Abstract
The principle of Summation, which is a technically sharpened version of the familiar claim that a whole is a sum of its parts, is presented by Peter van Inwagen as a trivial truth. I argue to the contrary, that it is incompatible with the natural assumption that a whole may gain or lose parts non-instantaneously. For, as I show, the latter assumption implies that something can be determinately a whole without being determinately a sum of parts, and this, in turn, indicates the falsity of Summation. I point out that the tension between Summation and the possibility of non-instantaneous gain or loss of parts compels us to rethink the relations between the concepts of whole and sum, and may have far reaching consequences for the mereology of physical objects.
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Notes
See van Inwagen (1990, p. 30).
van Inwagen’s (1990) affirmation of these theses, and especially of restricted composition, is a recurrent theme of his book. Uniqueness is denied e.g. by Simons (1987, pp. 112–127); restricted composition is denied e.g. by Lewis (1986, pp. 211–213); Rea (1998); Sider (2001, pp. 120–132); changeable composition is denied e.g. by Chisholm (1973).
It should be noted that Summation presupposes what might be described as a generic mereological sense of ‘sum’, which I define in D1 below. This should be distinguished from the more specific classical mereological sense, which is governed by assumptions that entail that composition is unrestricted, unique, and unchangeable. See Simons (1987, pp. 25–41). Yet another use of ‘sum’ occasionally features in discussions of emergence. In this sense a sum is a whole which lacks emergent properties. See Silberstein (2002, p. 81). van Inwagen criticises this emergentist use of the term at (1990, p. 30).
van Inwagen clearly accepts this assumption in his discussion of the way a carbon atom becomes a part of Alice after she drinks a cup of tea in which a lump of sugar has been dissolved. See van Inwagen (1990, pp. 94–96, 217). Merricks (2005, pp. 618–619) seems to accept this assumption, arguing that it entails vague composition but not vague existence. On my view it entails neither vague existence nor vague composition, but only vague summation.
In fact, van Inwagen (1990, p. 29), defines ‘the xs compose y’ as equivalent to ‘y is a sum of the xs, and no two of the xs overlap’.
The form of the definition of ‘overlap’ provided in the brackets here is somewhat more complex than usual, because I am using ‘part’ in the sense of ‘proper part’.
See Lewis (1986, p. 212): “… not all of language is vague. The truth-functional connectives aren’t, for instance. Nor are the words for identity and difference, and for the partial identity of overlap. Nor are the idioms of quantification, so long as they are unrestricted.” It seems to me that Markosian (1998, pp. 222–223) is wrong to suggest that only those who reject the ‘Linguistic Theory of Vagueness’ have reason to disagree with Lewis on this point. To claim that the predicate ‘a part of’ is vague is compatible with subscribing to the view that the source of the vagueness of this predicate is the character of language rather than the character of the reality described by language.
See Williamson (1994), especially Chaps. 7–9).
For a survey of the principal approaches to vagueness, displaying a variety of non-epistemic views as well as the epistemic view, see Keefe and Smith (1996).
It is often assumed by upholders of the non-epistemic view that vagueness is ‘linguistic’ rather than ‘metaphysical’ (or ‘ontic’), that is, that the vagueness of a predicate stems from the character of language (e.g. semantic indecision) rather than that of the reality described by language. It is arguable, however, that some forms of the non-epistemic view commit one to the existence of metaphysical vagueness (Tye 1994), or even that linguistic vagueness is itself a species of metaphysical vagueness. (Merricks 2001).
Someone might reject the possibility of indeterminate parthood without subscribing to the epistemic view, for example, by arguing that while (contrary to what the epistemicist assumes) there is no sharp cutoff between objects that are determinately parts and objects that are determinately not parts of A, no objects are determinately indeterminately parts of A; nor determinately such as to bear an n-th order indeterminacy of parthood towards A, for any finite n. For a discussion of the related notion of infinitely higher-order vagueness, see Graff (2003).
See Lewis (1993, pp. 28–31).
See Fine (1975) for an explanation of the supervaluational approach to vagueness and the notion of a ‘precisification’ (which corresponds to Fine’s ‘specification’).
P1 entails P3, because if y is a sum of the xs, each of the xs is a part of y (Suppose z were one of the xs but not a part of y. Then something might overlap one of the xs without overlapping y, which in view of D1 contradicts the assumption that y is a sum of the xs). By contrast, there is no obvious way to show that P3 entails P1. Note in this connection that y might have the xs as parts, without being their sum (E.g. my liver and kidney are parts of my body, but my body is not a sum of these parts: something might overlap my body without overlapping my liver or my kidney).
It seems that we might deny a sharp transition out of or into parthood without admitting that the part is at some time possessed indeterminately by the whole (perhaps along the lines indicated in footnote 11) only if we deny that the transitional period is determinately non-instantaneous. The plausibility of P5 relies on the plausibility of the assumption that the transitional period is determinately non-instantaneous, which I shall not attempt to defend here.
The implausibility of instantaneous transitions into parthood is arguably more extreme than the implausibility of a sharp cutoff between cases in which composition occurs and cases in which it does not (Sider 2001, pp. 122–124, describes continuous series of such cases and points to the implausibility of such sharp cutoffs). Merricks (2005) argues for such sharp cutoffs in composition, but seems to agree with van Inwagen’s (1990, p. 217) rejection of instantaneous transitions into parthood.
For simplicity, I am ignoring the possibility that the term ‘identical’ is vague. Recall the quote from Lewis in note 7 above, according to which ‘the words for identity and difference’ are among the elements of language that are not vague. To deny that ‘identical’ is vague is not to imply that identity statements cannot be indeterminate in truth value, in case these statements involve vague singular terms. It is to deny, however, what Garrett (1988, p. 130) describes as ‘the more radical thesis that the identity relation can be one of degree’. The vagueness of ‘identical’ would affect our analysis of indeterminate overlap (in D3), but not in a way which substantially affects my argument in this section.
I take it to be obvious that O cannot be at t a sum of objects some of which overlap neither the As nor B.
A variant of this proposal is that there are some parts (more than one) of B, the Cs, such that O is determinately a sum of the As and the Cs at t. An exactly parallel argument to the one I present above will show that this variant fails no less.
Here again I am assuming what I take to be a plausible version of a fundamental classical mereological principle, that of the Transitivity of Parthood, for cases involving indeterminate parthood: If x is determinately or indeterminately a part of y, and y is determinately or indeterminately a part of z, then x is determinately or indeterminately a part of z.
References
Casati, R., & Varzi, A. C. (1999). Parts and places: The structures of spatial representation. Cambridge, MA: MIT Press.
Chisholm, R. (1973). Parts as essential to their wholes. The Review of Metaphysics, 26, 581–603.
Garrett, B. J. (1988). Vagueness and identity. Analysis, 48, 130–134. doi:10.2307/3328215.
Graff, D. (2003). Gap principles, penumbral consequence, and infinitely higher-order vagueness. In J. C. Beall (Ed.), Liars and heaps. Oxford: Oxford University Press.
Keefe, R., & Smith, P. (1996). Introduction: Theories of vagueness. In R. Keefe & P. Smith (Eds.), Vagueness: A reader. Cambridge, MA: MIT Press.
Lewis, D. (1986). On the plurality of worlds. Oxford: Basil Blackwell.
Lewis, D. (1993). Many, but almost one. In J. Bacon, et al. (Eds.), Ontology, causality and mind: Essay in honour of D.M. Armstrong. Cambridge: Cambridge University Press.
Markosian, N. (1998). Brutal composition. Philosophical Studies, 92, 211–249. doi:10.1023/A:1004267523392.
Merricks, T. (2001). Varieties of vagueness. Philosophy and Phenomenological Research, 62, 145–157. doi:10.1111/j.1933-1592.2001.tb00045.x.
Merricks, T. (2005). Composition and vagueness. Mind, 114, 615–637. doi:10.1093/mind/fzi615.
Rea, M. (1998). In defense of mereological universalism. Philosophy and Phenomenological Research, 58, 347–360. doi:10.2307/2653513.
Sider, T. (2001). Four dimensionalism: An ontology of persistence and time. Oxford: Clarendon Press.
Silberstein, M. (2002). Reduction, emergence and explanation. In P. Machamer & M. Silberstein (Eds.), The blackwell guide to the philosophy of science. Oxford: Blackwell.
Simons, P. (1987). Parts: A study in ontology. Oxford: Clarendon Press.
Tye, M. (1994). Sorites paradoxes and the semantics of vagueness. In R. Keefe & P. Smith (Eds.), Vagueness: A reader. Cambridge, MA: MIT Press.
van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press.
Williamson, T. (1994). Vagueness. London: Routledge.
Acknowledgement
I wish to thank Jonathan Berg, Ruth Weintraub and two anonymous referees for very helpful comments.
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Meirav, A. The Principle of Summation. Erkenn 71, 175–190 (2009). https://doi.org/10.1007/s10670-009-9160-3
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DOI: https://doi.org/10.1007/s10670-009-9160-3