## Sunday, December 8, 2013

### A nominalist reduction

Suppose that there were only four possible properties: heat, cold, dryness and moistness. Then the Platonic-sounding sentences that trouble nominalists could have their Platonic commitments reduced away. For instance, van Inwagen set the challenge of how to get rid of the commitment to properties (or features) in:

1. Spiders and insects have a feature in common.
On our hypothesis of four properties, this is easy. We just replace the existential quantification by a disjunction over the four properties:
1. Spiders and insects are both hot, or spiders and insects are both cold, or spiders and insects are both dry, or spiders and insects are both moist.
And other sentences are handled similarly. Some, of course, turn into a mess. For instance,
1. All but one property are instantiated
becomes:
1. Something is hot and something is cold and something is dry but nothing is moist, or something is hot and something is cold and something is moist but nothing is dry, or something is hot and something is dry and something is moist but nothing is cold, or something is cold and something is dry and something is moist but nothing is hot.
Of course, this wouldn't satisfy Deep Platonists in the sense of this post, but that post gives reason not to be a Deep Platonist.

And of course there are more than four properties. But as long as there is a finite list of all the possible properties, the above solution works. But in fact the solution works even if the list is infinite, as long as (a) we can form infinite conjunctions (or infinite disjunctions—they are interdefinable by de Morgan) and (b) the list of properties does not vary between possible worlds. Fortunately in regard to (b), the default view among Platonists seems to be that properties are necessary beings.

Heath White said...

My suspicion is that this "solution" would deprive (meta)physical theories of explanatory power. For suppose we wanted to say, "For every property there is a (different) opposite property" -- we could perhaps come up with something logically equivalent (a long disjunction) but we would have abandoned any kind of explanatory ambition.

Alexander R Pruss said...

Agreed, and that's a nice example. I have van Inwagen in mind here as a target, though, and he has no explanatory ambitions of any kind in his Platonism.

But maybe your objection can still be run. One way is to say that my story may work for sentences where properties appear in two predicates, the instantiation predicate ("Smith has roundness") and the identity predicate ("green = blue"). But there are many more predicates one may apply to properties. How do we run the story about those? The worry is that for each such predicate, we need to hard-code which properties (or tuples of properties) satisfy it, and that makes the theory even more unwieldy.

Divide up predicates that apply to properties into two classes. Type I predicates are such that the applicability of the predicate to properties can be defined in terms of identity and instantiation and predicates applying to things. Type II predicates is everything else.

A potential example of a Type I predicate is "are incompatible". "P and Q are incompatible iff necessarily: (x)~(x has P and x has Q)." But there seem to be predicates of Type II. E.g., "are similar". That two properties are similar doesn't mean that their instances similar.

Maybe one can handle some or all Type II predicates with a "qua" connective, but that's pretty hairy. (Thus P and Q are similar iff necessarily:(x)(y)(x has P and y has Q implies x qua having P is similar to y having Q. But even here there are going to be technical difficulties. What if P and Q can't be instantiated in the same world?)

Alexander R Pruss said...

I think there is an important lesson here. The Quinean discussion of ontological commitment is focused on quantifiers and identity. But discussion of the applicability of predicates other than identity (e.g., "are similar" or "are opposite") may be just as important, or even more important, for particular reductive proposals.