Modal Epistemology

In my last post I worked out some details of a possible counterexample to the paradox of knowability. The thrust of the problem is stated by Jonathan Kvanvig on Certain Doubts:

The idea is that some truths might be knowable only to incompetent deducers, and knowledge that a given claim is an unknown truth might be just such a knowable claim–only the logically challenged could have such knowledge.

I have had some time to think about this problem in greater detail, and it will be the purpose of this post to work out and reject a potential work-around to the conjunctivity problems faced by Wasserman’s account.

The central notion of this problem is that, given a position-to-know closure schema, some logically challenged deducer would have certain second-order knowledge available that a competent deducer would find unavailable–due to the ability to deduce competently from one’s evidence to the implications of one’s knowledge. This strikes me initially as a variation on those solutions involving the rejection of closure related to the epistemic views of Nozick and Dretske, but given the details of the account it would be possible to work around Williamson’s variations which attack the restricted closure accounts. This ability, however, comes with the consequence that the agent involved must lack the concept of conjunction.

To read this consequence charitably, I will consider the possibility that this incompetent deducer might maintain the concepts of some basic logical notions in order to work around the conjunctivity issue. Particularly, as it is possible to build the basic logical operators ‘&’ and ‘~’ out of a single connective, the Sheffer stroke ‘|’, such that:

(S) ‘p | q’ is true iff ‘p’ and ‘q’ are not both true.

Provided (S), ‘~p’ may be expressed as ‘p | p’ and ‘p & q’ may be expressed as ‘(p | q) | (p | q)’. In this sense our incompetent deducer could work around his conjunctive deficiency whilst blocking the conjunction problems in Williamson’s example (provided in the previous post).

This move does not really work around the problems one confronts with the paradox, however, as the paradox itself may be cast in terms of the Sheffer stroke. The Fitch proposition may be stated, rather than ‘p & ~Kp’, as ‘(p | ~Kp) | (p | ~Kp)’, with the primary reductio subproof as shown:

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

(2) E(p | ~Kp) | E(p | ~Kp)                               (C)

(3) K(p | ~Kp) | K(p | ~Kp)                              (CD)

(4) (Ep | E(~Kp)) | (Ep | E(~Kp))                   (C)

(5) (Kp | K(~Kp)) | (Kp | K(~Kp))                  (CD)

(6) (Kp | ~Kp) | (Kp | ~Kp)                              (KIT)

(7) ~K((p | ~Kp) | (p | ~Kp))                           (1)-(6), Reductio

Given this, a strengthening of (WVer) with regard to the Sheffer stroke may be preformed:

(SVer) ((p | q) | (p | q))→◊((Kp | Kq) | (Kp | Kq))

From this, the modified reductio may be preformed without any instances of closure or competent deduction by substituting in the Fitch proposition ‘p’ and ‘~Kp’ for ‘p’ and ‘q’:

(1S) Assume: ((Kp | K(~Kp)) | (Kp | K(~Kp)))

(2S) (Kp | ~Kp) | (Kp | ~Kp)                            (KIT)

(3S) ~((Kp | K(~Kp)) | (Kp | K(~Kp)))         (1S)-(2S), Reductio

Following this, provided (SVer) with the Fitch proposition substituted for ‘p’ and ‘q’:

(FSVer) ((p | ~Kp) | (p | ~Kp))→◊((Kp | K(~Kp)) | (Kp | K(~Kp)))

we can strengthen the resultant (3S) modally:

(4S) □~((Kp | K(~Kp)) | (Kp | K(~Kp)))     (AN)

(5S) ~◊((Kp | K(~Kp)) | (Kp | K(~Kp)))     (MI)

From this, as (5S) is the negation of the right-hand-side of (FSVer), we are provided:

(6S) ~((p | ~Kp) | (p | ~Kp))                          (5S), (FSVer)

(7S) ∀p~((p | ~Kp) | (p | ~Kp))                   (6S), ∀-Intro

Given the preceding steps, (7S) is equivalent to the standard results of the paradox, ‘∀p(p→Kp)’, and our incompetent deducer has not been saved by the Sheffer-strategy. Thus, either the incompetence involved must penetrate to the most basic level of logical proficiency–the ability to grasp the fundamental concepts of the logical connectives–or the weaker deductive incompetence must be accompanied by a skepticism regarding the normative force of type 2 rationality. And neither of these options look tenable at the moment.

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).