Disarming Nominalism

[Aristotle]

I'm in the process of writing a paper about the Thomistic system of thought, which naturally should have a defense of moderate realism imbedded in it somewhere. My intention is to go through all the popular alternatives and show how they're all either self undermining or possess insoluble difficulties.

Conceptualism doesn't seem like it would be too difficult, given that it denies that the intellect actually grasps being, not understanding that the logical consequence of such a stance is an absolute inability to affirm anything whatsoever about anything in being - including the being of thought - leaving no room to say "this is logically consistent," "this is a contradiction," or even "I exist" given that these are statements about the being of something (thus rendering the thinker no more thinking than a plant, as I think Aristotle put once).

Nominalism seems a little more difficult to disarm in a similar way however (though I assure you that I am aware that nominalism is impossible). It seems to me that you can't take the nominalist claim "all concepts we know of are reducible to particulars juxtaposed to a name" and reveal it to be self undermining in a similar way as with conceptualism, despite the fact that it's definitely less true. I could just say "well here are concepts which can't be reduced to particulars in principle" which is fine, but I guess I'm looking to show that nominalism is self refuting - if that's possible, which I feel like it might be. Does anyone have any thoughts? And one more thing: does nominalism necessarily have to result in a form of empiricism, or could it in principle get beyond phantasms in some way?

[Aristotle]

One more thing. Does this syllogism hold true?
Thought is irreducible to particulars
All material being is particular
Thus, thought is irreducible to material being

Premise one would be true given nominalism is false (which it is). Premise two follows from the nature of matter as the principle of individuation of form. But I'm not sure if the conclusion is true given that our intellect is particular as well (given that we don't share one universal intellect), even if it isn't per se on the basis of matter. Wouldn't that mean that premise one rules out the fact that our intellect could be the "hub of thought"?

[tacf]

Maybe "some thoughts are irreducible to particulars"?
I thought we abstracted universals from particulars in the phantasm. Unless thoughts means only universals here. Although I would say that is not how it is used by people in general.
I know St Thomas gave arguments for  "no shared intellect" in the Summa Contra Gentiles but I haven't read it.

[Geremia]

I know St Thomas gave arguments for  "no shared intellect" in the Summa Contra Gentiles

You mean De unitate intellectus contra Averroistas?

[tacf]

Contra Gentiles part 2 chapter 73 was what I had in mind  but yours is probably where he goes the most in depth