A universal is just a quantification of a term in a claim or statement or predication. Is the "denial" of universals saying that no one can use the word "no" and other universal quantifiers? If so someone's flat out of luck, in the nature of the case, because of what that statement itself asserts.
Moreover---and I'm following Grisez and Boyle very closely here---once communicated, that universal denial becomes performatively and empirically inconsistent---not "just theory", as the presets of loop guruism would say. Notice that I'm no longer distinguishing between atheists and theists, except of course concerning The Grand Conclusion itself.
Because the set of communications is itself quantified so as to affect the set of objects with that name, "communication" (data transfer, sends, etc., call it what you like, doesn't change a thing)---including that communication itself---that same communication is thereby itself neutralized because of a self-imposed impossibility.
Kordig, the "Man With No Name" of the philosophy of logic, turned this kind of analysis into a science, and---sure enough---he was a theist. He knew well that divine mindfulness was inherent and necessarily assumed in the mind-ruling authority of how we treat our most basic and irreducible assumptions.