Just my interpretation. To briefly speak to the OP, and much like Pfhorrest, my opinion is that Kant used the term 'predicate' loosely as a mantra to poke-holes in the ontological argument (a priori/analytical judgements-of course). And that his ongoing mantra (critique) is simply to expose the inescapable truth that we can never really know existing things-in-themselves (and the true nature of their/our existence) through pure reason alone-a priori. Accordingly, he taught math, and believed that there were limitations to such a priori truths... .
Thus, from Pfhorrest's quote: "What Kant meant was that existence isn't a property of a thing. It's not like you can give a list of all of the properties of a thing and "existence" will be one of them. It's even more the case that you can't bake "existence" into the analytic definition of a thing"
Kant was right when he basically said existence can never be conceived by reason alone. (As you've stated, I think you already know that- 'the synthetic a priori'- but just wanted to offer another opinion to maybe arrive at some consensus here.)
His attack was on analytical philosophers... . — 3017amen
So to quote Kant directly: "Being is evidently not a real predicate, or concept of something that can be added to the concept of a thing". — 3017amen
That is a problem, because a thing that does not exist, cannot have predicates. — alcontali
Of course in thinking about it all, the question of whether mathematics is a human construct, or whether it is something already existing 'out there' rears its head... . One thing we do know is that; it is timeless a temporal, Platonic, metaphysical, a priori etc. much like the human concept of God. — 3017amen
Do you have any good links for Carnap? — 3017amen
The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem. — Wikipedia on Carnap's diagonal lemma
Concerning existence as a predicate, if existence were a predicate, something that does not exist would have the predicate of non-existence, i.e. the negation of the existence predicate, but that is not possible because something that does not exist cannot have any predicates at all. — alcontali
Now I know this is getting on slippery ground, but on first glance it seems like the products of our imagination have predicates. E.g. The word "unicorn" refers to an imaginary mythological creature that has various imaginary properties. — EricH
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.