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Say that I know why the caged tardigrade screams, KyS. Say that I know why pineapple on pizza is the key to unlocking the secrets of the universe, KyU. Then Ky(S & U)? My first reaction to this was to consider the PSR issue about knowing all the individual sufficient reasons for things in a sequence, but supposedly not knowing the sufficient reason for the sequence as a wholeL; so knowledge-why would not be agglomerative. But according to the default paradigm of metaphysical grounding, a conjunction is grounded in, hence metaphysically explained by, its conjuncts. So knowledge-why seems like it would be agglomerative, or that is, there is nothing to knowing the reason for a group of facts over and above knowing the reasons for each fact given.

Is knowledge-why both agglomerative and not agglomerative, or one or the other from time to time, or just one or the other all the time?

LI've also seen it said, in deep mathematics writings, that in some systems, we can prove for each n that something is true of them, but we can't go above and beyond them to prove the corresponding universal quantification, "All the n are that way." I think it's a matter of not "knowing" whether all the n have been considered; we have considered them all, but we can't prove that we've done this (in some systems).

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  • Metaphysically the world might be agglomerative; epistemically the knowing-why attitude may not be in a modal mode. Sometimes the reason for p defeats or changes the reason for q when both are considered jointly. So agglomeration can fail psychologically and even formally if the agent’s knowledge base isn’t closed under conjunction to avoid unrealistic omniscience. The gap is exactly between local provability and global quantification like those found in the 1st order arithmetic where there seem beings only of the nature of metaphysic not epistemic... Commented 18 hours ago
  • You are hinting at Sergei Artemov's criticism of Gödels consistency formula, which according to him is strictly stronger than the statement that PA is consistent? Commented 17 hours ago

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Yes. Knowing why X implies you know why X instead of ¬X. Knowing why Y implies you know why Y instead of ¬Y. Now, knowing why X, and having a sufficient reason for X, implies that you can now rule out ¬X. The same applies for ¬Y.

Now, ¬(X and Y) together consist of three possibilities:

  1. ¬X ∧ ¬Y
  2. ¬X ∧ Y
  3. X ∧ ¬Y

Since neither ¬X or ¬Y can be true due to us having a sufficient reason for X and Y individually, we can rule out all of these. But this is the same as saying we now know why (X ^ Y) instead of ¬(X ^ Y).

Thus, knowing why X and knowing why Y implies we know why (X ^ Y).

Notice that this all depends on “knowing why” being the same as possessing a sufficient reason that entails X and rules out ¬X. If instead “knowing why X” only meant “having some partial reason or local explanation for X” then those reasons might not jointly exclude the contraries.

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If knowing the reason why for a group of separate facts, p1, ... pn presupposes knowing a separate reason for why they all together are the case, or (something like) why they are all the case in the same world, or what links all of them together, then it might be that knowledge of the conjunction is "more" than just the knowledge why of each separate fact. Otherwise, you might assume that this knowledge of what links them, why they are all true, would already be presumed in the knowledge why of each separate fact. That may also be the case, depending on what kind of facts you're dealing with. But it may not necessarily be the case. So, I'd assume that both the precise interpretation or the logic of "we know why ..." and the mutual dependence (or independence) of the known facts plays a role. In other words, it would seem to matter a great deal what the closure properties are of the Ky operator. (And what those properties are may be rather arbitrary, i.e. depend on the purpose of this whole exercise of setting up a modal logic.)

In mathematics this issue seems more serious, since we may be dealing with an infinite series of formulas. "Knowing why" could in this case be "having a proof of". According to Sergei Artemov, the statement that PA is consistent should be read as as a schema for an infinite sequence of formulas: "the natural number n is not the Gödel number of the formula 0 = 1". Each instantiation of the schema can be expressed in PA and proven, and we can or should therefore, according to Artemov, also accept the schema (which is similar to a meta-theorem) as proven. However, Gödel's statement Con(PA), "For all n, n is not the Gödel number of the formula 0 = 1", is, according to him, strictly stronger, since, as Gödel has shown, it cannot be proven in PA (if PA is consistent). The core of the issue is that "For all n" in Gödel's Con(PA) statement

is not specified and its meaning depends on a choice of an arithmetical model.

Artemov's proof that PA-consistency and his consistency scheme are mathematically equivalent seems both elementary and extremely subtle. I still have not quite convinced myself that I understand all of it. If anything in my summary is incorrect or a bit "off", please let me know in a comment.

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  • If you understand deep inside you don’t understand, why not ask a question instead of posting an answer?… Commented 13 hours ago
  • I'm a rather shallow person, so there is no "deep inside". As far as I can tell, everything I wrote is correct. If you can show me it is not, I would appreciate that. Commented 13 hours ago
  • But it actually did occur to me too that I might turn part of what I wrote into a question. Later, perhaps. Commented 13 hours ago
  • Due to Godel's incompleteness theorems, first order PA cannot have such intended cohesive agglomerative property. To really understand the stipulated property of arithmetic one should be situated in Godel's favorite second order logic to range over semantically complete concepts per his categorical conceptual Platonism. The modern way to understand this is usually via the first order reflection principle of ZF to approximate the non-formal second order full semantics of the objective categorical arithmetic in a limited formal trace, and reflection cannot be attained from a shallow pond... Commented 7 hours ago
  • The philosophical question seems to me whether a kind of perfect reflection ever matters (though it's interesting to know about). Brouwer, who was not particularly shallow, wasn't very impressed with Gödel's results, since they were exactly what you'd expect from an intuitionist pov. Artemov seems to argue that being able to prove that every finitary fragment of PA is consistent is what we (or at least Hilbert) actually care about. The subtlety here seems to be that we overload the words "being able" and "every", while we may be tempted to always assume: Kp -> KKp... Commented 53 mins ago

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