Tuesday, December 27, 2016

Some weird languages

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: ∃JimxTally(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.


Heath White said...

On your suggestion for reducing all predicates to “instantiates”: Socrates instantiates wisdom but wisdom does not instantiate Socrates. In fact, necessarily, nothing instantiates Socrates. That would be an odd kind of necessity and we could re-identify the objects as those things which necessarily don’t instantiate anything. (I think this is what Wittgenstein was getting at when he said that the object/concept distinction in formal logic is shown rather than said.)

On your suggestion for reducing all predicates to identity plus a plethora of quantifiers: I’m not certain that this is still a quantifier. What does E[tall]x quantify over? The tall things, presumably; but that basically reintroduces predicates. Tall(x) is easily defined as “Vx[tall]x”. Is the difference between E[tall]x(x=a) and Tall(a) anything more than a notational difference? It’s a bit hard to figure because once you eliminate predicates, there is no such thing as universal instantiation or existential introduction, so I can’t tell if there are any proof-theoretic differences or not. (Maybe those are differences.) But if all we care about is model-theoretic differences, and the reduction is defined so that there are none of those, then it really looks like a notational difference.

Alexander R Pruss said...

1. There may be properties like being a square circle that can't be instanced.
2. That's a very good question. I think names introduced with existential elimination or used for universal introduction will be labeled with the type of quantifier, and universal elimination and existential introduction will require a variable of the right type. But I'm not sure this is enough for the logic to be complete.

Heath White said...

RE: the first suggestion. The difference, rather, is that you can perform logical operations on tallness (~tall, tall v short, tall * round) but not on Socrates. I remember reading this in Dummett a long time ago and could probably find the reference if you're interested.

Alexander R Pruss said...

That's a good point, but it depends on whether we have sparse or abundant Platonism. On abundant Platonism, indeed we can. On sparse, not necessarily.