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Part of the essence of the Hume-Kant counterargument against the ontological argument is that, as Hume put it, there is no being the nonexistence of which implies a contradiction. (Kant talks about how a contradiction is never "left over" after we suppress both a subject and all its predicates in thought, IIRC.)

My question is whether the description "the nonexistence of which implies a contradiction" is itself contradictory, then. It doesn't quite look like it; it doesn't seem to conflict with the concepts of existence or nonexistence or contradictions in such a way.

There's an article out there that looked (from its first page) as if it would be about this very topic, but I only had access to that one page.

Kristian Berry
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    If you are asking about this article specifically, the author charges Hume with a self-contradiction for holding both that "whatever we conceive, we conceive to be existent" and "whatever we conceive as existent, we can also conceive as non-existent". Basically, this is a derivative of what Quine called Plato's beard:"Nonbeing must in some sense be, otherwise what is it that there is not?" Ironically, the distinction between existence in thought and existence in reality that Anselm made is one way to take care of it. – Conifold Apr 25 '21 at 01:07

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First of all, Kant's rationalistic counterargument is a little different than Hume's empiric counterargument as referenced here. Kant clearly identified analytic and synthetic versions of the ontological argument and criticized the analytic version as follows:

If the proposition is analytic, as the ontological argument takes it to be, then the statement would be true only because of the meaning given to the words. Kant claims that this is merely a tautology and cannot say anything about reality.

So for Kant in this a prior case, "the nonexistence of which" is just those which are not entailed by any tautology, thus no classic logical contradiction here. The ontic argument is simply a self-claimed tautology at best...

While empiricist Hume argued that nothing can be proven to exist materially using only a priori reasoning:

...there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.

So clearly Hume's counterargument is similar to Popper's scientific empiric falsification principle, he regarded using a prior pure reason to prove some material existence fact is a category error. Since we cannot effectively falsify either such conceivable existence or conceivable nonexistence from our sense experiences, thus for Hume "the nonexistence of which" cannot imply contradiction either. But the reasoning is subtly different from Kant's above...

Double Knot
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  • Kant really has multiple counterarguments in play, here, one of which overlaps Hume's (to the extent that I've read Hume's). But now my question might be formulated as, is the sentence, "∄x → (A & ~A)," incoherent? It doesn't have to be that the contradiction implied by the nonexistence of x is something like "x exists and doesn't exist," I suppose. – Kristian Berry Apr 25 '21 at 02:15
  • Although then I'd be saying, "There exists an x such that if there did not exist this x, then some contradiction or other would be true," which sounds really weird. – Kristian Berry Apr 25 '21 at 02:18
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    @KristianBerry your above purely conceived proposition sounds like nothing but a definition of necessity in disguise. Unless we can directly or indirectly verify x's existence in reality (thus effectively falsify its nonexistence), there's no contradiction I can detect here... – Double Knot Apr 25 '21 at 02:29
  • @KristianBerry personally besides Kant and Hume's counterargument above, I see a few other weaknesses in Anselm's proof. For an idealist mind and reality is conflated somehow, one cannot assert definitely that a thing maximally exists both in mind and reality must be greater than that only exists maximally in mind... Another weakness is the assumed essentialism by Anselm, it's not certain we can assume to combine all essential properties into one existence, it's not chemical molecules which can be safely assumed so... – Double Knot Apr 25 '21 at 02:43
  • ∄x → (A & ~A) is not a wff. If you wrote instead ∄xFx → (A & ~A), that would be equivalent to ∃xFx. It is not incoherent, just a logically contingent proposition and dependent for its truth on the interpretation of F. That something exists is a logical truth of standard FOL, so the non-existence of everything would entail a contradiction, but it is logically contingent to predicate anything of that something, except perhaps its self-identity. – Bumble Apr 25 '21 at 03:02
  • @Bumble your point is exactly what Kant's synthetic counterargument leveraged which I've omitted in my answer. Kant famously emphasized that in the synthetic case, God's existence cannot be any predicate F, thus there's no logical way to prove such existence under any logical form in the synthetic case... – Double Knot Apr 25 '21 at 04:49