Lotteries and JK

JK: If you are justified in believing P, then you are justified in believing that you know P.

Argument against: Suppose that you know that your lottery ticket’s chances of winning are one in a million. Then you are justified in believing it will lose, but you are not justified in believing that you know it will lose. Hence JK is false.

I find this argument fairly plausible. Anyone see obvious flaws in it?


Comments

Lotteries and JK — 51 Comments

  1. I think a theory of normativity for cognition is going to have to leave open this possibility at least initially, the possibility that rational or justified or warranted belief may not reach the threshold needed for the kind of normative status necessary for knowledge (in addition to truth, belief, and some gettier condition). Even within the same category of normativity, I think this question needs to be left open by the theory: so, a rational belief and a belief sufficiently rational to be known (assuming the other necessary conditions are satisfied) might not answer to the same threshold. The same for justification and warrant.

    This point doesn’t quite sustain the above argument, though, since even if one begins theorizing by leaving such a question open, reasons may arise later on in the process of theory construction to close the question in such a way that the argument is rejected. If that happens, I would think it would happen in virtue of rejecting the claim that you are justified in believing that your ticket will lose.

  2. No, I didn’t mean to suggest that. I was thinking, instead, of Hawthorne/DeRose considerations about lottery situations, to the effect that justification needn’t track high probability alone, but is responsive as well to something like symmetries in regions of logical space. Neither of them draws the conclusion that no justification is present, but if the case is symmetrical for all tickets, and justification needs asymmetries in logical space for it to obtain, then there would be a reason to deny that you are justified in believing your ticket will lose.

    None of this is either clear or obvious, of course. It would all need to be sorted out in the process of theory construction.

  3. I’m not sure what I think.

    What about a related principle: if S believes p, then S believes that they know p. It seems related. I’m not sure what I think about that either.

    I’ll get back to you!

  4. The related principle is clearly false. Lots of theists believe that God exists but don’t believe that they know this. Lots of philosophers believe their philosophical theses but don’t believe that they know them. Lots of scientists believe their theories, but don’t believe that they know them to be true.

  5. Dennis,

    Take the following case:
    S knows that his ticket’s chances of winning are one in a million, and so is justified in believing that the ticket will lose. However, S is also persuaded by an argument (which he has considered carefully and competently) to the effect that one is typically in position to know that the ticket will lose if it is the case it will lose and one knows that the odds of it winning are one in a million. (As a matter of fact, I am somewhat sympathetic to this view myself…) It would appear that, regardless of whether the argument in question is a good one, it could provide justification for S’s belief that he knows that the ticket will lose.

    So JK appears to survive the argument you raise against it.

  6. Dennis,

    I’m not sure I buy the argument against the principle. First, I’m not sure that I share the intuition that you are justified in believing you’ll lose the lottery, so I don’t think lottery cases are going to be cases in which the antecedent of the conditional is satisfied.

    Second, suppose that’s wrong. While I think you don’t know lottery propositions, I can’t see why someone cannot be justified in mistakenly believing that you can know them.

  7. I think this principle is true and I don’t think the lottery case is a counterexample. What’s uncontroversial is that you’re justified in having a high degree of confidence that your ticket will lose. But it doesn’t follow that you’re justified in believing that your ticket will lose unless you adopt the Lockean thesis, i.e. you’re justified in believing that P iff you’re justified in having a sufficiently high degree of confidence that P. Personally, I’m inclined to reject the Lockean thesis.

  8. Pavel, that’s interesting. Let’s assume that S is persuaded, by argument A, which he carefully and competently considers, that he’s justified in believing he knows that his ticket will lose. You conclude from this, plausibly I think, that A provides S with some justification for believing that he knows that his ticket will lose.

    But it might turn out that S is nonetheless not justified in believing that he knows that his ticket will lose. Even though A provides him some justification for that belief, there might be other considerations that make the belief unjustified nonetheless. What other sorts of considerations? Well, perhaps something like this: if he knows the ticket will lose then he should throw it away; but he should not throw it away.

  9. I’d also prefer a version of the principle formulated in terms of propositional, rather than doxastic, justification, such as:

    JK*: If you have justification to believe P, then you have justification to believe that you know P.

    Unlike JK, this doesn’t imply that, for every proposition that I’m justified in believing, I also believe that I know that proposition, which seems unlikely to me.

  10. Declan and Clayton: For defending the claim that you are justified in believing that your ticket will lose, here’s the best I’ve got. First, your evidence that your ticket will lose is as strong as anyone’s evidence for any scientific theory. Second, some people have evidence for scientific theories that is strong enough to justify belief in those theories. Third, if evidence of a given strength justifies belief in any case, then evidence of that same strength justifies belief in every case. From these we get the conclusion that your evidence that your ticket will lose is strong enough to justify your belief that your ticket will lose. (Not an airtight argument, but still I think somewhat plausible.)

  11. Declan: I meant the original principle to be about propositional rather than doxastic justification. I was trying to use “justified in believing” talk in such a way that you can be justified in believing something even if you don’t actually believe it. So as I meant JK, it is identical to JK*. Sorry about this- I could have specified it from the get go, but I wanted to get the points across with minimal verbiage.

  12. Clayton: on your second point. I’ll grant that it is possible to be justified in believing that you know your ticket will lose. But consistently with this, it is possible to not be justified in believing that you know your tickt will lose, while being justified in believing your ticket will lose. Why does the latter possibility hold? Well, here’s one reason: you (Clayton) can’t be justified in believing something that you know you don’t know, and you (Clayton) know you don’t know your ticket will lose. Does that help?

  13. Thanks, Dennis. On your third step in 11 above, I’d start worrying about high/low stakes bank cases: same evidence in each case, but you’re only justified in believing that the bank will be open when the stakes are low. Or, if evidence is stipulated to be what justifies belief, then I’d start worrying about the first step instead. But, yes, I think that’s an argument to be reckoned with!

  14. Declan, yes, those sorts of considerations are just what I was thinking rendered that argument nonairtight. Maybe the strength of evidence needed to justify belief varies from case to case.

  15. Dennis,

    I didn’t mean to suggest that JK stands, only that your argument against it needs to be a little more specific.

    Indeed, I think there are even simpler arguments against JK, ones that do not rely on any controversial presuppositions about lottery cases or interest relativity.

    Consider: S is justified in believing that p, and also justified in believing (falsely, I hope) that knowledge of propositions in some domain to which p belongs (but not justifiedness about it) is an unattainable epistemic standard. If it is possible to meet this conjunctive condition (if it is possible to be justified in believing an epistemological thesis like this, and if knowledge and justification can come apart), then it is possible for S to be justified in believing that p, while not being justified in believing that he knows that p.

    Indeed, this could be taken as a schema for cases where JK fails.

  16. Hey Dennis,

    Apologies in advance if I’m missing something, but I’m still not yet convinced that the sort of considerations you’re offering cause trouble for JK.

    I’m awful fond of this principle:

    (P1) If you know you don’t know p, you can’t be justified in believing p.

    But, as we know we don’t know lottery propositions, I can’t see how that doesn’t show that lottery cases cannot be used to construct counterexamples to JK unless those examples involve subjects that are mistaken about the epistemic status of lottery propositions. It seems that the principle you appeal to #13 causes trouble for the argument in #11.

    In #13, you wrote:
    It is possible to not be justified in believing that you know your ticket will lose, while being justified in believing your ticket will lose. Why does the latter possibility hold? Well, here’s one reason: you (Clayton) can’t be justified in believing something that you know you don’t know, and you (Clayton) know you don’t know your ticket will lose. Does that help?

    If I’m reading you correctly, the problem is that while knowing you don’t know you’ll lose the lottery means it is possible to not be justified in believing that you know you’ll lose, in that circumstance, you cannot be justified in believing that you’ll lose. So, we have a situation in which the consequent and antecedent of JK are both false if we assume (P1) is a genuine principle and grant that we know we don’t know lottery propositions.

    I think Pavel’s strategy might be more promising, but I think cases that do not involve subjects wedded to mistaken epistemic principles are less interesting than cases involving well-informed and reflective agents.

    [This is all complicated significantly if we move to talking about propositional justification rather than personal or doxastic justification.]

  17. I think that JK is obviously false. I agree with Jon that some variant of a lottery example will work to show that it’s false. I also happen to think that my son is justified in believing the things that I sincerely tell him. Were I to sincerely tell him that Q, but he cannot know that Q, then he’d be justified in accepting Q, but not justified in accepting that he knows that Q.

    Also, there will of course be intelligent subjects who are justified in believing many things, but who lack the concept of knowledge, and so do not understand any claim to the effect that they know something. They’re not justified in accepting propositions that they don’t understand; thus, they’re not justified in accepting any proposition to the effect that they know something. Thus, they’re justified in believing many things, without being justified in believing that they know them.

  18. Can we avoid the controversies surrounding the lottery case by using the following counterexample to the principle that JB implies JBK?

    James’s brother disappeared in battle over a year ago. Clearly, James is justified in believing that his brother is dead. But is James justified in believing that he knows this?

  19. John: Why not think that you can have propositional justification to believe propositions that you cannot understand? Of course, you can’t be held responsible for failing to believe these propositions, since you lack the requisite concepts. But it might be useful for some theoretical purposes to have a relatively impersonal notion of justification that is not sensitive to one’s psychological capacities.

  20. Ram: suppose John reflects rationally and responsibly and comes to the view that he does not know his brother is dead. Wouldn’t it be kind of epistemically schizophrenic in this case to continue believing that his brother is dead? I’d say he should withhold belief, but retain a high degree of confidence that his brother is dead.

    Compare: you learn from locals that you’re in Fake Barn Country, so you come to the view that you do not know there is a barn ahead. In that case, it would be epistemically schizophrenic to believe there is a barn ahead. Rather, you should withhold belief, but retain a high degree of confidence that there is a barn ahead (assuming the number of fake barns is known to be small).

    I guess the suggestion is something like this: anything that defeats your justification to believe that you know P also defeats your justification to believe P.

  21. Declan,

    Sure, I think you could have such an impersonal concept of propositional justification. A version of JK might even be true for that concept.

    But I do not think the impersonal concept would track very closely our ordinary concept of justification or reasonableness. Normally at least, it would not be reasonable for someone to believe a claim she did not understand. (I think it could be reasonable for her to believe something which she only barely understood, principally through testimony and the acquisition of deferential concepts — but those are special sorts of cases.)

    BTW: I have no idea why that comment of mine appeared again at 11:26!

  22. I understand that the following makes a big metaphysical assumption, but it is still a case that seems interesting.

    Suppose that I believe that there will be a sea battle tomorrow, and have ample evidence that there will be such a battle. Suppose also that it isn’t now true that there will be, nor true that there won’t be, a sea battle, and that I know this (by philosophical argument perhaps?). Can’t I be justified in believing that there will be a sea battle, but not justified in believing that I know since knowledge requires truth?

  23. Clayton, you wrote “The principle you appeal to in #13 causes trouble for the argument in #11”. Um uh yeah, that seems right. Let me go think about it. Meanwhile, here’s a new principle: K-p entails -Jp. Substituting in “you know you lose”, we get

    K-(you know you lose) entails -J(you know you lose)

    Here the antecedent is true, and so given the new principle, the consequent is true too. Hence the new principle entails that you aren’t justified in believing that you know you lose. But now substitute “you lose” into the new principle. This gives us

    K-(you lose) entails -J(you lose)

    The antecedent here is false, and so the new principle does not via this substitution instance yield the consequent. I suspect that it does not yield that consequent in any other way either. So, I suspect, the new principle yields the result that you are not justified in believing that you know you lose, but does not yield the result that you are not justified in believing you lose. Maybe the best case for the view that you are justified in believing you lose but not justified in believing you know as much, then, consists in the conjunction of #11 with the new principle.

  24. Ram,

    I’m not sure what to make of this example:
    James’s brother disappeared in battle over a year ago. Clearly, James is justified in believing that his brother is dead. But is James justified in believing that he knows this?

    Why shouldn’t we say that he’s justified in believing he knows this? We’re granting that he’s justified in believing his belief is true. While I don’t think knowing puts you in a position to know you know, I’m not so sure that being justified doesn’t put you in a position to be justified in believing you’re justified. So, perhaps his being justified in believing p puts him in a position to justifiably believe he satisfies the justification requirement on knowledge. There’s Gettier cases to worry about, but bracketing the issue of having the justification to believe a belief isn’t Gettiered, I’m not seeing how that which puts James into a position to justifiably believe his brother is dead doesn’t put him into a position to justifiably believe (perhaps mistakenly) that he knows this.

    Running it the other way, what justifies the judgment that he’s not in a position to justifiably believe he knows his brother is dead? If the account we offer is an account on which he’s in a position to know he doesn’t know his brother is dead with minimal reflection, I’m not all that inclined to say that his belief is justified. If he were to believe that his brother were dead and his brother resurfaced, it seems that if the brother can establish he was in a position to know that he didn’t know that he was dead, the undead brother could rightly say, I think, that the mistake that had been made was not due entirely to factors external to the brother. In conceding that he knew he didn’t know or should have known better than to assume that he knew, it seems the brother would concede that he was responsible for the erroneous judgment. If you’re responsible for the erroneous judgment, I can’t see that you’re wholly justified.

    So, I think we’d need a story about how the brother could fail to have justification for the belief that he knows that his brother is dead on which the facts that prevent him from knowing are such that the brother cannot be held culpable for failing to factor them in when judging that his brother is dead. I think it’s controversial whether such a story can be told. While I’m not sure I think we ought to accept JK as true, it seems that the arguments against it are more often than not stories we’re rightly sceptical of.

  25. Hey Dennis,

    It’s an interesting suggestion. I think that the principle you offer is plausible:
    (P2) If you know ~p, you are not justified in believing p.

    While you’re right that this principle yields the result that (i) you’re not justified in believing you know you’ll lose without also yielding the result that (ii) you are not justified in believing you’ll lose, I take it that it’s not enough to point to a principle such as (P2) and the argument in #11 since there’s another principle, (P1), that causes trouble for the argument in #11. If (P2) plus the argument in #11 are really going to show that observations concerning the epistemic status of beliefs in lottery propositions show that JK is false, we’d need to see some reason to reject (P1).

    [Note: I don’t think you can argue from (P1) to JK, only that (P1) is a serious obstacle to some arguments against JK].

  26. Maybe what is plausible about P1 is captured by P2. If that is right, and if furthermore argument #11 has some force, then perhaps we should reject P1 and put P2 in its place.

  27. Dennis,
    I suppose that is a route to go, but I still think (P1) is pretty plausible. Anyone who satisfies the antecedent of (P1) believes for some proposition, p:

    (1) I don’t know p.

    The question is whether in such a state, the subject can be justified in believing p. If they are cognizant of the fact that they believe both (1) and that p, it seems they believe:

    (2) I don’t know p, but p.

    We grant that they know that they don’t know that p, so we assume that they are justified in believing that. The question is whether we ought to say that they are justified in the first-order belief that p as well. I’d say that they are not. I think I’m the camp with Huemer and possibly Adler and de Almeida in thinking that anyone who holds the beliefs associated with a Moorean absurd statement such as (2) holds beliefs that contradict the rational commitments that come with those very beliefs. But, if you consciously contravene the rational requirements that come with holding a pair of beliefs, I can’t see that you can be justified in believing the pair.

  28. Clayton and Declan:

    Suppose James believes that his brother is dead, but also believes that he does not know that his brother is dead. (He might sincerely assert “well, I believe that he’s dead, but I don’t know that he is”.)

    Two questions:

    (1) Have I so far told you anything about James that entails that he has an unjustified attitude?

    (2) If S is justified in believing that p, does it follow that S is not justified in believing that not-p?

    I think the answer to the first question is “no”, and the answer to the second question is “yes”, but I gather that you would disagree with at least one of my two answers. Which one?

  29. Hey Ram,

    Let me answer for Declan so that my answer doesn’t get me in trouble…

    I think the answer to the first question is ‘Yes’ and the second is probably ‘No’. I hesitate to answer the second in a straightforward manner because I suppose someone could have a sufficient justification for believing ~p and justifiably believe p, provided that the justification for believing ~p is not readily accessible.

    I gave a sketch above as to why I think you cannot justifiably believe p while believing you don’t know p. Assuming that these attitudes are before the mind, as it were, I think in believing you don’t know p you take yourself to be cognizant of something about the prospect of believing p that makes the believing of p wrongful. I think if you take yourself to be cognizant of something about the prospect of believing p that would make it wrongful if it obtained, in virtue of taking yourself to be cognizant of such a wrong-making feature it is wrongful to believe p in such a state.

    Having said that, I don’t want to deny that James could be justified in the attitues expressed by his sincere assertive utterance of “well, I believe that he’s dead, but I don’t know that he is”. In some contexts, to assert “I believe p” is to assert something weaker than that p and express an attitude other than belief in the truth of p. If in such a context, the assertive utterance of “I believe p” amounts to something along the lines of there is a good chance that p, it is likely that p, etc…, then I see no reason to think that the subject could not be justified in that attitude while acknowledging that he does not know p.

  30. Unless I’m just being stupid, it seem to me that Dennis and Ram have to be right (at least, if the principle is taken to be fully general). To see this, reason as follows. First, take the contrapositive of the principle:

    (1) -JKp => -Jp

    Now, let p = Kq, for some arbitrary q. Then, we have:

    (2) -JKKq => -JKq

    Then, by (2) and the original principle (in contrapositive form), we can deduce (by a simple chain argument):

    (3) -JKKq => -Jq

    Finally, contraposing (3), we get:

    (4) Jq => JKKq

    Since q was arbitrary, we also have (4) in full generality. Surely, THAT’s absurd, no? Since all the steps in this argument look good to me, except the original principle, I conclude that the original principle is false.

  31. Branden,

    I agree with you that JK is false. In response to your remark, I think defenders of JK might be well advised to restrict the principle’s scope. In particular, they should restrict it to claims that the subject can understand. Thus, if the subject cannot understand JKKq (or some subsequent iteration), then it’s not the case that Jq => JKKq. But so long as the subject understands the claim, they ought to be willing to call your intuition (re absurdity) and raise you one better.

    Clayton would, I think, have some sympathy with this sort of response, judging from remarks he made over at his blog. Declan, though, said something in response to an earlier comment of mine, which suggests he might prefer a different route (though I of course don’t presume to speak for him).

  32. Branden,

    I wanted to say that I’ve been defending JK not because I think it’s true, but because I don’t think the arguments offered previous to yours were conclusive. I’m officially agnostic. The arguments up until this point seemed to me to conflict with a version of (P1) from above, and that’s why I regard them as suspect.

    You don’t come out and say it, but I take it you think it would be absurd to hold a view on which hurdles that you must overcome to justifiably ascribe knowledge and knowledge that one knows were overcome once you put yourself in a position to justifiably believe the first-order belief. One quick argument against this view, I suppose, is this. Suppose you think it’s raining outside. Having justification for such a belief is a matter of having evidence about the state of things outside the apartment, not inside the head. But, having justification for the belief that one knows this requires having justification for saying both how things are in the head and how things are outside the apartment. There’s no necessary connection between having sufficient justification for making claims about matters external to the apartment and matters internal to the head, so JK is false.

    There might be problems such as this that require tinkering with the formulation of JK, but the reason I’m interested in the status of JK is that I’m interested in knowing whether there are additional substantive epistemic requirements on judging you know p beyond judging p. I don’t quite yet see the absurdity of the view that says that there aren’t.

  33. Can I suggest an improvement to the formulation of JK?

    JK: If you have justification to believe P, then: you have justification to believe that you know P, if you believe P

    Suppose you have justification to believe that P, but you do not in fact believe that P. In that case, you do not have justification to believe that you know P, since knowledge entails belief.

  34. Branden,

    I wonder if you could say more about why you think we should reject (4) Jq => JKKq.

    There’s a worry that goes like this: suppose you have justification to believe that P, and you do believe that P, but you do not form the second-order belief that you believe that P, since you do not consider the question. In that case, you do not have justification to believe that you know that you know that P, since second-order knowledge entails second-order belief.

    If that’s the worry, I think it can be handled by the qualification proposed in the previous comment, since it’s basically the same objection run at the second-order instead of the first-order.

  35. Branden,

    Could you say more about why you think we should reject (4) Jq => JKKq?

    Here’s a worry you might have: suppose you have justification to believe that P, and you do believe that P, but you do not form the second-order belief that you believe that P, since you do not consider the question. In that case, you do not have justification to believe that you know that you know that P, since second-order knowledge entails second-order belief, which you obviously haven’t got.

    If that’s the worry, then I think it can be handled by the qualification proposed in my previous post, since it’s basically the same objection run at the second-order, rather than the first-order.

  36. Here’s a different worry: it’s easy enough to show that JK entails not only (4) Jq => JKKq, but also (5) Jq => JKKKq, (6) Jq => JKKKKq, and so on ad infinitum.

    But Williamson (2000: Ch.5) argues persuasively that we don’t have knowledge to indefinitely many iterations. So, presumably, we’re not justified in believing that we do. But presumably we are justified in believing some things. So, JK must be false.

    I think this kind of argument does show that JK needs some restriction, although I don’t think the kind of restriction that John proposes is quite what is needed. It’s got to be a restriction that answers directly to Williamson’s argument. I’ve got it written down somewhere , but I think I left it at the office!

  37. Declan,

    Suppose that S has justification to believe P, believes P, but does not understand the concept of knowledge. Do you want to say that S has justification to believe that she knows that P — a claim she cannot even understand?

    Suppose that S has some justification to believe P, but not nearly enough to know P. S also believes P, but recognizes that her evidence falls well short of that needed for knowledge. In such a case, do you want to say that S has justification to believe that she knows P?

    That second case might not be entirely fair, depending on the details of your proposal.

  38. Hi John,

    I’m happy to say that you can have propositional justification to believe propositions that you do not understand. I didn’t get a chance to respond to your earlier comment, but there you said: “Normally at least, it would not be reasonable for someone to believe a claim she did not understand.” But in that comment you were talking about incomplete understanding, rather than a complete failure to grasp the concepts that compose the proposition in question. That’s a different topic isn’t it?

    On the second case, you pretty much anticipated my response: I don’t think there are any cases where you have justification to believe that P, but where you fail to know that P because you don’t have enough justification to believe that P. I guess that means I’m disagreeing with Jon Kvanvig in the first comment on this post. However, I do think there are lots of cases in which you have justification to be highly confident that P, but you don’t know that P because you don’t have justification to believe that P. So, I’m putting a lot of weight on the difference between belief and confidence.

  39. Hi John,

    I’m happy to say that you can have propositional justification to believe propositions that you do not understand. I didn’t get a chance to respond to your earlier comment, but you said: “Normally at least, it would not be reasonable for someone to believe a claim she did not understand.” But there you were talking about incomplete understanding, rather than a complete failure to understand the concepts in question. That’s a different topic isn’t it?

    On the second case, you anticipated my response: I don’t think there are any cases in which you have justification to believe P, but you fail to know P because you don’t have enough justification to believe P. I think that means I’m disagreeing with Jon Kvanvig’s opening comment on this thread. However, I do think there are lots of cases in which you have justification to be highly confident that P, but you do not know that P because you do not have justification to believe that P. So, I’m putting a lot of weight on the distinction between belief and confidence.

  40. Hi Declan,

    I had in mind a case where S has no concept of knowledge, completely fails to understand it, and so can’t entertain thoughts about knowledge.

    On your view, having (sufficient) justification to believe P entails having (sufficient) justification to know P. Do I have that right?

    And we’re not to think of belief as confidence passed a certain threshold, or even as full confidence?

  41. Here’s a slightly different strategy for attacking JK…

    Suppose you adopt a view on which justification is closed under known entailment. (I know this requires some controversial assumptions about justification and probability, but ignore that for the time being.) If being justified in believing p entailed being justified in believing I knew that p, if I also know that knowing p requires not being in a Gettier case, it seems that when I’m justified in believing (say) that this is a barn before me, it justifies me in believing that I know this is a barn before me. As I know knowing this requires not being in fake barn country, it follows that I have justification for believing I’m not in fake barn country!?!

    Thoughts?

  42. Hi Clayton,

    That seems right, but isn’t this just an application of a more general problem for anti-skeptical positions, e.g. if I have justification to believe there’s a barn ahead, then I have justification to believe I’m not a BIV in a lab. I’m inclined to opt for Pryor’s Moorean strategy here, but even if you don’t like that, can’t you just take your favourite strategy and extend it?

  43. Thanks, all. Declan is right that I was worried about the fact that we get an infinite sequence of iterations from the original principle (that was why I thought it was absurd). That has to be wrong (I agree with Williamson here). So, unless the principle is restricted in some serious way, it is false. Moreover, once we take Declan’s restriction, the counterexamples people have been discussing are avoided. But, at the cost of no longer being what Dennis and Ram had in mind, I think. Right?

  44. OK, I get it now (sorry for being slow in taking in the restricted principle). It would still be applicable to the examples Ram and Neta give. I’ll have to think more about the restriction.

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