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Pursuant to my question about a logic of "understanding" as a distinct, same-level concept vs. "knowledge", I have had cause to try situating at least ∀ and ∃ in relation to the distinction. I'm reading over Maria Aloni, "Knowing-who in quantified epistemic logic", for some basic pointers, here, but then so far, my main concern has been to model Kant's rejection of (categorical) analytical existence claims.

Let ◊ be generic for "it is possible that," k be generic for "it is known that" (I have seen that this tends to be indexed to specific agents, though, and paired up with a believes-that operator), and u be generic for "it is understood that." (I say "generic" to be evasive about the intended semantics, because I'm not competent to articulate how all these ideas might be accounted for using possible-worlds talk.) Then:

  1. ◊(k(∃x(Fx)) → u(∃x(Fx)))?

  2. ◊(k(∀x(Fx)) → u(∀x(Fx)))? (but I don't yet see that it'd be necessary for this kind of knowledge to lead to that kind of understanding)

  3. ¬◊(u(∃x(Fx)) → k(∃x(Fx)))? (this seems like a candidate for modeling Kant's claim)

  4. ◊(u(∀x(Fx)) → k(∀x(Fx)))? (analytical knowledge in general? at least if universal quantifiers are not made to carry existential import by default)

  5. ◊(k(∃x(Fx)) → u(∀x(Fx)))? (I suppose it could be possibly true, if there were exactly one x)

  6. ◊(k(∀x(Fx)) → u(∃x(Fx)))? (no clue)

  7. ◊(u(∃x(Fx)) → k(∀x(Fx)))? (maybe the analytic a posteriori)

  8. ¬◊(u(∀x(Fx)) → k(∃x(Fx)))? (alternative model of Kant's claim?)

Further possible "axioms": substitute ↔ for → in any of the above. I assume that this might have useful/interesting consequences per this section from a certain nLab entry:

Quite often, classical references will define ‘simple’ (or an analogous term) in naïve way, so that a ‘trivial’ object is simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is not simple. Usually this can be done by changing ‘if’ to ‘iff’ (or sometimes changing ‘or’ to ‘xor’) in the classical definition. [emphasis added]

Do either (3) or (8) model Kant's claim about no-analytical-existence-claims, or would we have to pick such a model out from at least one of (1) through (8) with the biconditional in place of the conditional? I tried out formulating (3) as a conjunction where the reverse conditional is negated, so that while it might be possible to know-then-understand while understanding-then-knowing, one cannot only understand-then-know (per some existence fact Fx).

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  • If the Kant's claim is "no existence claim is an analytic statement" then it clearly cannot be modeled in a first order system, even with modal operators, since it is a meta-claim about statements. Even if we take uS → kS as "S is analytic", which I do not think works either, putting ¬◊ in front does not do what's intended. The conditional is surely possibly true, at least, if it is the material conditional (e.g. with false premise).
    – Conifold
    Commented Jun 6, 2023 at 5:24
  • @Conifold per Bumble's answer to my previous question, I've discarded if-uS-then-kS as analyticity. I'm not even sure I have any plausible overarching way to separate out all the formulae into analytic/synthetic beforehand but might have to correlate each specific type of formula with a specific subdivision of Kant's scheme and then extrapolate the separation of formula into analytic/synthetic (if that's even worthwhile). My main reason for using modal operators was to try bypassing quantification from outside the conditionals, although it seems like there's a lot lacking in this attempt.
    – Kristian Berry
    Commented Jun 6, 2023 at 5:45
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    It seems like overthinking it to me. For one, Kant's assertion says nothing about knowledge or understanding. And it is captured by a simple meta-principle that no statement of the form ∃xF(x) can be a theorem. After all, while there is a problem with drawing analytic/synthetic distinction for languages in general, there is no problem drawing it in a fixed language. We simply list suitable axioms and declare all and only theorems "analytic". Kant's principle is then a meta-constraint on admissible languages.
    – Conifold
    Commented Jun 6, 2023 at 10:11
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    Number 3 doesn't work. It is equivalent to □(u(∃x(Fx)) ∧ ¬k(∃x(Fx))) which means all existence claims are understood but none are known to be true. If you wrote instead ¬□(u(∃x(Fx)) → k(∃x(Fx))) this would mean that it is possible to understand an existence claim and not know it to be true. You would still need a second order quantifier to say that this holds for all predicates in the F position.
    – Bumble
    Commented Jun 6, 2023 at 11:26
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    That wouldn't work either. ¬((uS → kS) ∧ ¬(kS → uS)) is equivalent to kS → uS, so that would amount to all existential sentences are such that if we know them we don't understand them. You need either a modal claim: it is possible to understand an existence sentence and not know it, or an a priori knowable operator APK such that ¬APK(uS, kS) for any existential sentence S.
    – Bumble
    Commented Jun 6, 2023 at 15:24

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