Composition as identity, Leibniz’s Law, and slice-sensitive emergent properties

Abstract

Moderate composition as identity holds that there is a generalized identity relation, “being the same portion of reality,” of which composition and numerical identity are distinct species. Composition is a genuine kind of identity; but unlike numerical identity, it fails to satisfy Leibniz’s Law. A composite whole and its parts differ with respect to their numerical properties: the whole is one; the parts (collectively) are many. Moderate composition as identity faces the challenge: how, in the absence of Leibniz’s Law, can one characterize what counts as a genuine kind of identity? This paper explores a promising answer: a genuine kind of identity must satisfy a version of Leibniz’s Law restricted to properties that ascribe qualitative character. Strong composition as identity holds that there is only one identity relation, that it satisfies Leibniz’s Law, and that the parts are identical with the whole that they compose. Strong composition as identity faces the challenge of showing that numerical properties do not provide counterexamples to Leibniz’s Law, and doing so in a way that is compatible with the framework of plural logic that is needed to formulate the theory. The most promising way to do this is to hold that plural logic is fundamental at the level of our representations, but not fundamental at the level of being. At the level of being, portions of reality cannot be characterized as either singular or plural. It turns out that the proposed moderate theory and the proposed strong theory are one and the same. In spite of its many attractions, I reject it. The main issue has to do with whether slice-sensitive emergent properties are possible. I argue that they are, making use both of specific examples and general principles of modal plenitude. I do not claim that my arguments are irresistible. But they cannot be evaded as easily as a related argument against strong composition as identity given by Kris McDaniel. I critically examine McDaniel’s argument to pave the way for my own.

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Notes

  1. 1.

    A similar argument was given in Sider (2007). But Sider (2011: pp. 208–215) now rejects the argument because he no longer takes the framework of plural logic to be fundamental. I claim below that there is a crucial ambiguity in what it means to take the framework of plural logic to be “fundamental.” In any case, however, I do not see how composition as identity can even be formulated properly without the use of plural logic. Using schematic letters in place of plural variables weakens the theory, and introduces an irrelevant dependence on language. Using quantification over sets (or classes) in place of plural quantification, as was done in early presentations of classical mereology (Tarski 1937) also weakens the theory (since not all pluralities form sets), and introduces an irrelevant dependence on sets, something I find especially problematic. For I prefer the reverse reduction of sets to plural logic over the reduction of plural logic to set theory. See Bricker (forthcoming b, chapter 1).

  2. 2.

    On abundant versus sparse conceptions of properties, see Lewis (1986: pp. 59–63).

  3. 3.

    Pronounce ‘xx’ as “the x’s” if you like. Although ‘plurality’ is grammatically singular, it is important not to think that, on my usage, a plurality is a single set-like object.

  4. 4.

    Those who do not accept plurally plural quantification will have to approximate this using quantification over classes (taking properties of pluralities to apply to the corresponding classes): for any classes, there is at least one property had by all and only those classes.

  5. 5.

    I use ‘slice’ broadly: whenever xx composes y, xx give a way to “slice” y, whether or not xx overlap. I use ‘fusion’ and ‘compose’ to denote the same relation: y is the fusion of xx iff xx compose y.

  6. 6.

    The expression ‘portion of reality’, as I use it, is a term of art that, although syntactically singular, is semantically neutral with respect to the plural/singular distinction. Talk of “portions of reality” could be regimented within plural logic using the schematic variables introduced above.

  7. 7.

    But cf. Cotnoir (2013), who can be construed as introducing a general identity relation distinct from numerical identity. It is unclear, however, whether he insists that any identity relation satisfy Leibniz’s Law.

  8. 8.

    I prefer to understand strong and moderate composition as identity broadly so that they include much more than what is stated here: they encompass a full theory of the composite nature of reality, including classical mereology. But the additional content will not be relevant here. See Bricker (2016) for the details.

  9. 9.

    For arguments that strong composition as identity leads to Collapse, see Yi (1999) and Sider (2007). For further unacceptable consequences of Collapse, see Sider (2014). Yi (2014) considers a related kind of collapse that results when composition as identity is combined with plural logic: generalized identity collapses into plural identity (if both satisfy substitutivity). It then follows that if xx are the same portion of reality as y, then xx are one. And that trivializes strong composition as identity.

  10. 10.

    For Baxter’s groundbreaking approach to composition as identity, see Baxter (1988a, b).

  11. 11.

    For Geach’s defense of relative identity, see Geach (1967). For discussion and critique, see Perry (1970).

  12. 12.

    Compare Jenann Ismael (2015) where this distinction is made in connection with Humean accounts of laws and chance.

  13. 13.

    See Bricker (2006) for how this plays out with respect to general propositions.

  14. 14.

    See Bricker (2016: pp. 280–283) for more discussion of Lewis’s view. See also Bohn (2011), who similarly thinks Lewis has been wrongly understood.

  15. 15.

    Note that this is strong emergence, requiring the failure of logical supervenience. On strong versus weak emergence, see Chalmers (2006).

  16. 16.

    Sider (2014) has argued, on different grounds, that McDaniel’s argument does not succeed against versions of strong composition as identity that accept Collapse. But Sider’s argument does not apply to a version such as Mods, which rejects Collapse.

  17. 17.

    Following McDaniel and Lewisian tradition, I use ‘perfectly natural’ in phrasing these definitions, instead of ‘fundamental’. But note that only properties and relations fundamental at the level of being should count as perfectly natural, and factor into the definition of duplicate.

  18. 18.

    This formulation is from Sider (2014). It is more intuitive than McDaniel’s formulation, and does not differ in any way relevant to the argument.

  19. 19.

    McDaniel does not explicitly make this “one-relation” assumption. He is content to leave the argument at an informal level, one that does not invoke any specific formulation of Leibniz’s Law.

  20. 20.

    As McDaniel notes, the mereological atomism presupposed by this argument is not essential. I am happy to go along with it.

  21. 21.

    Actually, what McDaniel asserts for the first premise is that “any reasonable formulation of [strong] composition as identity” will entail (PDP); but I take it that Mods would count as a “reasonable formulation,” and so McDaniel’s argument should apply to it.

  22. 22.

    See Bohn (2012) for a similar response to McDaniel’s argument.

  23. 23.

    The argument for the revised second premise is roughly this. Suppose slice-sensitive emergent properties are possible. Suppose xx and yy are plural duplicates, where xx compose w and yy compose z, with w and z composite. Because xx and yy are plural duplicates, w and z have the same emergent properties on one way of slicing. But on other ways of slicing w and z need not have the same emergent properties. One way of slicing w and z is the trivial one-membered slicing given by w and z themselves. With an appeal to plenitude, we can suppose that w and z do not have the same emergent properties on this trivial slicing. But that is just to say that w and z are not duplicates, contradicting (PDP).

  24. 24.

    One could question whether the values of C1 and C2 are legitimately “sums” of the infinite series, since these series do not converge; but I don’t think that matters to the force of the example. In any case, one could easily construct fancier examples involving conditional convergence. An infinite series is conditionally convergent iff it converges, but the series of absolute values of the terms diverges. The sum of a conditionally convergent series depends on the ordering of its terms; and different orderings corresponds to different slicings in the world where the infinite series is exemplified. For example, the sequence of atoms <a1a2a3, …> corresponds to the slicing given by the plurality: a1a1 + a2a1 + a2 + a3, ….

  25. 25.

    See Bergson (1910). But I use Bergson merely as a foil; he would not accept the assumption that a continuum is composed of dimensionless points.

  26. 26.

    Spacetime is gunky iff there are no atoms: every part of spacetime has a proper part. Actually, I am doubtful that gunk is possible. But since most philosophers seem to think otherwise, the example has dialectical force.

  27. 27.

    I am supposing that conjunctive and structural properties are not fundamental. Also, I take determinables rather than determinates to be fundamental; see Bricker (2017). If determinates are fundamental, one would want to add a clause saying that no determinate of the property’s determinable is instantiated at the world.

  28. 28.

    As Lewis (1986: pp. 69–80) held. See Bricker (1996) for discussion.

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Bricker, P. Composition as identity, Leibniz’s Law, and slice-sensitive emergent properties. Synthese (2019). https://doi.org/10.1007/s11229-019-02106-y

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Keywords

  • Composition as identity
  • Leibniz’s Law
  • Plural logic
  • Emergence
  • Modal plenitude