is in line with what Sider writes in Writing the Book of the World, where he makes clear that among the primitive terms in his language there are predicates from physics (2011, p. 292).
One can see now that if one accepts Core Mereology (but no theory stronger than that) and one wishes to define ‘=’ as in (D5), then one should not take ‘is a proper part of’ as the primitive term of Core Mereology. If ‘is a proper part of’ would be the primitive term, then ‘is part of’ should be defined thus: \(\hbox {P}{xy} =_\mathrm{df} \hbox {PP}{xy} \vee x=y\). But since this definition already contains ‘=’, definition (D5) would be circular. Hence, in Core Mereology one can only take ‘is part of’ as primitive if one wishes to define ‘=’. In stronger systems of mereology such as Extensional Mereology (Sect. 3 below) other notions can be taken as primitive, for example ‘overlaps with’ or ‘is disjoint from’. If ‘is disjoint from’ is taken as primitive, then one can define ‘is part of’ as follows: \(\hbox {P}{xy} =_\mathrm{df} \forall z \hbox {D}{zy}\rightarrow \hbox {D}{zx}\). This would not result in a circular definition of ‘=’ in terms of ‘is part of’. Hence, it matters what term is taken to be primitive and how strong a system of mereology one wants to formalize if one wishes to define ‘=’. For further discussion on primitive terms of mereology, see: Parsons (2014). Many thanks to an anonymous referee of this journal for asking me to clarify this.
Note, though, that a definition of ‘=’ in terms of ‘overlap with’ is only possible when the mereology is extensional, i.e., in the presence of an axiom like (P5) (see below) ‘overlaps with’ can be taken as primitive and function as part of the definiens of ‘=’.
To be sure, Sider (2013) does not suggest this definition.
Thanks to an anonymous referee for this journal for asking me to discuss this.
Which in turn may make the thesis that composition is the many-one identity relation equivalent to the claim that there are no composite objects. For discussion, see Calosi (2015).
Strictly speaking, it is an axiom-schema. Since the nihilist and the mereologist both need one axiom-schema they are on equal footing in that respect. Hence, for convenience, I use the term ‘axiom’ as meaning ‘proper axiom or axiom-schema’.
Thanks to Staffan Angere for helping me to see this more clearly.
This issue is further complicated by the fact that stronger theories of mereology, such as General Extensional Mereology, also known as ‘Classical Mereology’, can be formulated with only two axioms. See for discussion of the axiomatization of classical mereology: Hovda (2009).
Thanks to an anonymous referee of this journal for pressing me to discuss this objection.
To repeat: by defining identity in terms of parthood, one is not committed to the view that the concept of identity is reducible to that of parthood. The definition simply ensures that ‘=’ is not a primitive term in the language.
To say that a mereologist postulates more entities than the nihilist and that mereology is thus less parsimonious than nihilism is not necessarily true and, moreover, irrelevant to the objection under consideration. It is not necessarily true since, for one, the mereologist may hold that the world contains only seven objects (i.e., three atoms and four composite objects), while the nihilist may postulate the existence of eight atoms. (This is, admittedly, an unlikely scenario.) It is, more importantly, beside the point: the objection is that the mereologist who defines ‘=’ in terms of ‘is part of’ does worse with respect to a certain theoretical virtue. If this were the case, then that virtue should be (re-)gained by not defining ‘=’. Since the ontological commitments of mereology do not change whether ‘=’ is defined, ontological parsimony cannot be the relevant virtue in play.
Proof. It is only the consequent of (T1) that we need to prove, which is a bi-conditional. Left-to-right is an instance of \(x =y \rightarrow (\phi (x) \leftrightarrow \phi (y))\) with ‘PPz_’ for ‘\(\phi \)’. (Note that \(x =y \rightarrow (\phi (x) \leftrightarrow \phi (y))\) follows from the definition of identity (D5) together with the parthood-version of Leibniz’s Law (P3\('\)).) For the right-to-left part, we first prove that overlapping the same objects is sufficient for identity, i.e. \((\hbox {EO}) (\hbox {O}{zx} \leftrightarrow \hbox {O}{zy}) \rightarrow x=y\).
Assuming that \(\lnot \hbox {P}{yx}\) we get via Strong Supplementation (P5) that: (i) \(\exists z (\hbox {P}{zy} \wedge \lnot \hbox {O}{zx})\).
From (i) and the fact that \(\hbox {P}{zy}\) implies \(\hbox {O}{zy}\) we can derive: (ii) \(\lnot \forall z (\hbox {O}{zy} \rightarrow \hbox {O}{zx})\). Thus:
(iii) \(\lnot \hbox {P}{yx }\rightarrow \lnot \forall z (\hbox {O}{zy}\rightarrow \hbox {O}{zx})\); which we can contrapose to:
(iv) \(\forall z (\hbox {O}{zy}\rightarrow \hbox {O}{zx}) \rightarrow \hbox {P}{yx}\).
The antecedent of (EO) is: (v) \(\hbox {O}{zy} \leftrightarrow \hbox {O}{zx}\); which gives us, by (iv): (vi) \(\hbox {P}{xy} \wedge \hbox {P}{yx}\). From the definition of identity (D5) and (vi) we have (EO).
Notice that from the definition of proper parthood (D1) and the definition of overlap (D2), it follows that if x and y are non-atomic objects, then if x and y have the same proper parts, anything that overlaps x also overlaps y:
(vii) \(\exists z (\hbox {PP}{zx} \vee \hbox {PP}{zy}) \rightarrow (\forall z (\hbox {PP}{zx}\leftrightarrow \hbox {PP}{zy}) \rightarrow (\hbox {O}{zx}\leftrightarrow \hbox {O}{zy} ))\),
given (EO) and the transitivity of implication, the consequent of T1 (right-to-left) follows. (Proof is essentially the same as the one in Cotnoir (2014, p. 17).)
For an excellent discussion of these issues, see Varzi (2008).
Cowling (2013, p. 3906) also suggests an ideological interpretation of the Composition as Identity-thesis.
Franz Brentano may have had something similar in mind: ‘[I]t would be wrong to suppose that the two parts of a thing taken together constitute an additional third thing. For where we have an addition the things that are added must have no parts in common. Thus we may say, for example, that a triangle has three angles, but not that is has three pairs of angless [sic]: angles A and B form a pair, as do B and C, and also C and A, but each of these pairs has a part in common with each of the others.’ (Brentano 1981, p. 16).
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Acknowledgments
Thanks to participants of the Higher Seminar in Theoretical Philosophy at Lund (Autumn 2015) for valuable comments. Many thanks, in particular, to Erik J. Olsson, Wlodek Rabinowicz, and especially Tobias Hansson Wahlberg for discussion and helpful comments on earlier versions of this paper. Many thanks also to Staffan Angere, Katherine Hawley, Justine Jacot, George Masterton, Carlo Proietti, and Frank Zenker for valuable discussions of some ideas in this paper. Two anonymous reviewers for this journal also deserve many thanks for their comments, as does an anonymous reviewer for Erkenntnis whose remarks on a rather different manuscript led to this paper.
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Smid, J. ‘Identity’ as a mereological term. Synthese 194, 2367–2385 (2017). https://doi.org/10.1007/s11229-016-1056-6
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DOI: https://doi.org/10.1007/s11229-016-1056-6