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November 19th, 2009

Your Favourite Theory of Knowledge is Wrong

Consider this proposition:

N: Brian does not know that N.

Assume N is false. That is, I know that N. Knowledge is factive, so N. That contradicts our original assumption. So N must not be false. So it follows, at least classically, that N is true. So I don’t know N.

But I can follow the reasoning that showed N is true. And I accept that reasoning, so I believe N. And the reasoning justifies me in believing N. So I have a justified true belief that isn’t knowledge. So the JTB theory of knowledge fails.

My reasoning didn’t go via any false lemmas. It went via a false assumption, but making false assumptions for purposes of reductio is consistent with knowledge. So I have a JTB with no false lemmas, but no knowledge. So much for the JTB+No false lemmas.

I’m (generally) a competent logical reasoner. My belief in N, which is a true belief, was a product of my logical competence. Indeed, I formed the belief in N, rather than some alternative, because of that competence. So I should have Sosa-style animal knowledge of N. Indeed, I can reflectively, and aptly, endorse the claim that my belief in N is accurate because it was an exercise of competence. So I should have Sosa-style reflective knowledge that N. But I don’t; clearly I don’t know N.

It seems to me that pretty much any otherwise plausible theory of knowledge will fall this way. Whatever qualities or virtues a belief might have, short of knowledge, my belief in N has. But I don’t know N. Indeed, logic prevents me from knowing N. So any such theory must be false.

N also undermines various proposals people have relating knowledge to other things. Some people think knowledge is a norm of belief. But there seems to be nothing wrong with my believing N on the basis of the reasoning above, even though I don’t know N. So knowledge isn’t a norm of belief. Many people think knowledge is a norm of assertion. But I don’t see why I shouldn’t assert N. I have a deductive argument that it is true after all; I simply don’t know that it is true. So knowledge isn’t a norm of assertion.

I’m not sure whether N alone could knock out Williamson’s thesis that all and only evidence is knowledge, commonly known as E=K. But N’s good friend E can do the trick.

E: Brian’s evidence does not include E.

Assume E is false. Then my evidence includes E. Either evidence is factive or it isn’t. If it isn’t, then E=K is false for independent reasons. If it is, then it follows E is true, contradicting our assumption. So E is true. Since I can follow this argument competently, I know its conclusion is true. (Unlike the argument about N, logic doesn’t stop me knowing E is true.) So I know E, but E is, as it says, not part of my evidence. So E=K is false.

Note that this argument doesn’t touch the plausible view that evidence is all and only our non-inferential knowledge. Even if I know E via that argument, it is clearly inferential knowledge. So while I can refute all theories of knowledge with self-referential propositions, I can’t refute all theories of evidence.

Posted by Brian Weatherson in Uncategorized

42 Comments »

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42 Responses to “Your Favourite Theory of Knowledge is Wrong”

  1. Dan Greco says:

    Both arguments (the one concerning knowledge and the one concerning evidence) rely on very contentious premises.

    For the former, you need N to be true, for the latter, you need E to be true.

    But the self reference involved here is similar enough to that in liar cases that it’s easy to worry that N and E can’t be straightforwardly false. Maybe they’re neither true nor false, or meaningless, or don’t really express propositions, etc.

    I suspect that anybody sympathetic to saying something in this neighborhood about liar sentences will want to say something similar about N and E, and so will not accept that N is true but unknown, or that E is known (and so true) but not part of your evidence.

    November 19th, 2009 at 1:35 pm
  2. Dan Greco says:

    Meant “straightforwardly true” when I wrote “straightforwardly false.”

    November 19th, 2009 at 1:36 pm
  3. Matt Benton says:

    Nice post, Brian. On the knowledge-norm of assertion you say: “Many people think knowledge is a norm of assertion. But I don’t see why I shouldn’t assert N. I have a deductive argument that it is true after all; I simply don’t know that it is true. So knowledge isn’t a norm of assertion.”

    Here might be a reason why you shouldn’t assert N: N’s very content, when asserted by you, yields the problematic Moorean form “N and I don’t know that N” – although in your case it is easily abbreviated as just “N”!

    I’m wondering whether the argument you give would suffice for certainty: are you then certain of N without knowing that N?

    Question: what is it that you don’t know when you don’t know that N? Is it that you (Brian) don’t know that you don’t know that you don’t know that… Is it just an endless regress of iterated knowledge claims?

    November 19th, 2009 at 1:38 pm
  4. Brian Weatherson says:

    Dan,

    But why shouldn’t they be straightforwardly true? Their falsehood implies a contradiction. Their truth doesn’t imply the falsity of anything that we have compelling independent reason to believe. That makes them very different to the Liar.

    Matt,

    I’m not sure the Moorean sentences are that problematic here. After all, since N is true, I can’t possibly know it.

    It’s a good question about certainly. I’m not sure what the answer is. Possibly I am certain of N, but don’t know it. That would be a very odd consequence about the relationship between knowledge and certainty.

    November 19th, 2009 at 1:39 pm
  5. Matt Benton says:

    Brian: but if N were false, you also couldn’t know it, since knowledge is factive. Thus we’ve got an odd animal: whether N is true or N is false, you can’t know it. (Too bad for you!)

    Maybe that result is enough to make us eschew cases of self-reference…

    November 19th, 2009 at 1:47 pm
  6. Jonathan Ichikawa says:

    I think that in general, that I don’t know that p is a defeater for lots of good properties of believing p. It defeats justification, for example. Knowing that I don’t know p defeats justification in p.

    I think it also interferes with my ability to manifest skill by believing p. Imagine an archer who knows that he’s not going to hit the target by virtue of his skill; he oughtn’t to take the shot — if he does take it, even if it hits the target, this is not a credit to him in the kind of way that makes his shot apt.

    November 19th, 2009 at 2:02 pm
  7. Brian Weatherson says:

    Matt – right, I can’t know N. But that’s not a big deal. There are plenty of things I can’t know. I can’t whether the number of hairs on Caeser’s head when he crossed the Rubicon was odd or even. Depending on how I’m wired, there might be a Godel-sentence for me that I can’t know. That there are things I can’t know isn’t much of a mystery.

    Jonathan – I think the second paragraph here runs together two things that should be kept apart, namely knowledge and skill. I do reason deductively about N, and I come to the correct conclusion in virtue of my competence/skill. I just don’t get knowledge. That’s because knowledge involves something other than getting it right through skill.

    In other words, if I knew I wouldn’t come to the right belief about N through skill, I shouldn’t go in for deductive reasoning about it. But I don’t know that; in fact it isn’t even true. All that I know is that I won’t know N.

    November 19th, 2009 at 2:13 pm
  8. Matt Benton says:

    Hey, Jonathan! – side question: you say “Knowing that I don’t know p defeats justification in p.” Why think that?

    Can’t I have some reasons for believing p, which justify me in so doing, but if those reasons are recognized as not being good enough to make for knowledge, also not know that p? It doesn’t seem crazy that I could know that I don’t know that p, yet also believe that p on the basis of less-than-conclusive grounds. (Are you thinking that in such a case it wouldn’t be an all-out belief?)

    November 19th, 2009 at 3:37 pm
  9. Gabriele Contessa says:

    Brian,

    [I’m in a rush and haven’t had the time to read all comments so I don’t know if someone already raised this (if so, apologies)]

    What would you say if someone were to reply that there are no propositions like N and E, as no proposition is a proper part of itself?

    November 19th, 2009 at 3:59 pm
  10. Brian Weatherson says:

    I think you can get a version of N that involves self-description rather than self-reference. In that case the proposition won’t be part of itself and the problem won’t arise. But the argument against JTB etc goes through with these descriptive propsitions.

    November 19th, 2009 at 5:38 pm
  11. jon says:

    I think we’re very close to contradiction when we allow ourselves to form propositions like N above.

    Do you mean to treat knowledge as a predicate of propositions in N? If so, don’t we get a contradiction? For we can now define a knowledge-predicate for sentences:

    A sentence S is known iff there is a proposition p which is expressed by S such that p is known.

    But given minimal assumptions about knowledge this leads to contradiction by a result of Montague’s. (The assumptions are: knowledge (as a predicate of sentences) is factive; it’s known that knowledge is factive; if S is a first-order logical truth, we know S; closure under known implication.)

    If your framework is inconsistent, the defender of an analysis of knowledge probably isn’t too worried.

    You can block the contradiction, but I’m curious whether you curious whether you can block it without giving the defender of a particular analysis of knowledge a way out, too. E.g., one can block the contradiction by saying that some first-order truths aren’t known because they don’t express propositions. A general worry about that is that one would like a story about how and why certain sentences fail to express propositions. More to the point, if certain sentences fail to express propositions, why not sentences N above?

    November 19th, 2009 at 8:28 pm
  12. Andrew Bacon says:

    I’m kind of surprised by the idea (expressed in some of the comments) that you shouldn’t draw epistemological conclusions from the puzzles of self-reference such as the knower paradox. It’s a bit like saying we shouldn’t draw conclusions about truth from the liar.

    Incidentally, judging by the reaction philosophers generally had to the Lucas-Penrose argument I would have thought that people would be open to taking epistemological consequences from these kinds of arguments. (The Lucas-Penrose reasoning is formally very similar: given a person, A, you can produce a Gödel sentence for A, something like “G_A = A cannot know a priori that G_A”. Most people disagreed with Lucas and took the epistemological consequence that finite beings cannot know G_A.)

    November 19th, 2009 at 8:57 pm
  13. Geoff says:

    Great post!

    Maybe knowledge isn’t factive after all…

    November 19th, 2009 at 8:58 pm
  14. Dustin Tucker says:

    Jon: One possible story, I suppose, would go as follows. Sure, some sentences fail to express propositions, and N might be one such sentence. But I don’t care if I can express a proposition with N; all I care is that the proposition that I do not know this proposition (or, to avoid self-reference and with a tip of the hat to Anthony Anderson, the proposition that I do not know that my favorite proposition is true, which just so happens to be my favorite proposition; or whatever) is floating around somewhere for me to stand in the belief relation to, irrespective of whether I can express it with a sentence. I think that to get at the real problem, we need to move beyond talk of expression and sentences and start looking more at propositions, propositional attitudes, etc. themselves.

    Anyway, I’m tempted to think that whatever one thinks of similarly propositional paradoxes (I believe that something I believe is false and all my other beliefs happen to be true; I hope that everything you doubt is false and you doubt that everything I hope is true and everything else you doubt really is false and everything else I hope really is true; etc.) will apply here as well. Fond of genuine propositional truth-value gaps? Then proposition denoted by N lacks a truth-value (and so is not true, and so your belief in the proposition denoted by N is not JTB). Want to follow Arthur Prior’s paper on these paradoxes? Maybe the proposition denoted by N is just something that, no matter how hard Brian tries (after all, the proposition is about him), he cannot stand in the belief relation to. Got another story? Great!

    The consequences of any of these or other resolutions of the relevant paradoxes are going to be unpleasant, to be sure, but we need to handle them one way or another. And it seems like any such resolution had better have something to say about (the proposition denoted by) N, too. And I would not be surprised if, once we’ve heard what our favorite resolution has to say about N, it turns out to not be a counterexample to a theory of knowledge.

    November 19th, 2009 at 10:52 pm
  15. Brian Weatherson says:

    Geoff,

    Factivity becomes a tricky issue around here, because plausibly F1 and F2 are not identical.

    (F1) Everything known is true.
    (F2) Any argument of the form S knows that p; therefore, p is valid.

    When it is plausible that the T-schema fails, just how we define factivity gets delicate. I think both F1 and F2 are true, but they aren’t identical, and in the past I’ve probably run them together too much.

    November 19th, 2009 at 11:18 pm
  16. dtlocke says:

    Brian, you write:

    “But I can follow the reasoning that showed N is true. And I accept that reasoning, so I believe N.”

    Quick clarification question: are you using or mentioning N in the second sentence?

    November 19th, 2009 at 11:20 pm
  17. Brian Weatherson says:

    I was thinking I was using it. Does something turn on this? I’m worried that I might have slipped up here.

    November 19th, 2009 at 11:23 pm
  18. Brian Weatherson says:

    Jon,

    As Dusttuck says, I don’t need to claim that any sentence you can build from a certain set of resources expresses a proposition. I just need to claim that N is a proposition, and that I have a justified (or apt) true belief in it. And that might well be true even if there are some sentences that turn out not to express propositions.

    Dusttuck @14,

    I’m actually fairly sympathetic to the idea that belief is harder to get around here than it looks. I think the thing to say about B is that I don’t believe it.

    B: Brian does not have the true belief that B.

    I might try to believe it, but I’ll inevitably fail. Just like I might try, once I build my time machine, to kill one of my ancestors, but I’ll inevitably fail.

    But I’m not sure how far that should extend. One of the things that happens in the time travel case, as I understand it, is that there are certain limitations on what you can do that are needed to preserve consistency. (I mean that in somewhat plausible models of time travel, there turn out to be limits that look surprising considered locally, but which aren’t at all surprising if you consider what time travel is like.)

    But the limitations are fairly narrowly drawn. A ball can’t go through a wormhole and come out in a way that it knocks itself off the path into the wormhole. But it can go through the wormhole and come out in a way that it bumps into itself, as long as that bump is consistent with it being in the wormhole. Consistency is (I gather) the only limitation here.

    To a first approximation I say the same thing about the difference between N and B. I can’t consistently believe B, so I don’t. Arguably I don’t believe “Brian has a justified true belief in this proposition” either. But since N doesn’t have the concept BELIEVES as a constituent, I’m sceptical that we should say I can’t/don’t believe it.

    November 19th, 2009 at 11:26 pm
  19. Dustin Tucker says:

    I was trying to be non-committal about what it is that actually blocks the attitudes (certainly Prior is). But I was being sloppy, and I agree that, if that is the general route one wants to take, then (barring some fancy story about individuation of propositional attitudes), it is more plausible to say that knowledge, and not necessarily belief, is blocked in this case. And then, I suppose, we are back to having a counterexample to theories of knowledge. So much for the end of my first comment.

    That said, I think something in the spirit of that comment still holds. It sounded like you were suggesting that we might need an exception for propositions that contain the concept of knowledge, but this sort of exception, if it’s motivated by paradoxical cases generally, seems like it’s not going to undermine an analysis of knowledge in any serious way. That is, it seems like a fine analysis of knowledge (to me) that says, “S knows p iff A, barring funny paradoxical circumstances, whose identification is motivated independently over in this separate arena.”

    dtlocke: If N is a proposition, not a sentence, then I think it’s a category mistake to ask whether it’s used or mentioned. If, on the other hand, it’s a sentence, then, following what I and later Brian said, let’s deal instead with the proposition that Brian Weatherson does not know this proposition, call it (instead of the sentence) “N”, and work from there. I think everything will go through fine that way.

    p.s. While I don’t have a horse in the epistemological race, I am curious: Is there a theory of propositions that gives them concepts (perhaps among other things, I suppose) as constituents? That, I’d be interested in…

    November 19th, 2009 at 11:53 pm
  20. Brian Weatherson says:

    Eh, I wasn’t thinking clearly when I was saying BELIEVES could be the constituent of a proposition. I don’t know of any theory on which that is the case.

    I think I was running together two things. One is that concepts can be constituents of beliefs. The other is that something like concepts can play a role as modes of presentation of the properties that are constituents of propositions. But neither of those things mean that the concept is in the proposition.

    Sorry for being so sloppy.

    November 19th, 2009 at 11:59 pm
  21. Dustin Tucker says:

    To clarify the post script: I mean not just a general theory of structured propositions, or at least one that is not inconsistent in virtue of Russell’s paradox at the end of The Principles of Mathematics. I thought I had a sentence about that in there; sorry.

    November 20th, 2009 at 12:01 am
  22. Gabriele Contessa says:

    Brian @ 5:38pm,

    I think you can get a version of N that involves self-description rather than self-reference.

    I see how these distinctions make sense in the case of sentences but how would they work in the case of propositions in which presumably what is important are the referents of the referring expressions used to express them?

    Consider, for example, the proposition, if any, expressed by ‘Brian do not know that his favorite proposition is true’ as suggested by dusttuck @10:52pm. Now, it’s true that the sentence used to express this proposition does not use ‘N’ but that was not the problem with ‘Brian does not know that N’. The problem was that the proposition ‘Brian does not know that N’ supposedly expresses would be a proper part of itself and this remains true of the proposition expressed by ‘Brian do not know that his favorite proposition is true’ if those two sentences express both express the proposition referred to by ‘N’ in ‘Brian does not know that N’.

    November 20th, 2009 at 2:03 am
  23. Dustin Tucker says:

    First a general comment: I’d like to know more about propositions and what their parts are and why they can’t be parts of themselves. Some people (Barwise and Etchemendy, for instance) have allowed propositions that contain themselves, at least. And circularity is not always vicious. Is there a reason to prohibit it so universally? (There may very well be. I have no idea what propositions are.)

    Second, independent and more specific: Imagine two worlds, one in which Brian’s favorite proposition is p, the other in which it is q (distinct from p). In both worlds I believe that Brian does not know that his favorite proposition is true. (I think the previous sentence is fine. But maybe it is stacking the deck. Maybe I should instead say, “In both worlds, English is the same as it is here, and ‘Dustin believes that Brian does not know that his favorite proposition is true’ is true/satisfied/whatever.” Or something like that.)

    Do I really believe different things in the two worlds? That does not seem right to me, for the same reason that it seems wrong to me to say that the proposition denoted by the sentence “every dog barks” varies as the dogs in the world vary.

    Of course, that’s not an argument, and I might be missing something obvious. I imagine one could also tell a story about modes of presentation or something like that to account for these intuitions while insisting that the propositions themselves really are different, though then I would want to know more about modes of presentation and why we can’t set up the same problematic circumstances with them.

    Finally: If there’s just something funny going on with talking about favorite propositions, we can move to quantifiers, to make it more analogous to the dog case (and the other paradoxes I mentioned). Consider the proposition that Brian does not know any of the propositions he thinks about while writing such-and-such blog post, and suppose both (i) that he considers that proposition while writing the post and (ii) that in fact he does not know (and knows that he does not know) any of the other proposition he considers while writing the post.

    November 20th, 2009 at 3:45 am
  24. Brian Weatherson says:

    Gabriele,

    I basically just want to second what Dustin said. For the proposition < >, I think the things it has as parts are roughly the things it looks to have as parts in a standard Russellian first-order representation of it. So it has me as a part (perhaps twice). It has the properties/relations KNOW, FAVOURITE, PROPOSITION, TRUE (or something like them) as parts, and it has some logical connectives as parts. It doesn’t have itself as a part, though it has an existentially quantified part for which it is the only satisfier. But that’s not intrinsically paradoxical.

    November 20th, 2009 at 11:02 am
  25. Gabriele Contessa says:

    Brian and Dustin,

    Even admitting that propositions do contain quantifiers as their parts, it’s not clear to me that those quantifiers can range over the propositions that contain them. One argument for this view is that denying it seem to lead to all sorts of semantic paradoxes and now – thanks to Brian :-) – also all sorts of epistemological ones. But as we all know one philosopher’s modus tollens is another philosopher’s modus ponens.

    In any case, Brian, how do you see this fitting with your views about philosophical counterexamples? Wouldn’t you think that in this case like in the case of Gettier cases, if our intuitions give rise to a paradox that threatens some of our best epistemological theories, it may be the intuitions that gotta give?

    November 20th, 2009 at 12:26 pm
  26. Brian Weatherson says:

    Right, there are paradoxes looming. But for the reasons I gave in comment 18, I don’t think that’s a reason to deny N expresses a proposition. Any more than the fact that time travel might lead to paradox is a reason to deny all time travel.

    I’m less sure now than I was a few years ago about what intuitions are. But I’m pretty sure it isn’t an intuition that I don’t know N. It follows directly from some basic logical reasoning. And I don’t want to call modus tollens etc intuitions.

    November 20th, 2009 at 12:32 pm
  27. Gabriele Contessa says:

    Hi Brian,

    The intuitions I was mentioning are the intuitions one might have that the sentences you are using to set up the paradox express some proposition (just like one may have the intuition that there is a set of all the sets that are not members of themselves).

    In any case, my intuition is that none of those sentences expresses a proposition and this is supported by the intuition that propositions cannot be proper parts of themselves (just like sets, if there are any and notwithstanding some intuitions to the contrary, cannot be member of themselves) and that they cannot quantify over themselves. These intuitions are supported by the fact that denying them leads to paradoxes. Isn’t that as good an argument as it gets in philosophy?

    My view may be a minority view but wouldn’t it be attractive if it’s coherent and if by holding it I can avoid a number of paradoxes, including yours?

    November 20th, 2009 at 12:47 pm
  28. Dustin Tucker says:

    I am thinking that there would be two stories at work here, Brian: One that says propositions can’t contain themselves, and another that says they also can’t quantify over themselves. (Everything, I think, could be suitably reformulated so that it does not imply that propositions have propositions and quantifiers as constituents.)

    So there is no N as originally formulated, since it would contain itself, and the big quantificational proposition is fine, but cannot quantify over itself, so the argument will not go through.

    I am dubious of both restrictions on propositions. I would, for instance, still like to know more about why propositions can’t contain themselves. But I’m willing to grant that.

    The other one I am more worried about. Prohibiting propositions from ever quantifying over themselves amounts to ramification, at least as far as I can see. After all, we most prohibit cycles, too, on pain of paradox. To be sure, a ramified hierarchy of propositions will avoid the paradoxes, and I think we can avoid the trouble of reducibility if we just deal with mathematics extensionally or something like that. But ramification has always seemed heavy-handed to me: Why prohibit all circularity, even when it’s harmless? If the motivation is really avoiding paradoxes, we can do that without such drastic measures (even if we stick with quantifier domain restriction stories—it’s possible to flatten out the ramified hierarchy, as it were, only bumping propositions up to a higher order when it’s necessary to avoid a contradiction). And I do not know of any other solid motivation. (Russell had his Vicious Circle Principle, but Gödel pretty soundly demolished that.)

    As an aside, I think that it is interesting to note that all these problems go away if we allow propositions themselves to lack truth-values. Of course, then we have propositions that lack truth-values, which I don’t quite know what to make of. But I think it’s still worth keeping in mind.

    November 20th, 2009 at 12:59 pm
  29. Dustin Tucker says:

    Gabriele,

    Just a minor point: I still think it’s important to not merely talk about expression by sentences. The problem is not on the side of linguistic resources, but on the side of the proposition itself (or, at least, more generally on the side (how many sides are there?) of our ability to relate to the proposition by any number of attitudes).

    I don’t mean to say that you were trying to lay the blame on language rather than somewhere else. But I’ve frequently seen people say things about expression and then draw conclusions that are too narrow to actually resolve all of these problems, which is why I think continuing in these terms is potentially misleading. It is tempting to say, “Ah ha! Here are sentences that cannot express propositions on pain of contradiction. We’re done!” But given plausible (I think) assumptions about propositions themselves, we’ll be right back in the soup if all we’ve said is that certain sentences behave poorly sometimes.

    November 20th, 2009 at 1:07 pm
  30. Brian Weatherson says:

    I think I’m again just going to be echoing Dustin, but let me note one cost of the “propopsitions can’t quantify over themselves” line.

    If propositions can’t quantify over themselves, then we can’t have any quantifier that ranges over all propositions. And if we can’t have that, then we can’t have any absolutely unrestricted quantifier.

    Now some smart people have been happy with the latter claim because of the paradoxes. But it would make a very large difference to current metaphysical debates, I think, if it were accepted. It certainly seems like a big step to take.

    An alternative position (also radical!) would be to say that propositions aren’t things, i.e. in some sense there are no propositions, and we can’t quantify over them for that reason. And that’s true even though we can quantify unrestrictedly over all things.

    That’s all to say, the natural solutions to this puzzle about knowledge have very wide-ranging metaphysical implications.

    November 20th, 2009 at 1:28 pm
  31. Gabriele Contessa says:

    Dustin,

    I would, for instance, still like to know more about why propositions can’t contain themselves. But I’m willing to grant that.

    Because ‘x is a proper part of y’ is a paradigmatic case of anti-reflexive relation. Of course, one could deny this and one could also deny that a proposition need to contain other propositions as parts (for example if one takes propositions to be functions from worlds to truth-values). I think there are good reasons to reject both these proposals but I don’t want to hijack the thread more than I’m afraid I have already done (Btw, Brian, if you want us to stop all this discussion about propositions, just say, ‘Stop!’)

    Brian,

    I completely agree with you. This paradox has wide-ranging implications and this is the hallmark of all good paradoxes (nice one!!!). I would say that my position is conditional ‘if there are propositions, they cannot contain themselves as proper parts and if they can contain quantifiers, they can’t range over the proposition itself’. I agree this has the consequence that there are no absolutely unrestricted quantifiers, but I think I’d rather live with that than with the kind of paradoxes that the opposite view generates.

    And I agree that one could deny that there are propositions (In fact, I think I’m leaning more and more towards that view because of a number of other concerns I have with propositions but since they are so useful and most philosophers accept them I’m going to stick with them until I decide they do more harm than good like, I think, possible worlds do)).

    November 20th, 2009 at 4:03 pm
  32. Dustin Tucker says:

    Gabriele, I guess I just don’t have such strong intuitions about proper-part-hood. Especially when we’re dealing with propositions, since (i) I worry that theories that give them parts risk running into (other) paradoxes and (ii) I think self-referential propositions make perfectly good sense. (Indeed, it’s precisely because they make good sense that I can understand why they sometimes lead to contradictions.)

    Brian and Gabriele, I still think it’s a little misleading to characterize this as a puzzle about knowledge in particular. To be sure, it involves knowledge, and resolving it might force us to rethink something in our account of knowledge (though I am dubious that it will be anything substantial). But, again, I think that we can get the same metaphysical worries and consequences out of paradoxes that do not involve knowledge. Some involve other propositional attitudes. Some involve propositions that are circular in some way (though, following Yablo, not necessarily by being about/containing/whatever themselves). Some are along the lines of Russell’s paradox from Appendix B of The Principles of Mathematics.

    It’s not clear that all these paradoxes must be resolved in the same way. We’ve seen different stories arising for different paradoxes already in these comments. But they are diverse enough that I think it is dangerous to think of the present problems as crucially involving knowledge in particular.

    November 20th, 2009 at 4:45 pm
  33. Dustin Tucker says:

    Sorry, to clarify: When I said “I think self-referential propositions make perfectly good sense,” I did not mean to say that that is somehow irrefutable evidence that self-referential propositions are OK. It just makes me cautions about prohibiting them all without exception. And now I think I’m well past beginning to sound like a broken record, so I will shut up on this topic. Maybe. I hope.

    November 20th, 2009 at 4:49 pm
  34. Gabriele Contessa says:

    Brian,

    Some further worries about N. First of all, what is it exactly that you don’t know according to N? You don’t know that you don’t know that you don’t know… what?

    Second, is N an infinite proposition? And if so can we even entertain it?

    November 20th, 2009 at 5:09 pm
  35. Dustin Tucker says:

    What if, instead of N, we deal with the proposition that Brian does not know that this proposition is true. Maybe there is no such proposition, but it seems intuitively OK to me (insofar as I have intuitions about what propositions are OK, which is not very far, I think). Here, I’m just falling back to Barwise and Etchemendy.

    What is an infinite proposition?

    November 20th, 2009 at 6:22 pm
  36. Gabriele Contessa says:

    Dustin,

    The question is what is the proposition Brian does not know and your saying that is this proposition doesn’t help. Let me put the question differently: ‘What would Brian know were he to know that N?’. It seems that he would know that he doesn’t know that he doesn’t know that he doesn’t know that… My question is what is it that he doesn’t know he doesn’t know that he doesn’t know that he doesn’t know …?

    November 20th, 2009 at 7:05 pm
  37. Dustin Tucker says:

    Ah, I think I was (very possibly naively) treating propositions like any other thing floating around, there for propositions to be about. I see your point now. (I do not have something intelligent to say about it, but at least I think I understand it now!) Thanks.

    November 20th, 2009 at 7:31 pm
  38. Brian Weatherson says:

    Gabriele,

    I don’t think it’s wrong to say that if I didn’t know N, I wouldn’t know N. If you try to give a reductive account of this in terms of other things, it will be hard. But that’s just on a par with anti-reductionism about all sorts of things.

    If you want to model it, it’s going to be just like non well founded sets, as Dustin has been saying, so sets like {{{…}}}. That ‘set’ is a member of itself, and in a sense it is infinite, though it doesn’t look overly problematic.

    One related point. That a can’t be a proper part of a seems reasonable enough. But the principle you need here is that a can’t be one among many constituents of a, and that’s much more contentious I think. The ‘set’ {{{…}}} doesn’t have itself as a part, but it does have itself as its only member. And that makes, I think, a big metaphysical difference.

    November 21st, 2009 at 7:32 pm
  39. Brian Weatherson says:

    Quick follow up on that last point.

    I believe that I’m in New York. Plausibly, that belief is a part, or at least a constituent, of my mind. And I’m a constituent of the proposition that belief has as its content. But there’s nothing paradoxical or metaphysically dubious in any of this. This is all to say that constitution plays by very different rules to parthood.

    November 21st, 2009 at 7:33 pm
  40. Gabriele Contessa says:

    Brian,

    I don’t think your answer helps either. That is not what I was asking. I do not doubt that we can use a number of linguistic devices to refer to a proposition (including names and definite descriptions) but in order to have any propositional attitude towards it we need to be able to entertain it.

    My challenge is that, even if N existed, no finite being would be able to entertain it because N would be an “infinite” proposition. according to you, there would be nothing paradoxical in my knowing that N, right? But I have no idea of what I would know if where I to know that N. I guess I would know that you don’t know that you don’t know that you don’t know … But I don’t know what that means (what proposition it expresses)> Moreover, what is it that you don’t know according to N (or that you don’t know that you don’t know or that …)? Well, that you don’t know that you don’t know that you don’t know that… I have no idea of what it is for me or for you to know, believe or even entertain such an infinite proposition. Do you?

    Now, if I am right, then, if N exists, it is true and necessarily so, but that reasoning is no way for you (or me) to know that N because in order to know that N you (or I) would have to be able to entertain the proposition N and as I argued I think neither of us can.

    November 21st, 2009 at 11:38 pm
  41. Brian Weatherson says:

    I entertain lots of infinite propositions. I even know them. I know that there is a set {{{…}}} in non well-founded set theory, but not in ZFC. I know that there are arbitrarily long arithmetic progressions of primes. I know that all false propositions are not known, including the negation of this very proposition. Infinitude doesn’t seem like a problem here.

    November 22nd, 2009 at 10:32 am
  42. Gabriele Contessa says:

    Brian,

    As far as I can see, none of your examples is an example of an infinite proposition (in the sense I was using the term). They are only examples that quantify over infinite domains or domains that contain some infinite entities but none of the propositions themselves is infinite. Can you honestly claim that you can entertain your alleged proposition N? I don’t think you (or any finite being for that matter) can.

    November 22nd, 2009 at 1:59 pm

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