One version of nominalism holds that properties just are sets of things. There is no abstract universal red-ness. There just is a set of red things. The problem is that ordinary sentences of English look like they are talking about something that could only be something like a universal. There is no talk of sets.
Consider the following three sentences.
(1) The ball is red
(2) Red is a color
(3) Red resembles orange more than square-ness
Each of these appears to be talking about a property, and proponents of abstract universals maintain that their view makes more sense of the truth of (1)-(3) than nominalism.
Nominalism can handle (1) pretty easily, by offering an acceptable paraphrase.
(1*) The ball is a member of the red-set.
Apparent talk about universals turns into talk about the the ball, and how that ball is a member of some particular set. No universal talk is required.
Paraphrasing (2) is a bit trickier since it seems that red-ness is referred to in the subject slot, but nominalists have offered (2*)
(2*) All things that are members of the red-set are members of the colored-set.
Again, (2*) gets rid of talk about universals. All we have in (2*) is talk about particular objects and sets.
It’s when we get to (3) that Nominalists have difficulties. (3*) is clearly unacceptable.
(3*) All things that are members of the red-set resemble all things that are members of the orange-set more than things that are members of the square set
Why is (3*) unacceptable? Because (3) is true, but (3*) is false. A red square does not resemble an orange goldfish more than a yellow square. That’s enough to falsify (3*), and since you can’t translate a true sentence to a false sentence and have succeeded in preserving the meaning – we have an unacceptable paraphrase.
For a while, I thought that nominalists actually might have a good translation of (3). I thought that the nominalist could translate (3) as…
(3**) The red-set resembles the orange-set more than the square-set.
I thought that you might be able to talk about resemblance relations between properties as resemblance relations between the actual sets.
I’m now worried about this…More on that in a moment…
maybe your translation in (3*) is just a little unclear. if we were to say “the set of red things resembles the set of orange things more than the set of square things” this would get rid of the problem. by saying ‘all things’ you can bring up the problem of orange goldfishes, red squares and yellow squares and probably a whole host of other counterexamples ad infinitum. by referencing the set as a whole as opposed to the particulars it contains we avoid this problem and preserve the truth of the translation.
Hi Scott,
Yea. That’s what I had in mind with (3**)
I was treating ‘the red-set’ as a name for the set of all red things.