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Kant can be easily misread (or: I myself easily misread him, for a long time) as claiming that no "existence claims" are analytically knowable. Technically, though, his system has it that (positive) categorical existence claims are not analytically knowable, but some assertions about existence are. This is so because his system allows that the following existence claims are true (and knowable by logic alone, so to say):

  1. A exists or A doesn't exist.
  2. No A exists such that A = ~A.

He can even allow for existential conditionals like, "If a vacuum collapse bubble exists, the universe as we know it will probably be 'destroyed' after thousands of eons have passed." (It would take that long on account of the bubble only expanding at the speed of light.)

Now, as far as Kant's presentation goes, the analytic/synthetic distinction is exhaustive. Either "the concept of the subject logically contains the concept of the predicate," or it does not contain that. The concept of existence is not really descriptive or qualitative; the phrase "a unicorn" and "an existent unicorn" call to mind virtually the same imagery, and whatever the latter "adds" to the image is relational, outwardly, e.g. a sense (or wish) that we might encounter a particular unicorn someday, somewhere. (For a long time, I thought George Lakoff's invocation of the actionable paradox of "don't think of an elephant" was an ironclad ostensive demonstration of an extremely crucial fact about language and imagination. However, I later realized that we could distinguish "don't think of a unicorn in general" from "don't think of a particular unicorn," and the latter "works" contra the way the former doesn't. So maybe Lakoff's insight is not so penetrating.)

But now consider:

  1. Either X exists or the concept/question of X's existence/X existing is meaningless.

I don't know that Kant ever considered such a proposition. Perhaps in his meandering doctrines of noumena and things-in-themselves, there is room for claims like (3). They would be "hyperanalytical" existence claims, so to say. I prefer framing the issue in erotetic terms and I think that Kant would have done so, or in some way (subconsciously?) did so, when he said that, for example, "What is the constitution of a transcendental object?" is in some sense a meaningless question. In fact, transcendental logic really is an erotetic logic; Kant inherited talk of categories from Aristotle, and so he also inherited Aristotle's (purported) appeal to "forms of possible questions" in this connection, I suspect. And a transcendental argument shows that the meaningfulness of certain questions presupposes the existence of facts that ground this meaningfulness, either due to our erotetic capacity in itself, or more "locally" (per the local content of the transcendental argument in play).

Finally, consider a "degenerate"(?) option:

  1. Either X doesn't exist or the question of X existing is meaningless.

Is talk of hyperanalytical knowledge of certain "weird" existence claims consistent with Kant's scheme for the analytic/synthetic distinction, or if not his scheme, at least the thematism of the entire first Critique?

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  • "Assertions about existence" are not existence claims, existence claims are of the form ∃xP(x), and your 1, 2 are not. Kant worked within Aristotle's logic, so his analytic/synthetic distinction was far from exhaustive, the notion of "containment" does not apply to most logical forms allowed in modern logic. There are modernizations of it, e.g. analytic is that derivable from definitions without use of non-logical axioms. Denying meaningfulness of the transcendental object is part of the so-called "idealist reading" of Kant, which goes back to Fichte, but there is no "either or" in it.
    – Conifold
    Feb 8, 2023 at 3:28
  • @Conifold I know I misread Kant for a long time, here, I didn't see that the logical functions for categorical, hypothetical, and disjunctive judgments were interpolated (when transposed as categories) with possible forms of judgments involving the modal categories. And I only meant that in Kant's system the analytic/synthetic distinction seems exhaustive. Moreover, I have come to believe that many philosophers end up contradicting themselves at some point or another (the preface paradox), so I don't think it's possible to "prove" an interpretation of Kant just by using "prooftexts." Feb 8, 2023 at 3:36
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    Meaningfulness is clearly not a predicate, just like existence it attaches to concepts and not to objects. Either the concept is meaningful, and we can then ask about existence of its instances, or it is meaningless, and the existence question is moot. But I do not see how meaningfulness can be sensibly conditioned on existence/non-existence of instances, as in your 3, 4. The idealist reading is not popular these days, but see Hickey for a recent defense.
    – Conifold
    Feb 8, 2023 at 3:38
  • @Conifold yeah, I don't really have any idea about a possible "example" of (4). I do have an idea about (3), but it depends on a pretty extreme theory about free will and erotetic logic, which I haven't been able to axiomatize yet, so for now I would just offer that some theists (apophatic ones especially) might be enticed by (3)-like claims. Feb 8, 2023 at 3:42
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    I suppose, "if a concept is instantiated then it is meaningful" (is implies can) is analytic for the simple reason that instantiating a meaningless concept would be, well, meaningless. Thereby, meaningfulness is "contained" in instantiation. But that collapses your 3. Did you mean something like "either God exists or the question of his existence is meaningless"? Or maybe paradoxical plays on the inexplicable, "the Tao that can be told is not the true Tao"? "What can be said can be said clearly, and what cannot be said must be passed over in silence", as Wittgenstein quipped.
    – Conifold
    Feb 8, 2023 at 4:09

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Kant admits of something called the infinite judgement which is neither analytic or synthetic, because it asserts that the subject is essentially undetermined in regards to the predicate. This of course violates the law of excluded middle and Kant is aware of this - unsurprisingly, given his often semi-intuitionistic theory of mathematics. In any case, your assertion that the distinction is exhaustive, in Kant's eyes anyway, is incorrect.

Kant says that in regards to mathematical objects, we never consider them as actual (I really cannot remember where he says this but I am sure he does - I think either the Doctrine of Method or the Analytic of Principles - sorry for that), we work with mere possibility in the sense he discusses in the Postulates of Empirical Thought (unsurprisingly, given the formal nature of mathematics). However, in terms of modern mathematical logic, we, of course, do assert existence of various mathematical objects. For Kant, though, existence in this sense has to be coextensive with constructibility due to his commitment to semi-intuitionism (made plain in many places).

Mathematical intuitionists will of course understand any claim X as it is (constructively) provable that X. Then, the judgement of the form you name, i.e. neither X exists or X doesn't exist, can be understood as saying that X cannot be shown to be contradictory but is also not constructible. Excluded middle obviously doesn't hold because nothing guarantees that all descriptions are either contradictory or specify a constructible object (in Kant's terms: one for which an intuition can be given of it). All of this is quite easily assimilated into Kant's account of mathematical construction (which I think is quite robust). If we look at, ex. the Preface to the second edition of the first Critique (other places too), we will find that Kant also distinguishes empirical contentful knowledge from mere thought of an empty logical possibility by the givenness of an object in an intuition (which, as Kenneth R. Westphal points out in multiple articles, is a challenge to both skepticism and infallibilism and strongly supports Newton's hypotheses non fingo methodology) so I think the very same holds for substantial empirical knowledge, although experience replaces a priori mathematical construction here. This also means that Kant is one of the first serious direct (but not - causal) reference theorists, cf. Amphiboly of the concepts of reflection.

In other words, if we substitute for your "meaningless" "indeterminate", then yes - Kant admits of statements that are neither true or false in a mannor similar to contemporary intuitionism. And this is a significant, and, I believe, not at all irrelevant today, aspect of his epistemology.

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