Knowability and Position-to-Know Closure

January 22, 2009 in Epistemology, Logic | 5 comments

Ryan Wasserman1 recently suggested a new line of argument against the Paradox of Knowability (brought up on Certain Doubts by Jonathan Kvanvig), and as I have just done some work on the paradox involving a position-to-know closure schema, I thought I might work out some of the details of what I took to be the thrust of his proposal2.

From what I can tell, Wasserman’s proposal was that, given a position-to-know closure schema, the paradox might be blocked through the proposal of an incompetent deducer. Assuming a generic position to know closure schema, where knowledge of p provides one with evidence of p’s implications:

(C) (Kp & (p→q))→Eq

And given the weak assumptions necessary for the paradox of knowability:

(KIT) Kp→p

(AN) ⊢ p ⊢ □p

(MI) □~p ⊢ ~◊p

Where (KIT) states that knowledge is logically stronger than truth, (AN) states that if a proposition is provable from no premises, it is a necessary truth, and (MI) a modal identity provided the interdefinability of the modals possibility and necessity. Given (C) the paradox would need an additional competent deduction principle:

(CD) CD(Eq)→Kq

This step states roughly that one may deduce from one’s evidence knowlede of q, provided that one has not lost one’s knowledge of p and p→q. The paradox itself begins with the weak assumption of the verificationist that truth is in some sense connected to one’s epistemic ability:

(WVer) p→◊Kp

The Fitch proposition (p & ~Kp)–that there is an unknown truth–is then assumed to be known for the purposes of a reductio:

(1) Assume: K(p & ~Kp)

(2) Ep & E(~Kp)                     (C)

(3) Kp & K(~Kp)                    (CD)

(4) Kp & ~Kp (KIT)

(5) ~K(p & ~Kp)                    (1)-(4), Reductio

(6) □~K(p & ~Kp)                  (AN)

(7) ~◊K(p & ~Kp)                  (MI)

At this point, substituting the fitch proposition into (WVer) provides ((p & ~Kp)→◊K(p & ~Kp)), and this in conjunction with (7) results in the negation of the consequent of (WVer). As such, the antecedent must be false: ~(p & ~Kp). And by trivial logic this provides (p→Kp). Generalizing on this:

(OP) ∀p(p→Kp)

Hence, if all truths are knowable, all truths are known (at some time, by some person).

Wasserman seems to propose an instance where some person has deficient deductive skill, in effect blocking (CD). This would seem to prevent the proof in the same manner as a wholesale rejection of knowledge-based closure. Given this, the solution must overcome Williamson’s revised proofs3 which propose a method of circumventing the rejection of distribution under conjunction made available by a closure schema: K(p & q)→(Kp & Kq). This circumvention comes in two forms. First, Williamson proposes a strengthening of (WVer) to the premise that if a conjunction holds, it is possible that each of its conjuncts is known to hold:

(CVer) (p & q & … & r)→◊(Kp & Kq & … & Kr)

It is here that Kvanvig suggests Wasserman’s solution stalls, as the incompetent deducer in question would have to lack the logical acumen to handle conjunction. For in this version of the paradox, the contradiction ensues from the instantiation of the fitch proposition into (CVer) itself, rather than from any later deduction:

(1W) Assume: (Kp & K(~Kp))

(2W) Kp & ~Kp (KIT)

(3W) ~(Kp & K(~Kp))                       (1W)-(2W), Reductio

As seen here, the reductio is available without use of (CD), and thus Wasserman’s incompetent deducer would need to be additionally incompetent regarding the basic rules of logic. This is beginning to border on a risky approach, however, as once one stipulates the knower as unable to grasp the most basic rules of logic the solution begins to look somewhat trivial. That is to say, it is unclear at this point how the potential knower would be a candidate for any knowledge whatsoever. The only knowledge available to the “knower” would be of the non-inferential variety, and it could be argued that there is no such knowledge (ie., from the position of the coherentist).

  1. Wasserman just presented here at Baylor, and I have to say his presentation was quite interesting. While Ethics is not my area, I found some interesting parallels between his position and subject-sensitive invariantism in Epistemology. Good show! ↩
  2. Now, I may be completely misinterpreting Wasserman’s approach, and if so consider this simply an exercise in a possible solution to the paradox. ↩
  3. Williamson, Timothy (1993), ‘Verificationism and Non-Distributive Knowledge’, Australasian Journal of Philosophy, Volume 71, No. 1, 78-88. ↩

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5 comments

  1. Ryan Wasserman’s avatar

    Ryan Wasserman on February 14, 2009 at 1:11 pm

    Hey Jon

    Thanks for taking the time to work through my initial idea about the knowledge paradox. I agree with you that my objection does not apply to Williamson’s proof and so it doesn’t get to the heart of the issue (I have to confess that I hadn’t heard about Williamson’s move until you mentioned it to me).

    Let me just clarify one point about the “incompetent” deducer: He doesn’t need to be incompetent. He just needs to be someone who knows that (p & ~Kp) and yet can’t deduce each of the conjuncts. ONE way to be such a person is to be incompetent. Another way is to be unlucky. Perhaps the person who knows the conjunction always slips on a banana peel and is distracted when he tries to do the disjunction. Or perhaps he always has a change of heart when he sets out to do the deduction. Or whatever.

    Analogy: It’s possible to travel back in time and meet one’s grandfather. And it’s possible to try and kill one’s grandfather in such a situation. But one is always going to fail, since it’s impossible to kill your own grandfather (or to change the past at all). In some cases, the failure may be the result of an incompetent assassin. But in other cases the assassin might just have a change of heart or slip on a banana peel or whatever.

    In the same way, it’s possible to know (p & ~Kp). And it’s possible to TRY and deduce the conjuncts. But one is always going to fail, since it’s impossible to know both p and ~Kp.

    Thanks again.
    RW

  2. Ezra Cook’s avatar

    Ezra Cook on February 14, 2009 at 2:23 pm

    Hi Ryan,

    This is Ezra, I am one of Professor Kvanvig’s students, and I just recently worked up a writing sample on the paradox. Thanks for the comment.

    The luck modification does eliminate the conceptual problems previously discussed, but I think that there is an additional problem with this proposed solution. Simply put, it does not generalize properly. Given that the knowledge operator ‘Kp’ is an abbreviation for the existentially closed three place relation ‘there is some subject S who knows some proposition p at some time t’ or ‘∃s∃p∃tKspt’, the availability of a single counterinstance is not going to block the paradox. The counterinstance must generalize, and I think here you are going to have problems. I do not think that, generally, all inferences from K(p & q) to Kp & Kq are blocked, either by logical incompetence or luck.

    Thanks,
    Ezra

  3. Ezra Cook’s avatar

    Ezra Cook on February 15, 2009 at 4:49 pm

    Hi,

    I have thought a bit more about this problem, and I would like to clarify a point to see if there might be a way out for your solution. The problem as I saw it was that of instancing an agent which was unable to make the inference from ‘K(p & ~Kp)’ to ‘Kp & K(~Kp)’, and I don’t think this works. This is due to the existential generalization in the K operator over agents. If some agent in the domain is unable to make the inference, there is nothing blocking some other agent making the inference without failure. This lead me to the conclusion that some schematic theorem must be rejected (such as the decomposition of conjunction previously mentioned), but I think something more restricted might be feasible.

    What I am thinking here (and this may be what you are getting at with the time travel example) is that there is something unique about the specific conjunction ‘p & ~Kp’ that disallows the move in question generally. To use an example of Williamsons, I know at t_1 that either ‘at some point t_0 previous to t_1 the number of books on my bookshelf was even’ or ‘at some point t_0 previous to t_1 the number of books on my bookshelf was odd’. One of these two propositions is true and thus I know an unknown truth. Now at this point it is unclear to me how the agent in question would make the jump from knowledge of an unknown truth to knowledge of the conjuncts within the claim that one knows an unknown truth. This might not be due to luck or incompetence, but perhaps due to some tension between the first-person and third-person perspective involved. The competent deduction schema builds in some first-person restraints to the extension of knowledge under deduction, but I am thinking that perhaps the decomposition of the conjunction involved in the unknown truth here might presuppose a logically omniscient perspective. I could say that, in this case, there is some proposition that I both know and know that I don’t know, but I don’t know which proposition it is (either even or odd). Is this, perhaps, where you are going with this solution? I would have to take a look back at the literature to see how Williamson works out this problem, but I don’t think he took into account position-to-know closure schemas, which might make a decided difference.

    Ezra

  4. Ryan Wasserman’s avatar

    Ryan Wasserman on February 17, 2009 at 7:34 pm

    Hi Ezra,

    First, sorry for mistaking you for Jon!

    Second, you write:

    The problem as I saw it was that of instancing an agent which was unable to make the inference from ‘K(p & ~Kp)’ to ‘Kp & K(~Kp)’, and I don’t think this works. This is due to the existential generalization in the K operator over agents. If some agent in the domain is unable to make the inference, there is nothing blocking some other agent making the inference without failure.

    My thought was that, while it may be possible for someone to know (p & ~p) and it may be possible for someone to know (p & ~Kp) AND try to deduce p and ~Kp. But it’s not possible for ANYONE to know (p & ~Kp) and SUCCESSFULLY deduce p and ~Kp, since that would result in contradiction.

    Here’s another analogy (from Ted Sider’s paper on time travel). Call someone a ‘permanent bachelor’ just in case he never was, is or will be married. I am possibly a permanent bachelor; i.e., there are some worlds where I am a permanent bachelor. In some of those worlds, I might try to get married. But we know, given the definition, that I’ll fail at those worlds. I might fail for different reasons at different worlds (banana slips, cold feet, etc), but the bottom line is that it’s not possible for me to be a permanent bachelor while successfully getting hitched.

    What goes for me goes for everyone (at least every guy). So it’s nothing specific about me. It just about the property of being a permanent bachelor.

    Same thing for the knowability paradox: It’s not about any particular person, it’s just about the proposition itself. Or, at least, that was my thought.

    RW

  5. Ezra Cook’s avatar

    Ezra Cook on February 18, 2009 at 10:06 am

    Ryan,

    Let me see if I am getting this distinction right. It is possible for someone to hold inconsistent beliefs, and it might even be possible to know a contradiction (if, perhaps, dialetheism were correct). What is not possible, however, is for someone to know a conjunction of propositions in which there lies a contradiction and successfully derive the contradiction from the conjunction. As you stated here: “My thought was that, while it may be possible for someone to know (p & ~p) and it may be possible for someone to know (p & ~Kp) AND try to deduce p and ~Kp. But it’s not possible for ANYONE to know (p & ~Kp) and SUCCESSFULLY deduce p and ~Kp, since that would result in contradiction.”

    I have two worries about this proposal as it stands:

    First, a worry about Frege. Frege would have known his Basic Law V and the naive comprehension axiom which it entails, but from this he is simply a competent deduction away from the set of all and only non-self-membered sets, as noted by Russell. Now, I don’t think it is feasible to say that it would be impossible for Frege to derive the contradiction on the basis of his knowledge of the axioms, for Russell, I would think, would have had to know the axioms in order to derive the contradiction himself. This leads me to think that the impossibility you are proposing must be restricted to the Fitch proposition ‘p & ~Kp’ itself.

    Second, if the strategy you are proposing is restricted as previously mentioned, then, on the face of it, it seems similar to the restriction strategies proposed by Neil Tennant (1997, 2001). He proposed various strategies for restricting the verificationist claim to Cartesian propositions (which are just those propositions not provably inconsistent) or restrictions in terms of what is metaphysically possible to know. A wealth of literature has built up around these strategies, and I can say that both Kvanvig and Williamson have posed serious problems for strategies of this kind. Off the top of my head, I would say that you are going to need some motivations for this restriction external to the paradox itself to avoid claims that the solution is ad hoc in nature (See Kvanvig and Hand 1990).

    Ezra

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