# A Priori KK?

Does the KK thesis hold for a priori knowledge? That is, if p is a priori knowable, is it a priori knowable that p is a priori knowable? It isn’t obvious to me how to translate Williamson’s arguments against KK into an argument against this thesis, but that’s hardly a ringing endorsement of the thesis.

## 15 Replies to “A Priori KK?”

1. I can think of a way that Williamson might run it. Mr. Magoo knows that if the least number that isn’t small is n, then he does not know that the least number that isn’t small isn’t n+1. I imagine that knowledge of which numbers are small is a priori. By Williamson’s argument, we get that Magoo knows that the least number that isn’t small is not m, where m is the least number that isn’t small. Contradiction, so KK for a priori knowledge should be rejected.

Set that aside. Does this case sound coherent?

Ramanujan can intuit many mathematical truths that no ordinary human can intuit, indeed some that are not provable in any reasonable formal system. Knowledge of those truths can come only via Ramanujan’s amazing faculty, or via testimony from Ramanujan. Ramanujan’s faculty has limits, however, but Ramanujan doesn’t have a priori access to where those limits lie, or even that his faculty has limits. (Imagine that he in fact believes his faculty has no limits, although he never oversteps his bounds in forming mathematical beliefs.) Imagine that experts have determined roughly how complex a truth can be for Ramanujan to be able to reliably intuit it. There will be p such that p is a priori knowable, but where it is not a priori knowable that p is knowable – since p may be of a certain complexity such that it lies close to the penumbra between those truths than Ramanujan can know and those he cannot. It’s knowable that the truth is a priori knowable (the experts can see that the truth falls well enough below the penumbra) – but not a priori (since their knowledge of where the penumbra lies is not a priori; it’s empirical knowledge of Ramanujan’s faculties).

(I think the example also probably works sans experts.)

2. I think the KK principle holds for a priori knowledge. Consider that (1) one knows a priori that P. Then suppose it can be the case that K(1) – where K is the knowledge operator— is not a priori. Then it has to be that some of the concepts occurring in (1) are so that one has to appeal to experience to know that (1) is true. But it can’t be P, as it is known a priori, so it has to be the concept of a priori knowledge. But this is I think false, as the meaning of ‘a priori’ is stipulated by definition. So it seems the supposition that K(1) is not a priori is false.

3. Jonathan Weinberg says:

I agree with Allan generally here, and find persuasive his case of Ramanujan’s empirical penumbra. There seem to me to be several other ways in which one could know p a priori, but not know a priori that one knows p a priori. For example, suppose (plausibly, I hope) that for me to know a priori that there are infinitely many primes, it is sufficient that (i) I can construct a valid proof of the infinity of primes, without relying on empirical premises; (ii) there really are an infinity of primes; and (iii) there be no misleading evidence in my environment such that, if I were aware of it, I would take my justication from (i) to be undermined. (E.g., a paper from a mathematician I trust purporting to demonstrate for some n that n is the greatest prime.) For me to know all that a priori, however, would require my knowing not just (i) and (ii) a priori (which seems plausible) but also (iii) a priori (which seems highly unlikely).

More generally, there may be conditions for Kp that can only be ascertained empirically. This shows what Istvan missed, I think — the added empirical component may be in the concept of knowledge itself, even leaving the ‘a priori’ part of ‘a priori knowledge’ aside!

It occurs to me that there may be more wrong with Istvan’s argument, because even if the meaning of “a priori” is stipulative, it does not follow that one can determine whether particular cases fall in its extension on an a priori basis. Stipulate “bachelor” to mean “unmarried man”, nonetheless for any particular person I need to do some empirical legwork to determine whether they do or do not satisfy that stipulated definition.

One more thought: it may also be only a posteriori determinable whether a particular belief is supported by purely non-empirical reasons. If we accept the cogency of intuition as a source of a priori justification, then it might be in some instances an empirical psychological fact that a particular belief that p is based on such an intuition, and not on (say) a now-forgotten chain of reasoning some of whose premises were empirical.

4. Regarding Allan’s case: I usually take “a priori knowable” to mean roughly “knowable just through a chain of ideal reasoning without any contingent premises”. If we accept this definition or something like it, then it seems that there are many a priori knowable truths that Ramanujan can’t know because he isn’t capable of doing the relevant chains of ideal reasoning.

Hence, I guess I’m not seeing what relevance Ramanujan’s ignorance of his own limits is supposed to play. On the definition of a priori I suggested, his own limits are irrelevant to what is a priori knowable.

It seems that, on the definition I suggested, the APK-APK thesis does hold. Suppose (for conditionalization) that P is a priori knowable (APK). Since P is APK, there is a line of ideal a priori reasoning yielding the conclusion that P. Hence there is a slightly longer line of a priori reasoning that combines the first line of reasoning with the definition of APK to yield the conclusion that P is APK. So, by the definition of APK, it is APK that P is APK. QED, Or no?

5. Jonathan Weinberg says:

My worry with your argument, Justin, is that one of the things you need for the ‘longer line of reasoning’ for the propisition that P is APK, is that all of the premises in the original line of reasoning for P are justified non-empirically. Let me use ‘NEP’ as an abbreviation for ‘justified with no empirical premises’. It seems to me entirely possible that one could have
NEP
without having
NEP)
since it may be something of an empirical question just what the basis of the premises are, in the line of reasoning for p.

6. Jonathan Weinberg says:

Huh. Some bits of text in parentheses got deleted or rendered invisible there. Trying again, with a different notation:
It may be possible that one could have
NEP{p}
without having
NEP{NEP{p}}.

7. Jonathan,

Brian asked whether it would follow from the fact that P is APK that it is APK that P is APK.

So, suppose that P is APK. By my definition of APK it follows that there is an argument for P that requires no empirical premises, and that would satisfy an ideal-enough reasoner. I guess you would write this NEP{P} — I myself would write it as “P is APK”. I don’t understand the need for the second acronym.

Anyway, call this argument A1. You agree that there is a longer argument A2 that joins A1 (which is a demonstrative proof of the existence of an ideal argument with no empirical premises leading to the conclusion P) together with the definition of APK (which says that if such an argument does exist then P is APK), and concludes (by modus ponens) that P is APK. I proposed that A2 is itself an ideal argument relying on no empirical premises for the conclusion that P is APK. Hence, using the definition of APK one last time, we may conclude that it is APK that P is APK.

Your objection, I take it, is that even while A1 required no empirical premises, perhaps A2 does. But, I don’t see what empirical premise A2 is supposed to require.

I do grant that one might plausibly hold that the definition of APK is an empirical premise, but, at least in contexts where people are taking the idea of a prioricity seriously, definitional meanings must be admitted into the the base for a priori arguments. If you deny this (as I admit one perhaps should) then I take it you aren’t even playing the game that people are playing here.

Other than this, I don’t see what empirical premises A2 is supposed to depend on. Perhaps you can help me.

8. Jonathan Weinberg says:

“NEP” is not equivalent to “APK”, since the former is just part of a justification condition, but the latter is a full-blown knowability condition (which I believe entails the former). I was trying to highlight a particular aspect of being a priori knowable that I thought made trouble for the “APK-APK” thesis. In particular, it is meant to illustrate that there is not necessarily any easy way to establish the antecedent of the conditional that you talk about in your last comment. It is one thing simply to have an argument whose premises are appropriately non-empirical; but it is quite another to know — or even be justified in believing — that the argument has that characteristic.

So I am contending that, contra what you’ve been saying here, we cannot merely extend the original argument with some conceptual considerations. We would also need to have some way of establishing that the original argument is of the right sort, and (I am claiming) that may not be an a priori matter.

9. I agree with Justin (and with myself a bit earlier). I don’t understand the case presented above by Jonathan:

Passage 1:

“For example, suppose (plausibly, I hope) that for me to know a priori that there are infinitely many primes, it is sufficient that

(i) I can construct a valid proof of the infinity of primes, without relying on empirical premises;

(ii) there really are an infinity of primes; and

(iii) there be no misleading evidence in my environment such that, if I were aware of it, I would take my justication from (i) to be undermined.”

Passage 2:

“For me to know all that a priori, however, would require my knowing not just (i) and (ii) a priori (which seems plausible) but also (iii) a priori (which seems highly unlikely).”

Passage 1:

Satisfying (iii) is required not for a priori knowing (i), but for knowing (i) simpliciter. Therefore, since as Justin points out, the question was whether the KK holds for a priority, (iii) is orthogonal to this issue. Rather, to answer the original question, one must start with the premise that P (in our case the infinity of the series of primes) is known with a priori justification. Then, one has to show that if this is true, there can be the case that it is not known with a priori justification. In other words, condition (iii) is from the start supposed to hold.

Passage 2:

“To know all that a priori” is, as far as I see, the same as to know (i) to (iii) a priori. This is why I don’t see how this is relevant… Because, if you say that to know a priori that the series of primes is infinite is to know “all that a priori” [i.e. (i) to (iii)], and you further say that you can’t know, or, as you say, highly unlikely to know, (iii) a priori, then you, in fact say, that you can’t know the original proposition a priori. But this was not the issue.

As regards the Ramahujan and other similar examples, I think they are irrelevant. If an agent is able to deduce something a priori, them it seems to me, if he is a normal, reflective, human being, that he will know that the he deduced it a priori, without having to check whether some emprical conditions have been satisfied. Contraposing, if he needed any such empirical check, he would not know or deduce the original proposition a priori.

10. Sorry, “(i)” in the first sentence of the first comment, namely in:

“Satisfying (iii) is required not for a priori knowing (i), but for knowing (i) simpliciter.”

should be replaced with the proposition under consideration, P, that is, “There is an infinite number of primes”.

11. Jonathan Weinberg says:

I take it that we all agree that the question is whether there is an entailment from
(1) S knows a priori that p
to
(2) S could know a priori that (1).

My condition (iii) is a condition that must be true in order for (1) to be true, but it is something that must be known to be true in order for (2) to be true. And my claim is that the no-misleading-evidence condition could be true without being known to be true. So Istvan is right that condition (iii) is ‘from the start supposed to hold’, but that’s not at issue. What’s at issue is whether S needs to know that (iii) holds, and indeed whether S needs to know a priori that (iii) holds. (It might be the case that in general we know that there is no misleading evidence lying around — it’s a fairly low probability sort of occurrence, I hope! — but that’s a bit of empirical knowledge about the epistemically favorable circumstances of the world we live in.)

We can see the disagreement here clearly, in Istvan’s comment on the second excerpt from my comment, he writes: “if you say that to know a priori that the series of primes is infinite is to know “all that a priori” [i.e. (i) to (iii)], and you further say that you can’t know, or, as you say, highly unlikely to know, (iii) a priori, then you, in fact say, that you can’t know the original proposition a priori.” I am explicitly denying the identification made in the first antecedent here. To know a priori that the series of primes is infinite, I conjectured that it is sufficient for (i) – (iii) to all be true. But it is not at all required, to know that there is an infinite number of primes in this way, that we also know anything at all about the truth of (i) or especially (iii), and in particular it is not required that we know anything about (i) or (iii) in an a priori way. (It is of course still required that we know (ii) a priori, since (ii) is just the statement that there are an infinity of primes.)

12. Thanks for the clarification!

So, (iii), that is:

(iii) there be no misleading evidence in my environment such that, if I were aware of it, I would take my justication from (i) to be undermined.

Must be:

(*)true in order for (1) to be true, where (1) is “S knows a priori that p”

(**) known to be true for (2) to be true, where (2) is “S could know a priori that (1)”

The problem is the follwing. Suppose (*) is true. Question: does (**) follow?

I answer in the affirmative, for the following reason.

a. If (*), then (iii) is satisfied.

b. (*)

c. S knows a priori that (a.)

d. S knows a priori that if (iii) is not satisfied, then not-(*)

e. S knows (simpliciter) that (*).

f. If S knows (simpliciter) that (*), then S knows a priori that (iii) is satisfied.

g. If S knows a priori that (iii) is satisfied and S knows simpliciter that (*), then, if S knows a priori that (a.), S knows a priori that (*).

h. (**)

I think this argument is sound.

13. Jonathan Weinberg says:

I don’t see where e. and f. come from. It might be because I’m misunderstanding the notation is your (*) equivalent to (iii) itself, or is it equivalent to “(iii) must hold in order for ‘S knows a priori that p’ to be true”? Either way, some of the premises come out fishy.

Take the first interpretation, that (iii) = (*). a. comes out fine as an instance of identity, and therefore c. and d. should be unproblematic as well. b. is stipulated hypothetically, since we’re trying to see what is the case if S knows a priori that p. But I don’t see where e. and f. come from. As I noted above, it’s not entailed from S’s knowing that p that S knows that the conditions for that knowledge are met. So e. stands in need of an argument. f. also strikes me as odd — why is the knowledge simpliciter that a condition for knowledge is met sufficient for a priori knowledge of the same? What does seem plausible is

f’. S knows a priori that, if S knows that (*), then S knows that (iii) is satisfied.

but that’s not enough to do the necessary work in the argument; nor is it enough if we strengthen all the ‘knows’ to ‘knows a priori’.

So let’s consider the other interpretation, in which (*) stands for the proposition that (iii) is a necessary condition on S’s knowing a priori that p. In that case, a. seems unmotivated — it doesn’t follow from ‘X is a necessary condition for Y’ that ‘X is true’. If a. is in trouble, then so are c. and d. (I must confess to not really seeing what work d. is meant to do, so that might be contributing to my confusion here.) b. is fine, and I’ll grant e. (though that presumes a certain epistemological sophistication on the part of S.) But f. fails for the same reason a. does. (My best bet would be that you mean the first interpretation of (*), if for no other reason than, on the second interpretation, I don’t think that (**) would need any special argument from (*) in the first place!)

So I just don’t see how the argument is supposed to run. It might help me, btw, if you give some indication of which lines of the argument are meant to follow from which others, and which ones are meant to be basic premises. Perhaps I’m just not seeing how the pieces are meant to fit together.

14. Ooops! Notation problems again. You are right about not seeing where some things come from! This is what happened: I started with using a. and b. for your conditions S knows a priori that p and S knows that (iii) holds, then I used 1., 2., etc. for the premises of my argument. But then I realized I have to change it to a., b., etc., because of your using them for the two claims. Then i realized I have to adjust further my original a. and b., and use * and **, and so on. At a point I made the mistake of confusing the two conditions regarding (iii) with the two propositions under consideration about S’s epistemic relation to them. All this in the context of my constant preoccupation for catching my bus to arrive home at a normal time …

OK, so let us keep my argument as it is, just consider that (*) is S knows a priori that p, and (**) S knows a priori that (*).

When on the bus I came to distrust a bit my argument, because of premises e. and f.. Let us start with f.. I have to develop a bit the point. With the notational adjustments now in hand we can say that if S knows simpliciter that S knows a priori that P, then S knows simpliciter that (iii) is satisfied. But if this is conjoined with the premise that S knows a priori that if not-(iii), then not (*), then S knows, it seems to me, without any empirical justification that (*).

As regards (e), I supposed this was supposed to hold for the the original question of Brian’s to be answered. But while on the bus it occurred to me that one can have three a priori KK questions in fact, a strong, a weak, and a middling a priori KK, and my argument is for the middling to hold:

Strong: “S knows a priori that p” implies that “S knows a priori that S knows a priori that p”.

Weak: “S knows a priori that p” implies that “S knows that S knows a priori that p”.

Middling: “S knows a priori that p” and “S knows that S knows a priori that p” implies that “S knows a priori that S knows a priori that p”.

15. Jonathan Weinberg says:

Ah, that does make a lot more sense for (*) and (**)! And all of a. – d. are acceptable on that reading (a., c., and d. are all likely true, and b. is the hypothetical premise). I see that we might be talking past each other with regard to e. — unlike b., I definitely did not see it as part of the hypothetical, because I thought we were considering the strong version of the APK-APK principle. (Brian, you should maybe come clear up your initial question a bit, buckaroo!)

But f. is still problematic, on any reading of the APK-APK principle under consideration. Here are the things that we can unproblematically say that S knows a priori:
—p
—If S knows a priori that p, then (iii) is satisfied.
—If (iii) is not satisfied, then S doesn’t know a priori that p.
—If (iii) is not satisfied, then S doesn’t know p simpliciter.

So, using these premises (plus any other unproblematically a priori premises), can S derive the conclusion: S knows a priori that p? More to the point, can S derive the conclusion: (iii) is satisfied? I don’t see how. Even if, granting e. arguendo, that doesn’t help here, because e.‘s being true wouldn’t add “S knows that p” to the list of the a priori-acceptable premises. If it were — if S could determine entirely a priori that S knows that p — then I can see how maybe S could know a priori that S knows a priori that p. But without S’s bit of self-knowledge itself being a priori, it can’t be used to secure other pieces of a priori knowledge. So we’re back looking at that short list of a priori-acceptable premises, which doesn’t, I think, entail (iii); and so doesn’t entail that S knows that p, or that S knows a priori that p.