Platonism would allow one to reduce the number of predicates to a single multigrade predicate Instantiates(x1, ..., xn, p), by introducing a name p for every property. The resulting language could have one fundamental quantifier ∃, one fundamental predicate Instantiates(x1, ..., xn, p), and lots of names. One could then introduce a “for a, which exists” existential quantifier ∃a in place of every name a, and get a language with one fundamental multigrade predicate, Instantiates(x1, ..., xn, p), and lots of fundamental quantifiers. In this language, we could say that Jim is tall as follows: ∃Jimx Instantiates(x, tallness).
On the other hand, once we allow for a large plurality of quantifiers we could reduce the number of predicates to one in a different way by introducing a new n-ary existential quantifier ∃F(x1, …, xn) (with the corresponding ∀P defined by De Morgan duality) in place of each n-ary predicate F other than identity. The remaining fundamental predicate is identity. Then instead of saying F(a), one would say ∃Fx(x = a). One could then remove names from the language by introducing quantifiers for them as before. The resulting language would have many fundamental quantifiers, but only only one fundamental binary predicate, identity. In this language we would say that Jim is tall as follows: ∃Jimx∃Tally(x = y).
We have two languages, in each of which there is one fundamental predicate and many quantifiers. In the Platonic language, the fundamental predicate is multigrade but the quantifiers are all unary. In the identity language, the fundamental predicate is binary but the quantifiers have many arities.
And of course we have standard First Order Logic: one fundamental quantifier (say, ∃), many predicates and many names. We can then get rid of names by introducing an IsX(x) unary predicate for each name X. The resulting language has one quantifier and many predicates.
So in our search for fundamental parsimony in our language we have a choice:
- one quantifier and many predicates
- one predicate and many quantifiers.
Are these more parsimonious than many quantifiers and many predicates? I think so: for if there is only one quantifier or only one predicate, then we can collapse levels—to be a (fundamental) quantifier just is to be ∃ and to be a (fundamental) predicate just is to be Instantiates or identity.
I wonder what metaphysical case one could make for some of these weird fundamental language proposals.