The Operator Theory of instantiation ∗

∗This is a version of my paper ‘What Is It For a Particular to Instantiate a Universal?’ given on 6 July 2005 in ‘The Philosophy of D. M. Armstrong’ stream of the Australasian Association of Philosophy Conference. I would like to thank all who commented especially David Armstrong himself. Many thanks also to an anonymous referee.

Abstract

Armstrong holds the Supervenience Theory of instantiation, namely that the instantiation of universals by particulars supervenes upon what particulars and what universals there are, where supervenience is stipulated to be explanatory or dependent supervenience. I begin by rejecting the Supervenience Theory of instantiation. Having done so it is then tempting to take instantiation as primitive. This has, however, an awkward consequence, undermining one of the main advantages universals have over tropes. So I examine another account hinted at by Armstrong. This is the Operator Theory of instantiation, by which I mean the theory that universals are operators, and that a particular instantiates a monadic universal because the universal operates on the particular, resulting in the state of affairs. On this theory the state of affairs supervenes on the instantiation rather than vice versa. In the second part of the paper I develop this theory of universals as operators, including an account of structural universals, which are useful for accounts of modality and of mathematics.

Notes

∗This is a version of my paper ‘What Is It For a Particular to Instantiate a Universal?’ given on 6 July 2005 in ‘The Philosophy of D. M. Armstrong’ stream of the Australasian Association of Philosophy Conference. I would like to thank all who commented especially David Armstrong himself. Many thanks also to an anonymous referee.

¹As Kim 1993: 165 – 74] points out, to say that there is dependent supervenience is not to narrow down the kind of supervenience but merely to make the further claim that the subvenient explains the supervenient.

²Armstrong 1978a: 110] suggests that it might be perspicuous to write a typical monadic universal as F( ). To be sure he also suggests writing the particular as ( )a, which privileges monadic over polyadic universals, contrary to his expressed position.

³This is a rather different argument from Tooley's 1977, which, being averse to fictionalist accounts, I also endorse in spite of Armstrong's spirited defence 1983: chap. 8].

⁴Such as John Bacon 1995: 17 – 18], who advocates a single ‘likeness relation’, while noting that D. C. Williams 1953 and Keith Campbell 1981 consider tropes to have degrees of resemblance.

⁵An aspirin a day decreases the probability of a fatal heart attack. Decreasing the probability of a fatal heart attack increases the probability of death in other ways, and in particular by cancer. It does not follow that an aspirin a day increases the probability of death by cancer. For aspirin also helps protect against some cancers. Increasing probability is not, therefore, transitive.

⁶Suppose, however, we adopt a non-classical mereology, as Armstrong is committed to because he holds that properties are parts of states of affairs. Then we may in fact give a mereological account of singletons. See Forrest 2002.

⁷I am ashamed to say I overlooked this until it was pointed out to me by Michaelis Michael inconversation.