Goodman and Quine's nominalism and infinity

[This post is largely wrong. -- Note added in December, 2024]

The argument in this post is highly compressed. It's even more a note to self than other posts are.

Goodman and Quine have a very clever nominalist metalanguage that lets them handle first-order logic. There seems, however, to be a serious problem in their system. As it stands, the system will be inconsistent in certain infinite worlds, if it allows excluded middle (which it does, being classical). And they do not have the resources for specifying that the worlds they're using are finite.

The problem stems from the fact that Goodman and Quine nowhere specify that the sentences of their target language are finite. Because they fail to specify that, they cannot rule out infinite sentences corresponding to something like "~~~......~~~p". Think of the front part of it as two infinite sequences of smaller and smaller tildes, with the infinite tails touching. (Goodman and Quine work with Sheffer strokes, and it's a touch harder to explain how such an infinite sentence is done with Sheffer strokes, but I think it can be done, too. I will work with ordinary FOL.)

Now, why would you want to rule out such sentences? Well, write N for the doubly infinite sequence of negations, as above. Then by excluded middle, we have Np or ~Np. But ~Np is the same sentence as Np. Hence, we have Np or Np. Hence we have Np. But again ~Np is the same sentence, so we have both Np and ~Np. And that's pretty bad, because everything now follows by explosion, again because we have a classical logic.

Could Goodman and Quine cleverly exclude such infinite sentences? It seems that they can't do it using their present metalanguage primitives and axioms, nor by means of any straightforward extension of them. For their metalanguage is also classical and first-order, and hence unable to express sentences that "logically imply" that there are finitely many Fs (say, letters in s)—i.e., sentences that are true in all and only all interpretations on which finitely many things satisfy F (this is easy to prove by compactness, and I think does not even require the Axiom of Choice).

That looks pretty much fatal. Except that Goodman and Quine might be able to help themselves if they could use some heavy duty metaphysics to establish that our world either has only finitely many objects or is a single continuous plenum. For given that the world is a single continuous plenum, we might be able to express the idea that a sentence is finite by saying that for every mereological sum of curvy arrows in the world (arrows being certain arrow-shaped parts of the plenum—we need mereological universalism for the system to work) such that every letter of the sentence is at the tail end of exactly one arrow, and every arrow points to a letter of the sentence, and no two arrows point to the same one, every letter of the sentence is pointed to. But this only works given a continuous plenum where there is enough stuff to make enough arrows to ensure this isn't spuriously satisfied. And I doubt there is a good way to express the fact that we have a continuous plenum in the Goodman and Quine system. So the system can only be made to work on a quasi-empirical assumption that the system, apparently, cannot state. And it is bad that whether a logical system is consistent depends on how matter is arranged in the world—if it is arranged in a plenum or there are only finitely many objects, it's consistent, otherwise possibly not.

Another move would be to require that all the letters be exact copies of each other and that they be all in a straight line. There is no way to form "~~~......~~~p" in an Archimedean universe where all the letters are in a straight line. But, again, their logical system will depend for its consistency on the assumption that our world is Archimedean. And that's weird.

Goodman and Quine mention something related to the finiteness issue in footnote 14, in the context of the alternative framed ingredients method. I think the alternative framed ingredients method also requires an assumption of finitude.