Homework V.3

First of all, sorry to Dan for bumping down his new and excellent post. Here's my homework for the week:

Here is a revised version of the objection from alien natural properties. I believe I’ve incorporated the changes suggested by Dan and Chelsey (with respect to the original version of the objection I had presented). Here goes:

(1) There is at least one world in which there is an individual that instantiates an α-alien natural property.
(2) If there is at least one world in which there is an individual that instantiates an α-alien natural property, then there is an infinite sequence of α-alien instantiable natural properties.
(3) If there is an infinite sequence of α-alien instantiable natural properties, then it is not the case that the GR analysis of the modal concepts is complete.
(4) If it is not the case that the GR analysis of the modal concepts is complete, then it is not the case that the ontological component of GR (postulates O1-O11 and the principle of recombination) provides an accurate analysis of the modal concepts.
(5) It is not the case that the ontological component of GR (postulates O1-O11 and the principle of recombination) provides an accurate analysis of the modal concepts.(1-4)*

Divers provides a justification of the move from (1) above to the conditional in (2) with the following sub-argument. Let ‘L1’ designate the exhaustive list of actually instantiated natural properties.

(1a) L1 is the list of all the actually instantiated natural properties.

We have it pre-theoretically that, whatever natural properties are included in L1, it could have been the case that L1 contained one more natural property. In other words, (2a) and (3a):

(2a)(1a)-->by ‘alienation’ and recombination there exists a world W1 such that the individuals in W1 instantiate all the properties in L1, plus some natural property X1 not contained in L1. Call this list L2.
(3a)By ‘alienation’ and recombination there exists a world W1 such that the individuals in W1 instantiate all the properties in L1, plus some natural property X1 not contained in L1. Call this list L2.

The move from (1a)-(3a) above establishes (1) in the main argument. Next, since the (individuals in) the actual world could have instantiated L2 (instead of L1), we have:

(4a)(3a)-->the actual world could have instantiated all the properties in L2.
(5a)The actual world could have instantiated all the properties in L2.

However, if the actual world could have instantiated L2, then it is also the case that the actual world could have instantiated some new list of natural properties, distinct from (L1 and L2). In other words, reiteration of the move in (2a) will generate an infinite sequence of worlds containing more and more alien natural properties:

(6a)(5a)-->successive applications of alienation + the principle of recombination will generate an infinite sequence of worlds containing more and more alien natural properties.
(7a)Successive applications of alienation + the principle of recombination will generate an infinite sequence of worlds containing more and more alien natural properties.

And, if that’s right, then there exist infinitely many instantiable alien natural properties:

(8a)(7a)-->(9a)
(9a)There are infinitely many instantiable alien natural properties.

To justify (3) Divers argues that (i) the ontological postulates of GR are unable to generate a set of worlds such that an infinite sequence of α-alien natural properties are instantiated among individuals existing at those worlds and (ii) that completeness requires GR to provide a world to match all of the possibilities we intuitively think there are. And finally, Divers justifies (4) by arguing that, since completeness is one criterion of extensional accuracy, it follows that if GR does not deliver completeness, then it does not deliver an accurate analysis of the modal concepts.

I proposed that we conceive of the natural properties as tropes. This suggestion did not receive a response, so I will just restate it briefly. Please note that I don’t necessarily think this is a satisfying response to the problem, only that it seems perfectly consistent with what Lewis says on the topic of natural properties. For Lewis says the following:

I noted that one way to characterize natural properties was to help ourselves to Armstrong’s theory of universals. But in ‘Against Structural Universals’ I note that one part of that theory gives us trouble: the part about structural universals. Those are universals built up somehow- the problem is how?- out of simpler universals. I note that a similar theory of structural tropes-particular property instances- has no parallel problem; and, further, that accepting a sparse theory of tropes would be another good way to give ourselves the distinction between natural and gerrymandered properties.**

So: Lewis thinks that it is plausible to suppose that the natural properties are tropes. I suggested that the standard metaphysical account of the nature of tropes leads to the following two principles:

(T1)∀x {[Wx-->∃y (Ty & Iyx)] & ∀z (Tz & Izx)--> z = y}
(T2)∀x {[Wx -->∃y (Ty & Iyx)] & ∀z [(Wz & z ≠x)-->~Iyz]}

(where Wx is ‘x is a possible world’, Tx is ‘x is a class of natural-quality tropes’ and Ixy is ‘x is instantiated by y’).

And, if this is right, then the argument given by Divers in support of (2) is blocked, since the conditional in (2a) would be false. I think it would go something like this:

(1t) All and only the perfectly natural properties are tropes.
(2t) (1t)--> ~[(1a)-->(3a)]
(3t) ~[(1a)-->(3a)] (1t,2t)
(4t) ~(1a) v ~(3a) (3t)
(5t) ~ ~ (1a)
(6t) ~ (3a) (4t, 5t)

(5t) seems right if we think it is plausible to suppose that there is some sort of ideal list of the natural properties that are actually instantiated. So it seems like the above is one argument Lewis could give on behalf of GR. It doesn’t seem to be a happy one, but it would work. The view would appear to be one according to which extreme modal haecceitism is true, since this is a consequence of identifying the natural properties with tropes.