Mon 2 Nov 2009
A Refutation of the “JJ” Principle
Posted by wedgwood under formal epistemology, justification
[18] Comments
This blog post offers a refutation of the following “JJ” principle:
(1) If you are justified in believing p, then you have the highest possible degree of justification for believing that you’re justified in believing p (in other words, you can be certain that you’re justified in believing p).
The refutation will be based on broadly Williamson-inspired considerations about “margins for error”. Nonetheless, the argument is also designed to be completely compatible with internalism about justification (or at least with the “mentalist” form of internalism).
My refutation of (1) is based on the following “margins for error” principle for justified belief:
(2) If you’re justified to degree d in believing p, then in all relevantly close cases, you’re justified in believing p to at least degree d – ε (where ε is some small difference in degree of justification).
(2) can be motivated by considering the same sorts of cases that Williamson considers in supporting his “margins for error” principle for knowledge (Knowledge and Its Limits, Chap. 5).
E.g., the degree of justification that you have for believing the proposition ‘That man is less than 6 feet tall’ varies smoothly over a spectrum of cases: the shorter the man looks to you, the more justification you have for believing the proposition. Among the cases where you at least have more justification for ‘He is less than 6 feet tall’ than for its negation ‘He is not less than 6 feet tall’, the closer you are to the tipping-point where you cease to have more justification for the proposition than for its negation, the less justification you have for that proposition.
Now let us use the phrase ‘a very high degree of justification for believing q’ to mean: a degree of justification for q that is at least 1 – ε (where 1 is total certainty, and ε as before is some small difference in degree of justification).
Then (1) and (2) entail:
(3) If you’re justified in believing p, then in all relevantly close cases, you have a very high degree of justification for believing that you’re justified in believing p.
The second premise in my refutation of (1) is:
(4) If you have a very high degree of justification for the higher-order proposition that you’re justified in believing p, then your justification for that higher-order proposition must itself rest on (or incorporate) your justification for p – and so you must indeed be justified in believing p.
In effect, (4) is a ‘JhighJp → Jp’ principle: if you’re highly justified in believing that you’re justified in believing p, then you are justified in believing p.
(4) should seem appealing to any philosopher who is attracted to anything like (1). After all, how could you have such a high degree of justification for the (higher-order) proposition that you’re justified in believing p, unless your justification for that higher-order proposition somehow rested on or incorporated your justification for the lower-order proposition p?
Taken together (3) and (4) entail:
(5) If you’re justified in believing p, then in all relevantly close cases, you’re justified in believing p.
But if we iterate (5) sufficiently many times, we can infer from the obviously true premise that there are cases in which you are justified in believing ‘That man is less than 6 feet tall’ (when you are looking at someone who is clearly less than 5 feet tall) to the obviously false conclusion that you’re still justified in believing ‘That man is less than 6 feet tall’ when looking at someone who is clearly more than 7 feet tall!
The obvious conclusion to draw is that the ‘JJ’ principle is false and must be rejected – indeed, it must be rejected for fundamentally the same reasons as the better-known ‘KK’ principle.
18 Responses to “ A Refutation of the “JJ” Principle ”
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Interesting argument. But why should a proponent of JJ allow arbitrary iterations of (5)?
The argument looks analogous to arguments against Lockean views of rational belief that rely upon closure under arbitrary intersection (aka, the rule of adjunction). To some it appears obvious to ditch threshold belief; but to most Lockeans, it appears obvious to ditch unrestricted closure under adjunction.
While I think you may be right that something like (4) should seem appealing to those who are attracted to (1), I don’t see why (4) itself should. At least, not when (4*) is available:
(4*) If you have a very high degree of justification for the higher-order proposition that you’re justified in believing p, then your justification for that higher-order proposition must itself rest on (or incorporate) your justification for p – and so you must indeed have a very high degree of justification for p.
(4*) is just the same as (4), except the consequent states only that you have a very high degree of justification for believing p, rather than that you are justified in believing p. My prima facie intuitions about these matters aren’t fine grained enough for me to think that (4) might capture them, while (4*) might not. As far as I can tell, if we’re attracted to (4), we should also be attracted to (4*) for the same reasons.
However,(4*) has the advantage that it won’t combine with (3) to generate (5). Rather, it will only generate (5*)
(5*) If you’re justified in believing p, then in all relevantly close cases, you have a very high degree of justification for p.
But of course, no absurd conclusions result from iterating (5*). In fact, it can’t be iterated. If we take a situation in which you’re justified in believing p, and use (5*) to conclude that in close cases you have a high degree of justification for p, we can’t then apply (5*) to generate conclusions about those cases, since the antecedent is no longer guaranteed to be true in them.
I worry about the (seemingly inessential) part of 4 that says “then your justification for that higher-order proposition must itself rest on (or incorporate) your justification for p”
I don’t see why that should always be the case. It looks like the justification for the higher-order proposition that you’re justified in believing p can come from a different source than your justification for p. That seems to allow room for the possibility that whatever justification the higher-order proposition has for you may not rest on (or incorporate) your justification for p.
Hi Ralph,
Excellent stuff! I have a stake in JJ, so your argument worries me. But the version of JJ that I like is weaker than the version you argue against. It says that if one’s degree of justification to believe p is d, then one’s degree of meta-justification to believe that one has justification to believe that p is greater than or equal to d, although it need not be maximal. It seems to me that when you plug in this weaker version of the JJ principle, you can’t generate the soritical conclusion. Do you agree? As far as I can see, all you get is the following:
(1) If one’s degree of justification to believe p is d, then one’s degree of meta-justification to believe that one has justification to believe that p is at least d
(2) If one’s degree of justification to believe p is d, then in any close case, one’s degree of justification to believe p is at least d-minus-a-bit
(3) So, if one’s degree of justification to believe p is d, then in any close case, one’s degree of meta-justification to believe that one has justification to believe that p is at least d-minus-a-bit
(4) If one’s degree of meta-justification to believe that one has justification to believe that p is at least d-minus-a-bit, then one’s degree of justification to believe that p is at least d-minus-a-bit
(5) So, if one’s degree of justification to believe p is d, then in any close case, one’s degree of justification to believe that p is at least d-minus-a-bit
Thanks for all your comments!
1. Gregory — You seem to be making a mistake here. When I spoke of “repeated iterations of (5)”, I just meant that we, the theorists, could make repeated modus ponens inferences involving (5) and various propositions of the form, “In case Ci, you’re justified in believing p.” This enables us (the theorists) to go from the obviously true “In case C1, you’re justified in believing p” to the obviously false “In case Cn, you’re justified in believing p.” No principle about the “closure” of justified belief under logical implication is assumed — just the unrestricted validity of modus ponens!
2. Jeff — Well, you’re right that (4) is slightly stronger than I need. All that I need is some principle of the form ‘JJp -> Jp.’ I’ll think about whether your suggestion is better.
3. Declan — Suppose that you only have barely more justication for p than for its negation. (E.g. a fair coin is going to be tossed 3 times, and p is the proposition that it will land heads at least once.) Surely the enthusiast for the “JJ” principle will think that you have a lot of justification for the higher-order proposition that you have barely more justification for p than for not-p? But your principle (1) implies you have only about the same level justification for believing that higher-order proposition (about your level of justification for p) as you have for the negation of that higher-order proposition!
Don’t “access internalists” typically think that our access to the facts about our current level of justification for propositions comes from (i) introspection and (ii) a priori reflection — and don’t they typically regard these methods of access as the royal roads to the most sterling quality of justification? So I think that if you just settle for your principle (1), you’re basically selling the farm!
Ralf,
That’s a bit dramatic. Modus ponens preserves truth, not almost the truth; your refutation works by conflating the two. That’s the obvious part. What to do in response is less obvious.
If a proponent of JJ is sincere about thresholds, she will have something in mind to ensure that ‘high degree of justification’ is preserved. One of those options is to block unrestricted iteration of (5). And there various ways to achieve this end.
So, a better engagement with JJ would start where your argument ends by looking at various options for bounded deduction and critically evaluating them. One suggestion is an MPε rule that stops when you pass below a 1-ε threshold; But that’s messy, tinkering with the rules of the logic. Instead, one might model the premises as 1-ε accepted statements within the classical modal system EMN, “The Logic of Risky Knowledge” (Arlo-Costa, Kyburg & Teng) i.e., one in which the box (acceptance) modality does not freely distribute across conjunction. Here, MP is intact and the restriction is managed in the semantics. But there are curious theorems of EMN pertaining to disjunctions. Are those reasonable? And so on.
That would be a nice contribution!
Gregory — I’m not at all sure what you mean, but I don’t think that there’s any “conflation” (let alone an “obvious” conflation!) in my argument.
At all events, I was not relying on any special assumption about the logic of degrees of justification. (Talking about degrees of justification does not commit me to making sense of “degrees of truth”; after all, talking about degrees of temperature does not commit us to degrees of truth, so why should talking about degrees of justication?)
I suppose that I was relying on some very very elementary arithmetic (e.g the assumption that 1 – ε < 1), but I don't see how you've identified any flaw in my argument whatsoever.
Ralph – (sorry about ‘Ralf’: I correspond with a Rolf, and I remember you writing in German some time ago…which must have seemed enough in my mind to strike the ‘ph’ for an ‘f’. (Sigh).)
I understand that you aren’t relying on a logic for degrees of justification–or degrees of anything; the question is why block that option for a defender of JJ who allows for a margins of error principle.
The temperature example is an excellent one, since temperature is not an additive quantity. If hotel room 1 is 20 degrees C, room 2 is 20 degrees C, and room 3 is 20 degrees C, one doesn’t get one big 60 degree room by knocking down the adjoining walls. You get a 20 degree room. Plus some dust.
Risk of being wrong, however, is additive: the thermostat in room 1 is 20 plus or minus ε degrees C, the thermostat in room 2 is 20 plus or minus ε degrees C, the thermostat in room 3 is 20 plus or minus ε degrees C, does not guarantee that in combination will be a room which is 20 plus or minus ε degrees C.
Let me try putting the matter another way: Once I see (2) in your argument, the margins of error principle, I immediately think that I will need some way to restrict the logic to avoid your conclusion, for I could put anything in place of (1) and run the same type of argument against that alternative, (1*). That’s why it isn’t obvious that (1) has to go. And why some work squashing non-adjunctive strategies, like the ones I sketched, seems a necessary step before dropping the hammer on (1).
Daniel — Surely, if I have “a very high degree of justification” for p, I must be “justified” in believing p?!
As I was understanding a “very high degree of justification”, it includes the degree of justification that one has for logical truths, conceptual truths, introspectible truths about one’s own mental states, etc. etc. Surely, I have enough justification for believing those propositions to count as “justified” simpliciter!
But of course if I’m right that ‘You have a very high degree of justification for p‘ entails ‘You are justified in believing p‘, then your conclusion (5*) entails my conclusion (5).
So your conclusion (5*) still enables us to get from the true premise that you’re justified in believing ‘That man is less than 6 feet tall’ in case 1 (when you’re looking at the man who is clearly less than 5 feet tall) to the obviously false conclusion that you’re justified in believing it in case n (when you’re looking at the man who is clearly more than 7 feet tall).
Hi Ralph, My version of the JJ principle says that if my degree of justification to believe p is d, then my degree of meta-justification is greater than or equal to d, but it’s left open that it could be *much* greater! So, why am I committed to the claim that my degree of justification is always about the same as my degree of meta-justification?
Your main point, I take it, is there’s no good motivation for weakening JJ in this way. But I’m motivated in part by the thought that we never have maximal justification to believe anything! Even if I have a priori or introspective justification to believe p (the internalist gold standard!), my justification may be sub-maximal insofar as it is irrational for me to bet my life against a penny on the truth of p. I’m not sure how seriously these sorts of worries should be taken, but prima facie, they seem to motivate some sort of weakening of JJ.
Declan — You’re right, I’m sorry, I misspoke. Your principle (1) doesn’t entail that one’s degree of meta-justification is equal to (or slightly less than) one’s degree of justification for the lower-order proposition. Still, your principle is surely much weaker than the idea that “access internalists” have typically defended, since it doesn’t entail that one’s degree of meta-justification is ever greater than one’s degree of justification for the lower-order proposition (not even in cases where it seems obviously much greater).
Moreover, there is also another response available to me, even if I concede your view that “access internalists” should only ever have accepted a significantly weaker version of the “JJ” principle than the one that I criticized here. Even if the “JJ” thesis is weakened, my argument will still work, with a correspondingly strengthened version of my premise (4).
For example, suppose that we replace (4) with the premise that if you have a reasonable degree of justification for believing that you’re justified, then you are justified. I take it that a great many “access internalists” will accept such strengthened versions of (4). But then this revised version of my argument will go through even if we weaken the “JJ” principle in the way that you suggest.
Ralph,
I’m not sure the argument will go through with your revised version of premise (4). I’d accept (4) if it means something like this: if one’s degree of meta-justification to believe that one has justification to believe p meets the threshold for justification simpliciter, then one’s degree of justification to believe p also meets the threshold for justification simpliciter. But now plug this into my reformulated version of your argument and suppose for the sake of argument that the threshold for justification is d. As we move along the sorites series of cases, there’s going to come a point at which we move from a case alpha in which one’s degree of justification to believe p is d and so one’s degree of meta-justification is at least d to a case beta in which one’s degree of meta-justification is at least d-minus-a-bit, so one’s degree of justification is at least d-minus-a-bit. But that may not be enough to meet the threshold for justification simpliciter. So we avoid your soritical conclusion (5).
All of this is presupposing my weakened version of JJ, so I’d still need to explain why I think it is strong enough for a reasonable version of access internalism. But setting that aside, it does seem to me that this version of JJ gets around your argument.
Declan — All that my argument needs is that the degree of (meta-)justification that features in the consequent of (1) is at slightly greater (d-plus-a-bit) than the degree of justification (d) that features in the antecedent of (4).
Given (2) — which you have not disputed — the conjunction of (1) and (4) will push you along, from saying that you’re justified “simpliciter” in believing p in one case, to saying that you’re also justified simpliciter in believing p in the next case.
So all that I need is the claim that there is some degree of justification d such that (1) is always true with “justified to degree to d-plus-a-bit” in the consequent, and (4) is always true with “justified to degree d” in the antecedent. I’d be quite surprised if you could produce a convincing argument for the conclusion that there is no such d!
Interesting.
The boost one gets going from Jp to JJp (according to this access internalist) is at least as great as the loss one takes from going to nearby cases.
Suppose in case alpha, p is just over the threshold for justification. In alpha, Jp is maximally justified, by (1).
In nearby case beta, Jp loses only a little justification, by the margin of error principle.
In beta, p is still over the threshold for justification, by (4).
To fix the things, one would have to propose versions of (1) and (4) that ensured that one always loses more justification for JJp by shifting to a nearby case, than one gained in the trip from Jp to JJp. Since the postulated loss between cases is tiny, the boost would have to tiny as well, and that doesn’t sound like traditional-access-internalism-JJ to me.
I do think (4) needs qualification, however. Access internalists are often fallibilists about justification in general, and (4) asserts an infallibility of sorts. Granted, it doesn’t say we’re infallible about whether we’re justified, but it does say that whenever we have a high degree of justification about our own justification, we’re right about it too. And I can see internalists balking at that.
I’m not sure exactly whether testimony or memory can ever endow a “high degree of justification”, but suppose it can.
Suppose I explain my reasoning about p with Tim Williamson, who, even though he doesn’t like talking about justification, assures me that I have suffices for (what people call) justification for believing p. I’ve found Tim to be extremely reliable on the question of when I’m justified, but as it happens, Tim is mistaken on this occasion, and I’m not justified in believing p.
That seems like a counterexample to (4), and maybe what Jeff had in mind.
Wait. Why wouldn’t Tim’s testimony itself constitute justification for P?
No. Because Tim never commented on p itself or on his belief regarding p. Nor did I assume anything of the kind. Tim often (truthfully) tells me I’m justified in believing something when he knows that I’m, in fact, mistaken.
Perhaps you should state that p holds for cases where one’s justification for Jp is nothing in addition to (a) one’s justification for p and (b) reflection on *that*. (b) is tricky, of course.
Ralph, this is helpful. If you’re right, I need to deny a version of JJ that says if my degree of justification to believe p is d, then my degree of meta-justification is even marginally greater than d. This inclines me to retreat to an even weaker version of JJ that says if my degree of justification to believe p is d, my degree of meta-justification is also d.
You criticized this version of JJ on the grounds that my degree of meta-justification can be higher than my degree of justification, e.g. to believe that the coin tossed three times will land heads at least once. But although I certainly have justification to assign this proposition a high degree of credence, I’m not sure it follows that I have any degree of justification to believe it outright. So, maybe the weaker version of JJ is ok if understood in terms of degrees of justification for outright belief?
Troy — Yes, you’re right that my premise (4) is a kind of “infallibility” principle. According to (4), if you have a certain sort of justification for the proposition that you justified in believing p, then that proposition must be true. Why should “access internalists” accept (4)?
There are two reasons, I think. First, access internalists may think that you can’t really have a high degree of justification for the proposition that you’re justified in believing p on the basis of someone else’s testimony or the like, but only a basis that somehow incorporates your justification for p itself. Secondly, these access internalists may accept a version of the idea that “evidence of evidence is itself evidence” — namely, the idea that a high degree of justification for believing that you’re justified in believing p just is, in itself, justification for believing p. Of course, ‘I’m justified in believing p‘ doesn’t entail that p is true. But it seems plausible that if I’m rational, and I think ‘I’m justified in believing p‘, then I will also believe p; and this seems to me to support (4).
Declan — All that I need is to pick a proposition p that has the following two features: (i) you are justified in having an “outright belief” in p; (ii) it is clear that you have less justification for p than for many other propositions that you’re justified in having an outright belief in. (To give an example that I have used elsewhere, p might be the proposition that Dushanbe is the capital of Tajikistan.) Now suppose that you have reflected intensely about the basis of your belief in p, about the nature of the justification that you have for it, and so on, and you conclude that you are justified in believing p. Unlike the lower-order belief, this meta-belief is based on introspection and a priori reflection. Surely, in cases of this sort, you have more justification for the meta-belief than for the lower-order belief!
Ralph — I agree that internalists may be drawn to (4) as it stands for either of your reasons. The second reason (evidence of evidence is itself evidence), however, could admit of weaker and stronger forms. The strong form works just as you say. The weaker form would only hold that evidence of one’s evidence counts as *some* evidence, without thereby implying that the evidence rises to the level of justification. The corresponding version of (4) would say that if you’re highly justified in believing Jp, then there is some justification for p. This leaves it open that the degree of downwardly-inherited justification falls below the threshold for justification. And it seems a lot less risky.
It *is* difficult to imagine a reflective creature highly justified in believing Jp while at the same time suspending judgment about p itself. But what lesson should we draw? Isn’t Moore’s paradox a similar case? It’s hard to imagine a rational being believing that she believes p while suspending belief about p itself. Yet, one is not obviously justified in believing p on the basis that one is justified in believing that one believes p.
Yes, I know there are important disanalogies here, but I’m not yet convinced that the rational pressure to believe p if JJp must count as — or support the idea that there is — (sufficient) justification for p itself.