Abstract
This paper provides a defence of the account of partial resemblances between properties according to which such resemblances are due to partial identities of constituent properties. It is argued, first of all, that the account is not only required by realists about universals à la Armstrong, but also useful (of course, in an appropriately re-formulated form) for those who prefer a nominalistic ontology for material objects. For this reason, the paper only briefly considers the problem of how to conceive of the structural universals first posited by Armstrong in order to explain partial resemblances, and focuses instead on criticisms that have been levelled against the theory (by Pautz, Eddon, Denkel and Gibb) and that apply regardless of one’s preferred ontological framework. The partial identity account is defended from these objections and, in doing so, a hitherto quite neglected connection—between the debate about partial similarity as partial identity and that concerning ontological finitism versus infinitism—is looked at in some detail.
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Notes
As they bestow analogous characteristics on their bearers, but differ in the specific manner in which they do so.
This is because he argues that if there are universals at all, then there must be structural universals. For a counter-argument, see Williams (2007).
In the case of resemblance nominalism, clearly, what is postulated to have internal structure are concrete particulars, not properties.
The reasons for this caveat will become clear later.
The reason why the sketch of a more precise treatment of partial similarity as partial identity is confined to a short Appendix is that, although an increase in the mathematical precision of the claims made in the context of discussions of PIT might be deemed desirable, the present paper is intended as a ‘philosophical preliminary’ to any attempt at performing such task.
Armstrong’s example involves Greenness as opposed to Electron-hood: while we cannot speak of ‘a green’, we can meaningfully say ‘an electron’, and this is because Electron-hood is a particularising universal while Greenness isn’t. Indeed, we talk of the property of being green and of the property of being an electron.
For a recent discussion and defence of Armstrong’s views, see McDaniel (2009).
Importantly, the information encoded in the relational properties being invoked is, on the second construal, purely structural.
One may worry that it is more natural to think that relations ground relational properties than the other way round (see, for instance, Hawley (2009)). In that case, the argument just offered would need to be backed up by additional considerations. There is no space to do this here, but their suggested role in characterising at least certain structures unambiguously might in itself count as a reason for taking relational properties to be more fundamental than relations (at least in some cases).
One could also take structures as non-analysable primitives. However, this would force one to take partial similarities grounded in structural differences to also be primitive and non-analysable, which is clearly at odds with the basic idea that underlies PIT.
The reason why the magnitude of a vector property may be encoded in more than one component is also the reason for the magnitude component not being comparable with the angle components: the magnitude of the vector may consist of more than one ‘determination dimension’, and the magnitude and angle components of vector properties in fact correspond to distinct determination dimensions. For more details on the notion of a determination dimension see below, “Denkel and Gibb’s Objection(s)”.
See, for example, Beisbart (2009). Busse (2009) voices another worry: he argues that the view just suggested fails to do justice to vectorial quantities such as those dealt with, e.g., in classical electrodynamics, which are not described as composites of several autonomous components. Here we cannot get into details, but it looks like Busse’s worries can be obviated by specifying constraints that the component scalar quantities must obey when combining into vectorial ones. To be sure, Busse doesn’t show (nor claims) that what he calls the ‘multiple quantity conception’ of vectorial properties is inconsistent.
Forrest points out that fields need not be composed of points, but it is not necessary to discuss this here, especially in view of the fact that Forrest shows that his point-based proposal extends quite straightforwardly to non-point-like vector quantities.
Notice, incidentally, that Eddon’s point b) ignores the fact that the impossibility to isolate the parts of point particles may in fact have an explanation. Consider for example the phenomenon of quark confinement: the constituent quarks of elementary particles cannot exist on their own for physical, not metaphysical, reasons.
The parenthetical qualifications are needed because two properties can be partially similar even if one of them is simple: the simple property will be identical—qualitatively or numerically—to one of the two or more constituents of the other property.
Trope theorists will argue that (some) objects can exemplify exactly similar tropes, and resemblance nominalists will distinguish between, say, the classes of similarity composed by ‘P things’, by ‘P&P things’ etc.
Set aside for a moment the suggestion about conjunctive properties made at the end of the previous section.
This clearly assumes that if the two aspects were identical, the initial resemblance would be exact rather than partial, i.e., that the considerations about purely structural differences made earlier in connection to Pautz’s objection are temporarily bracketed.
Strictly speaking, this is not correct. The non-identical aspect is T while the identical aspect is 2T. But what Gibb means is clearly that the identical aspect and the non-identical aspect have themselves an identical aspect and a non-identical aspect which are identical: T. In the terms of our schematic example, this is to say that P and Q can be partially similar without being of the form A-B and A-D but, rather, of the form A-A-A and A-A. That is, one of the properties may contributes nothing to the non-identical aspect.
For further discussion, see Morganti (2009).
Notice that, quite importantly, the infinitist strategy doesn’t involve an assumption that the world actually is infinitely composed (as Denkel, Gibb and, to some extent, Eddon—following Armstrong, seem to think), but just that it might be. The Gibb strategy, instead, requires that every property that resembles another actually is ultimately composed of basic identical simples.
It is interesting to emphasise that the Gibb strategy does not turn out to be preferable upon any plausible conception of properties. Consider Schaffer (2004) distinction between the ‘scientific’ and the ‘fundamental’ view of properties (both based on the idea that empirical science is the best guide for identifying the properties that truly are ‘out there’—Armstrong’s ‘scientific approach’—but the former allowing for several levels of genuine properties and the latter taking (some) physical properties to be ontologically basic). On the fundamental conception, at present we should consider basic the properties described by the Standard Model of elementary particles. However, although they are quantitative, these properties are not analysable in terms of simple basic constituents (clearly, to simply claim that even more basic properties will be discovered by physicists in the future is question-begging). On the scientific account, the infinitist strategy comes out at least on a par with the Gibb strategy, for (as Schaffer argues well) the assumption that there is no unique level of fundamental entities and/or properties naturally leads to the idea that there are no simple entities and/or properties at all because there is no fundamental level. As a matter of fact, it could even be legitimately argued that, all things considered, the alleged vicious regress individuated by Denkel and Gibb points to the best option available to the supporter of PIT!
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Appendix
Appendix
The claims in the present paper can be usefully summarised employing a slightly more rigorous approach. The simplest case is of course that of a complex property with only one determination dimension and a finite number of basic constituents that are either mere ‘aggregated’ or structured in simple ‘linear’ fashion. An example of such a property is mass. In this case, a measure of similarity is straightforwardly obtained via a count of constituents. If the number of constituents is infinite, the same holds, and the only difference is that there is no basic level of analysis and the ‘units of measure’ can be arbitrarily small. For complex properties with n determination dimensions (each one composed of basic constituents), similarity can be measured employing the tools of analytic geometry. Set, for instance, n = 2: given an xy-plane and taking each axis to correspond to one determination dimension, two points (x 1, y 1) and (x 2, y 2) in it will represent two determinates of the relevant (two-dimensional) determinable, and their degree of similarity will be given by the distance between the points in the plane, which is \( d\left( {\left( {{x_1},{x_2}} \right),\left( {{x_2},{y_2}} \right)} \right) = \surd \left( {{x_2} - {x_1}} \right)\left( {{y_2} - {y_1}} \right) \). The same holds for any value of n, of course: one just needs to add other elements to the square root to the right of the above equality ((z 2-z 1) etc.). Notice that, given what was said earlier about positive and negative quantities being determinates of different determinables, one doesn’t need to employ the full Cartesian (or at any rate n-dimensional) space, but just one sub-space with positive values along all axes. Something similar works for vector properties analysed in terms of spherical coordinate systems as suggested in the paper. Considering the latter, however, one may envisage cases in which the determination dimensions are not ontologically on a par, i.e., one of them is ‘more fundamental’ as a component of the relevant complex property. In particular, the magnitude of a vector property may legitimately be deemed ‘more important’ than its two components determining spatial orientation. In this case, either one weighs the relevant constituents accordingly (which, admittedly, may be irredeemably arbitrary), or one has to posit two or more specific similarity spaces that are, strictly speaking, mutually incomparable. As we have seen, if vectors are instead understood á la Forrest in terms of pairs of mutually related field magnitudes, they become analogous to structural properties. As for the latter, as argued in the main text, similarity with respect to constituents must be considered alongside similarity with respect to structural facts. This second sort of similarity can be expressed, as we have seen, either in terms of relational properties that make specific reference to monadic constituents of the structural property, or of relational properties expressing purely structural facts. In both cases, the structural information is encoded in multi-dimensional properties that (independently of whether or not one takes them to be genuine constituents of structural properties) can again be analysed by having recourse to the notion of a determination dimension and to analytic geometry along the lines just sketched.
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Morganti, M. The Partial Identity Account of Partial Similarity Revisited. Philosophia 39, 527–546 (2011). https://doi.org/10.1007/s11406-010-9290-5
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DOI: https://doi.org/10.1007/s11406-010-9290-5