Abstract
Armstrong’s combinatorial theory of possibility faces the obvious difficulty that not all universals are compatible. In this paper I develop three objections against Armstrong’s attempt to account for property incompatibilities. First, Armstrong’s account cannot handle incompatibilities holding among properties that are either simple, or that are complex but stand to one another in the relation of overlap rather than in the part/ whole relation. Secondly, at the heart of Armstrong’s account lies a notion of structural universals which, building on an objection by David Lewis, is shown to be incoherent. I consider and reject two alternative ways of construing the composition of structural universals in an attempt to meet Lewis’ objection. An important consequence of this is that all putative structural properties are in fact simple. Finally, I argue that the quasi-mereological account presupposes modality in a way that undermines the reductionist aim of the combinatorialist theory of which it is a central part. I conclude that Armstrong’ quasi-mereological account of property incompatibility fails. Without that account, however, Armstrong’s combinatorial theory either fails to get off the ground, or else must give up its goal of reducing the notion of possibility to something non-modal.
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Notes
It is unclear whether Armstrong still subscribes to combinatorialism. In his recent book Truth and Truthmakers, Armstrong argues that the relation between a particular and the universals it instantiates is necessary. This robs particulars and universals of the combinatorial freedom they require to yield all the possibilities (see Armstrong 2004 and Kalhat 2008a).
There are also incompatibilities (and entailments) among relations, but I shall not dwell on those here. For Armstrong’s handling of incompatibilities among relations, see Armstrong 1989: 84–6.
Armstrong does not think that the relation between a complex property and its constituents is merely the part/ whole relation. For example, the conjunctive property P&Q is something more than the mereological sum P+Q-the ‘something more’ being the requirement that the same particular which instantiates P also instantiates Q. But having said this, Armstrong thinks that ‘this extra condition does not abrogate the simple mereological relations that hold between [them]’ (1997: 53). So, while Armstrong’s conception of properties is only ‘quasi-mereological’, I will nevertheless follow him in speaking of the relation between a complex property and its constituents as the part/ whole relation.
A more straightforward explanation for the incompatibility of the properties being exactly two kilograms in mass and being exactly five kilograms in mass is that since the former is the property of being at least two kilograms and not more than two kilograms in mass and the latter is the property of being at least five kilograms and not more than five kilograms in mass, the incompatibility is an outright logical contradiction. But not all property-incompatibilities reduce in this way to logical contradictions, e.g., shape incompatibilities (see below for further examples).
For his naturalism, see Armstrong 1997: 5–6.
Armstrong gives an account of the resemblance of determinates of a determinable in terms of partial identity (1997: 55; 1978: 116–124). Determinates of a determinable resemble one another (to varying degrees) because they are partially identical. The closer the resemblance between any two determinates, the more they overlap, and hence the closer they get to complete identity.
Armstrong is aware of this difficulty; he writes: ‘It may be conceded at the outset that the unity of the class of shapes is a much messier affair than lengths, durations, masses which all arrange themselves as simple one-dimensional arrays’ (1997: 55).
This is not to say that all complex properties are conjunctive properties. A complex property is conjunctive only if it is the very same particular that instantiates each of the conjuncts; otherwise, it is structural.
Lewis also objects to structural universals on the grounds that they violate the Principle of Uniqueness of Composition, which says that for a given number of parts, there is only one whole that they compose. Kalhat 2008b refutes this objection.
This is, in essence, Armstrong’s ‘One over Many’ argument for universals. He writes: ‘I would wish to start [by saying that] many different particulars can all have what appears to be the same nature and draw the conclusion that, as a result, there is a prima facie case for postulating universals’ (1980: 102).
Lewis’ objection does not pose a similar problem for conjunctive universals. Since the very same particular which exemplifies a conjunctive universal also exemplifies its conjuncts, the situation where that particular exemplifies a universal F, and exemplifies another F, and so on, simply cannot arise.
It might be objected that on Bigelow and Pargetter’s account, being hydrogen is a part of being methane, since it is a part of the property of having a part that instantiates being hydrogen, and the latter property is indeed a part of being methane. However, if being hydrogen is taken to be a part of the property of having a part that instantiates being hydrogen, then being hydrogen will also be a part of the property of having a part that is distinct from the first and instantiates being hydrogen, and having another such part, etc. But then the property of being hydrogen occurs four times over in the structural universal being methane, which is impossible. Bigelow and Pargetter must therefore deny that being hydrogen is a part of the properties of having a part that instantiates being hydrogen, and having another such part, etc. This means that they must either deny the transitivity of the parthood relation, or deny that the more complex property of having a part that instantiates being hydrogen is a part of being methane at all.
Couldn’t Armstrong understand ‘following from’ syntactically, in terms of derivability in some rich logical system? The matter is complex and cannot be pursued here, but let me indicate why I think that this possibility is not open to Armstrong. The notion of derivability itself appears to hide modality, i.e., ‘X is derivable from Y’ means ‘it is possible to derive X from Y’. To avoid the modal construal of derivability, the combinatorialist should, I think, insist that ‘X is derivable from Y’ means that there is a derivation of X from Y. But since, of course, not all derivations have in fact been carried out, this suggestion requires a commitment to Platonism: derivations are abstract objects. A naturalist like Armstrong, however, cannot countenance abstracta. Therefore, I do not think that he can pursue the syntactic approach, in which case he must embrace the standard and modal understanding of ‘follows from’ as entailment. (See Lewis 1986b: 150–57 for related difficulties).
I thank the referee for this journal for the suggestion that follows.
For what it is worth, I do not think that that necessity can in fact be reduced.
References
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Acknowledgments
I wish to thank Hanjo Glock for valuable comments on previous drafts of this paper.
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Kalhat, J. Is There A Quasi-Mereological Account of Property Incompatibility?. Acta Anal 26, 115–133 (2011). https://doi.org/10.1007/s12136-010-0090-0
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DOI: https://doi.org/10.1007/s12136-010-0090-0