Monday, May 22, 2006


No Factivity of the Matter

Some homework for the logically inclined amongst you:

I've been reading Chris Kelp and Duncan Pritchard's draft for the forthcoming OUP collection on the Knowability Paradox. The Church-Fitch proof I mentioned in a recent post starts out by assuming that knowledge is factive ((FAC): Kp -> p) and that it distributes over conjunction ((CON): K(p&q) -> (Kp & Kq)). Fine assumptions one might think, but Chris and Duncan argue that the anti-realist has good reason to reject factivity, and that (CON) is only plausible given a prior committment to (FAC). The Church-Fitch proof is thus exposed as relying entirely on thoroughly classical/realist assumptions.

I've found myself in pretty intense disagreement with almost every point in the paper, which doesn't happen very often. I have some stuff written about why I don't think the anti-realist should be too thrilled by their proposal, but here I just want to concentrate on the claim that the plausibility of (CON) rests on an assumption of (FAC). In the body of the paper (p8) they give a extremely puzzling argument to this effect. Then in an endnote (p17n6) they write:

'A related reason to be suspicious about (CON) given prior doubts about (FAC) is that (CON) can be shown to be a theorem of (FAC) plus the closure principle for knowledge (which states, roughly, that if one knows one proposition, and one knows that this proposition entails a second proposition, then one knows the second proposition).'

Now for the homework. Where does one require an appeal to (FAC) in the derivation of (CON) from closure? This is meant to be a proposal on behalf of the anti-realist, so make sure all moves are intuitionisitically acceptable.

Now, I would have thought we could just do the following. Assume the antecendent of (CON):

1. K(p&q)

Presumably we also know basic logical truths, so:

2. K((p&q) -> p)
3. K((p&q) -> q)

The following are instances of the closure principle:

4. [K(p&q) & K((p&q) -> p)] -> Kp
5. [K(p&q) & K((p&q) -> q)] -> Kq

It looks like the consequent of (CON) should now follow by propositional logic (conjuction introduction and modus ponens, both of which are intuitionistically kosher). I just don't see the play with factivity at all, but then I'm pretty fallible on such matters.


Since I am partly responsible for the paper Aidan finds troubling, I would like to take the opportunity to post a few words by way of (admittedly rather sketchy) response:

If the anti-realist rejects factivity in favour of the weaker principle Duncan and I have suggested, he will not only have reason to revise conjunction distribution but also closure. It is pretty plausible that factivity underlies closure since there is good reason to believe that epistemic notions that are not factive, such as warrant, justification etc., are not closed under known logical entailment. If factivity underlies closure and the anti-realist has reason to revise closure, then, provided that (CON) is indeed a theorem of closure, there is reason for the anti-realist to revise (CON) as well. (I have convinced myself that the best way to go about this is slightly different than the way Duncan and I did in the paper. However, it seems to me that the general strategy still works.)

Sorry that the response is so sketchy. Although I have stuff written on this more global issue, it is not yet in presentable form. I am happy to provide material as soon as I have polished things up...
Thanks for taking the time to respond, Chris. (And let me note that I posted this before the email swapping started - if I'd known I get a chance to ask you about this stuff more directly, I wouldn't have posted this question up here).

On the points you raise here, a couple of quick responses. Firstly, it's hard to see from what you've said why factivity underlies closure (and I presume that initially you mean epistemic closure), since you've just outright asserted that closure is implausible for non-factive operators. What would be nice would be to find out quite generally what needs to happen model-theoretically to validate closure for a modal operator, and whether it requires the accessibility relation to be reflexive. I'm not claiming that would settle the matter, but I would like to be sensitive to these sorts of technical issues. Otherwise it feels like we've regressed to a pre-Kripke method of establishing which inferences in modal logic are plausible, and I don't see we'll make progress like that.

I also can't see why the general shape of the argument you give for the anti-realist to revise (CON) isn't entirely suspect. Let me grant temporarily that factivity underlies closure and that the anti-realist has reason to revise closure. not-not-(P v not-P) is a theorem of closure, so there is reason for the anti-realist to revise not-not-(P v not-P). There's glory for you!
Actually, I plead jetlag. K models will validate closure won't they? So I'm not clear on what it means to claim that factivity 'underlies' closure.
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