I have been thinking a lot. I am considering (and inclined toward) the view that we must restrict ourselves to a first-order language. As I recall, Quine believed this. I think it destroys most of math. But science can, I think, get along with just rational numbers. It is possible to do math as just manipulation of symbols, which have no meaning – except in our imaginations. Also, it seems to destroy my ideas about belief.
After more thought: I am not sure it affects my philosophy of belief. I quantify over symbols for properties and relations – not properties and relations themselves!
Note: (later yet). There could be many properties and relations foe which there are no symbols.
Another Note (Still later). Could you permit quantifying over names of properties and relations of properties and relations and so forth (without quantifying over the properties and relations themselves?
Another Note (Still later) I read in Goble – _The Blackwell Guide to Philosophical Logic_ (page 34) that thhere cannot be adequate descriptions of natural (hence rational) numbers in first-order languages. I had forgotten that — there is so much to remember. I need to work out axioms for my idea – which is not either exactly first-order or higher-order. (at least as I understand these).
Another Note (7/28) I should have known 1st order logic was inadequate to arithmetic – as Goedel’s theorem proves any logic adequate to arithmetic is incomplete, and he also proved st order logic was complete. (At least if my foggy memory & reasoning are correct).