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Sunday, May 14, 2006

More kung-fu

Here's a question that popped into my head today. Consider the following argument:
1. Maybe I know kung-fu.
2. I don't know kung-fu.
---
3. Maybe I know kung-fu.

Is this argument valid? It strikes me as invalid, when I ignore my logical indoctrination. But it looks like 3 is just a reiteration of 1, and reiteration is a valid rule of inference if anything is. My options seem to be to explain how/why it's really invalid by explaining what rules of valid inference are being broken, or to say that it's valid and explain away my impression that it's not. This second option will probably require that we flesh out the situation in which the argument is made, but I'm not sure what kinds of details would be relevant. Do all-y'alls have any thoughts about this?

8 Comments:

  • At 7:09 PM, Blogger Josh said…

    Wow, that's an interesting case. I share the feeling that it's invalid, yet I see that it's valid to merely repeat a premise as the conclusion. There is the option of explaining that the intuition that the argument is invalid is just wrong, but it just seems to me that there is a legitimate reading of this that makes it invalid.

    So let me take a shot at explaining how this argument could be interpreted as invalid. First, I think 'maybe' is best interpreted as modal in this kind of argument. It seems that 'maybe I know kung-fu' means 'it's possible that I know kung-fu'. On that reading the argument can be re-written as follows (with 'P' being the possibility operator):

    1. P(I know kung-fu)
    2. ~(I know kung-fu)
    ---------------------
    3. P(I know kung-fu)

    This doesn't vindicate the feeling that it's invalid though, because 'it's possible that I know kung-fu' just means that there is some possible world in which I know kung-fu, and that need not be the actual world. But I think this version doesn't reveal as much logical or modal structure as it could. Perhaps, 'maybe' in this context means 'it's possible that I actually know kung-fu', which I take can be re-written as 'it's possible that it's actually the case that I know kung-fu'. With that in mind, maybe we can vindicate the intuition that it's invalid with the following reading of the argument (with 'A' as an actuality operator):

    1. P[A(I know kung-fu)]
    2. A[~(I know kung-fu)]
    ---------------------
    3. P[A(I know kung-fu)]

    I think this reading vindicates the interpretation that the argument is invalid, but that depends on how the modal logic is cashed out. First, 'possibly actually X' must mean something other than 'possibly X', otherwise the argument would be valid (as we already saw). It must come out that (1) and (2) contradict each other. (Although, in that case wouldn't it still be valid!? Doesn't a contradiction imply anything? Well, maybe the intuition is not so much that it is invalid in the formal sense, but that it is just a bad argument.)

    At this point I would need help with how to cash out the argument in some modal system. I suspect that different systems cash the logic out in different ways and that this is where the confusion is. Statements of the form 'possibly actually X' are difficult to figure out. 'Actually possibly X' seems to just mean 'possibly X', but, as this argument shows, 'possibly actually X' seems different.

    -Josh

     
  • At 10:47 AM, Blogger douglys said…

    Josh seems to be on the righ track. "maybe" here seems to be some kind of epistemic possibility (EP), i.e. "For all I know ...". If so, I think I can get the premises to contradict each other.

    1. (For all I know) I know kung-fu.
    2. I don't know kung-fu.

    From (For all I know) I know kung-fu, it follows that (For all I know) I don't know kung-fu. right?
    EP(Q) -> EP(~Q)

    And from EP(Q) and EP(~Q) it follows that ~K(Q) and ~K(~Q), that is

    1a. (I don't know that) I know kung-fu.
    1b. (I don't know that) I don't know kung-fu.

    Now on the second premise, it seems to follow from 'I don't know kung-fu' that (I know that) I don't know kung-fu. (some kind of KK principle)
    ~K(R) -> K~K(R)

    Now using (1b) we can get a contradiction:

    1b. (I don't know that) I don't know kung-fu.
    2a. (I know that) I don't know kung-fu.

    so still valid but the badness revealed. seem plausible? I'm a bit worried about using the KK principle where 'know' is more of a 'know how' ability thing. anyway that's my attempt...

    B

     
  • At 12:00 PM, Blogger Luke Manning said…

    Thanks for the comments, guys. First off, I don't think a KK principle can be used on the second premise. If I know how to do something, it doesn't follow that I know that I know how. Children acquiring their native language know how to form grammatical sentences, but needn't know that they know how to do it. So I don't think we can get (2) to imply that I know (2), since it's just a subject-predicate proposition [F(a)].
    Second, I don't think the problem here is that the premises are inconsistent, although they may be. The problem is that the argument seems invalid, not just "bad" in some respect or other. Compare:
    1. P
    2. ~P
    ---
    3. P
    Here, the premises are inconsistent, but the "weirdness" of this argument has a different flavor: it's more like it's just a case we wouldn't ordinarily consider informally, but that's covered by our formal systematization of deductive inference. The argument I offered in the post just seems invalid.

    You both give interesting readings of 'maybe' as a sentential operator, but I'm not sure that that strategy is going to work out, largely because I can't see it explaining how this argument isn't just a simple application of reiteration. I think we may have to confront that issue more directly.
    Here's a brief suggestion about that: (2) seems like a retraction of (1), such that (1) is no longer available for use in reiteration. But how could we make formal sense of this notion?

     
  • At 9:25 PM, Blogger Josh said…

    I didn't realize that you (Luke) were thinking that the problem was that it is literally invalid. I don't know that that was my intuition from the start, but it seems like another reasonable reading of the argument. As far as the intuition that Brian and I were following, I think Brian (and me, I guess) found a pretty good way to make sense of it.

    As regards Luke's intuition that it is formally invalid, I think that the retraction hypothesis makes sense. I'm not sure how to cash that out formally, but it seems like it wouldn't be too difficult to do. However, we need to make (1) and (2) consistent, right? If (1) and (2) contradict each other, then the argument is hopelessly valid, even on the retraction hypothesis, right?

    -Josh

     
  • At 11:57 PM, Blogger Luke Manning said…

    I agree, I think you're both roughly on the right track if we take 'maybe' to be a sentential operator. Perhaps that captures what maybe does in some cases. However, in this argument, no sentential operator is going to make the argument invalid with consistent premises, because it will just be a case of reiteration.
    What I guess is most noteworthy about this argument (as I'm conceiving it) is that it's not even an argument according to the traditional formal definition, i.e., a sequence of sentences. Instead, we have to think of them as assertions (or somesuch), because assertions (not sentences) can be retracted. In that case we need a new definition of validity, which probably would be something like the following. A sequence of assertions and retractions is a valid argument iff each assertion in the sequence is either a premise, axiom, or follows from previous assertions that have not at that point been retracted. This last point is a little tricky; consider the following, where Ret(P) is the retraction of P:
    1. P
    2. P -> Q
    3. Q
    4. Ret(P)
    ---
    5. Q v R

    Here each assertion follows from previous assertions that have not yet been retracted, but the argument is intuitively invalid, right? The point is that for each inference (line n), we must go back to the beginning to see whether, using only the assertions that do not get retracted before line n, we can infer the assertion of line n by one of our inference rules (assuming it's not an axiom or premise).

    I think this is part of the story. What I haven't figured out is what 'maybe' is contributing. It seems to be something weaker than assertion, and not assertion of metaphysical possibility. Now, I'm a little wary of adopting Brian's strategy of construing the non-maybe line as a knowledge claim. It's just an assertion, not a claim that I know that P (where P = "I know kung-fu"; the 'know' in that isn't relevant to the formal structure of the argument). I think the argument I first presented shares some features with arguments about possibility and knowledge, but I think it's stretching things to call 'maybe' an epistemic operator. I think its function is much more likely one of suggesting than one of asserting something (e.g., something less than truth or knowledge).
    As a very informal characterization, I'm sort of inclined to say that "maybe P" offers P for possible refutation, and "~P" refutes it; the second "maybe P" brings up something that has already been shot down. Of course, this is beginning to sound a little different from the above "retraction" reading. What do you think, guys? What role does 'maybe' play in ordinary discourse, and how can we describe its formal features?

     
  • At 10:51 AM, Blogger douglys said…

    I retract my earlier post, in light of discussion with Ian and Josh and Luke’s post. Yet, I think the spirit of my original post is aligned with the True.

    Luke says "...it's not even an argument according to the traditional formal definition, i.e., a sequence of sentences. Instead, we have to think of [the premises] as assertions...". I agree, so I will propose some norms of assertion.

    M1: S is permitted to assert "maybe P" iff S is permitted to assert "I don't know that P" AND S is permitted to assert "I don't know that not P".

    A1: S is permitted to assert "P" iff S believes that P.

    B1: S believes that P iff S believes that S knows that P.

    I think these are intuitive principles (...is B1 right?) and that they are enough to reveal the oddity in the Luke-style-kung-fu arguments. I will change the example, although I like kung-fu the "know" kept distracting me. I will use the following but it shouldn't make a difference.

    1. Maybe P.
    2. Not P.
    ------------
    3. Maybe P.

    Starting with assertion (1):
    By (M1) it follows that if S asserts "Maybe P" then S is permitted to assert "I don't know that not P".

    By (A1) it follows that if S is permitted to assert "I don't know that not P" then S believes that S doesn't know that not P.

    Thus, by using our norms of assertion we conclude from S's assertion that "Maybe P" that:

    (1*) S believes that S doesn't know that not P.

    Turning to assertion (2):
    By (A1) it follows that if S is permitted to assert "Not P" then S believes that not P.

    By (B1) if S believes that not P, then S believes that S knows that not P.

    So, by using our principles we conclude from S's assertion that "Not P" that:

    (2*) S believes that S knows that not P.

    Putting (1*) and (2*) together S has contradicting beliefs, i.e. S believes both that she does and doesn't know that not P.

    To fill in the P's let's use this.

    1. Maybe Finland has a king.
    2. Finland doesn't have a king.
    ------------------------------
    3. Maybe Finland has a king.

    I have argued that from S's assertion that "Maybe Finland has a king" we can conclude that S believes that she doesn't know that Finland doesn't have a king.

    However, from S's assertion that "Finland doesn't have a king" we may conclude that S believes that she knows that Finland doesn't have a king.

    So we can conclude that our S who asserts these sentences commits herself to conflicting beliefs (if the principles are right).

    Anyway, that's all the fun I get for now. I think there is a way to make sense of the weirdness of it; whether it is "invalid" or not seems to be a terminological issue. It violates some norm of rationality, the "norm of validity" seems to be a subset of that broader class of norms...but now I am not sure what I am talking about...

    [Perhaps this could all be fancied up with some stuff about retraction; that seems like an important norm of assertion. So i think the direction I am aiming is consistent with where Luke is aiming.]

     
  • At 1:29 PM, Blogger Dr. Funk said…

    I would rephrase the argument as follows:

    1. The probability I know kung fu P>0
    2. The probability I know kung fu P=0
    ---
    3. The probability I know kung fu P>0

    So I see the premises as being inconsistent. This is my interpretation of the word maybe.

     
  • At 11:30 AM, Blogger Luke Manning said…

    Thanks for the further comments, all. I have also talked to Ian and Carl about this, and pretty much nobody has been sympathetic to my view that the argument is invalid. So let me restate the problem:

    Consider the argument:
    1. Maybe P
    2. ~P
    ---
    3. Maybe P
    Though there are readings of this argument on which it's valid (and there may even be sound interpretations, depending on how we construe 'maybe'), I think there's a good and perfectly ordinary understanding of it as invalid. Consider a discourse in which someone uttered those three sentences. The most common reason for uttering (2) after (1), I submit, would be to retract (2). On that understanding, after step (2), (1) is no longer available for reiteration, and so the argument is invalid.
    I'm suggesting, somewhat subversively, that we have a notion of valid argument that's broader than class of sequences of declarative sentences. Now we could simply take the above argument as not the argument that's presented in the discourse I'm imagining. In that discourse, the argument is:
    1. ~P
    ---
    2. Maybe P
    because when "Maybe P" was retracted, it stopped being a part of the argument. Then we can apply the canonical formal notion of validity to show that it's invalid.
    I'm suggesting that we're better off taking the whole thing (including the retracted premise and its retraction) as an argument. The advantages of this probably aren't evident in this simple case, but I'm thinking that there might be ordinary argumentative-discourse-moves beyond retraction and assertion, some of which might operate on assertions that have previously been retracted. So if we want a general framework for representing arguments in ordinary discourse (and not mere sequences of propositions), we may want to leave open the possibility that retracted assertions might come back into play. This, plus the possibility of other argumentative-discourse-moves capable of formal treatment, gives us pretty good reason to take the whole thing (1-3) as the argument.

    So it turns out that what I found interesting about the argument is not what anybody else found interesting about it. "Maybe", for me, was (unintentionally) a red herring. I think it's interesting to consider what "maybe" does, and I think the proposals that have been offered (metaphysical or epistemic possibility, epistemic probability) are all pretty plausible, at least in certain cases. I also think that it can serve a discourse-specific role (i.e., one that's not merely a contribution to the truth-value of the sentence containing it, and in this case one that plays a role in argument structure), perhaps of introducing a proposition for possible confirmation or refutation, or maybe something else. Of course, if it doesn't have some assertive content, then this isn't an argument, because the conclusion of an argument has to be an assertion (or, perhaps, a plan about what to do, which this isn't either).

    So I've mostly shown my cards. I think we're left with two issues:
    1. Is there a "broad" notion of validity like I described? Do we recognize it in ordinary discourse, or is validity really only about sequences of propositions, whereas the other stuff in argumentative discourse is mere placement and coloration? I've been influenced by David Kaplan's recent work to find the latter view dogmatically restrictive. It's certainly not impossible to treat this stuff formally, and people do. The fundamental philosophical issue is the scope of the notion of validity.
    2. What does "maybe" do? The readings that we've suggested are plausible in various cases, but if possible I think we should find a unified semantic account of this term. Can we generalize from the given suggestions, or decide on one function that "maybe" has? Or alternatively, can we give good reasons to think that it has no single function?

     

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