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I am aware that Nominalism comes in at least two flavors, one in particular is the denial of universals. Under this paradigm of Nominalsim, is it possible for a mathematician or even a logician to be a Nominalist? Can s/he outright deny universals while simultaneously accepting that given a universe U containing all objects under consideration, the proposition ∀x in U : p(x) is true for some propositional function p? If so, is it due to the individual merely carrying out the logical consequences of a given statement, while withholding belief concerning its truth with regard to reality broadly construed (e.g. physical reality, mathematical reality, etc)?

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Yes, a nominalist logician can do so, and even preserve classical logic and while denying existence of both abstract objects and universals. What gives? A nominalist changes the standard semantics instead, how truth values are assigned to predicates and quantifiers. Traditional assignment formalized by Tarski in 1936 requires a universe of objects with properties, and P(a) is evaluated true if object a has property P. But according to nominalism access to these universes is mysterious and their existence is redundant, because all that is actually involved in doing mathematics is mundane manipulation of symbols. In a way this reverses the Plato's cave metaphor. If all we ever see are shadows on the wall (symbols) there is no reason to leap to ideal forms that supposedly cast them (abstract objects and universals). Whether they exist we can never know, and the real task is to describe what we actually deal with anyway.

It seems strange at first that one can quantify without anything to quantify over, but consider an example. Suppose P is "divisible by 17". If we want to evaluate P(243) we would divide 243 by 17 and see if it is an integer, or something like that. To do the division we could use a pocket calculator, or paper and pencil algorithm, or even perhaps do it in our head. But nowhere in the process do we make contact with Number 243, the abstract object, or Divisibility by 17, the universal, only signs and symbols are involved throughout, even when simulated in our head. Once we know how to assign truth values we can quantify in a standard way: ∀xP(x) simply means that P(a) is evaluated true for all relevant symbols a. Moreover, in cases of "infinite" universes evaluation of ∀xP(x) and ∃xP(x) may not even involve specializing to any P(a), as the standard semantics would have it, but rather giving general proofs, a more complex manipulation of symbols. In particular, ∃xP(x) does not mean that there is some a with property P, but rather that some composite symbol P(a) is evaluated true, or that ∃xP(x) is proved true. In other words, despite the name the "existential" quantifier has nothing to do with existence.

This is called deflationary nominalism. Another popular form of nominalism is fictionalism, which evaluates P(a) in "as if" fashion of fictional narratives, similar to "Pegasus is a flying horse" being true-according-to-Greek-mythology despite the non-existence of Pegasus or flying horses.

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