Wednesday, October 25, 2006


Needing closure

I've been thinking some more about the role of the epistemic closure principle in Hawthorne's presentation of issues generated by the lottery paradox. Hawthorne argues, and with great force, that respecting (at least single-premise) closure is an adequacy constraint on any proposed solution to the lottery proposition. But it's hard not to feel that some of the proposals he discusses respect the letter of closure whilst doing considerable violence to its spirit.

Hawthorne writes:

'As Williamson remarks, such principles articulate what is an extremely intuitive idea, namely that 'deduction is a way of extending one's knowledge' (2000:117). (Consider, for example, the paradigmatic status of mathematical knowledge that proceeds by way of deductive proof).'
(Knowledge and Lotteries: 33. Author's fn suppressed. The reference to Williamson is, of course, to Knowledge and its Limits)

Once some solutions to the lottery paradox are tabled, however, this 'extremely intuitive idea' simply goes AWOL. To fix ideas a little, let me explicitly state the formulation of (single-premise) closure that Hawthorne thinks we need to hang on to:

SPC: Necessarily, if S knows p, competently deduces q, and thereby comes to believe q, while retaining knowledge of p throughout, then S knows q. (34)

And here's an informal statement of the lottery paradox:

I know that I won't have enough money to go on a very expensive dream cruise this year. My lacking sufficient funds for the cruise entails that I won't win the major lottery that I have just bought a ticket for. But until the draw is made and the outcome announced, I don't know that my ticket is a loser.

Paradox ensures once we realise that SPC seems to straightforwardly deliver the result that I do know that my ticket will lose.

Contextualism about 'knows' gains a great deal of appeal from its elegant attempt to do justice both to the Moorean intuition that I do in fact know a great many things about my near(ish) future (such as that I won't be able to afford the cruise this year), and the intuition that I can't know that my ticket will lose in advance of the draw, whilst preserving SPC. The basic thought is that what appears to be a failure of closure is really a context shift (in which, perhaps, certain possibilities become salient), so that the semantic value of 'knows' when it appears in the first premise is different from the semantic value of 'knows' when it appears in the third premise. (82)

So far from competent deduction extending one's knowledge, in this case it actually shifts you into a context where you know less. (Well, this is kinda careless. But I don't really want to get bogged down stating the point as it strictly speaking should be stated).

If anything, things are even worse when we consider the sensitive invariantism Hawthorne tentatively endorses's attempt to preserve SPC. (Actually, on second look, I can't find any discussion in that section specifically on how to preserve SPC. But what follows draws on 176-80 and 160-1). As will be perfectly familiar to readers of this blog, these kinds of invariantist positions make whether or not a subject's true belief counts as knowledge depend in part on their practical enviroment (with different views suggesting different accounts of that dependency). The idea for Hawthorne's version of the view seems to be that I start off reasoning from a known premise, perform a competent deduction, and thereby come to believe that my ticket won't win the lottery. But this latter belief doesn't count as knowledge, since it would be unacceptable for me to employ it as a premise in my practical reasoning (176). Although this again looks like a failure of closure, in fact my knowledge of the premise is detroyed; since SPC requires competent deduction and belief-formation throughout which I retain my knowledge of the premise, there's no violation of SPC.

And again, the intuitive idea that competent deduction extends one's knowledge is clearly a casualty. Now, Hawthorne does give a bunch of motivations for preserving SPC that are entirely independent of the intuitive idea, so I still accept his point that solutions to the lottery paradox that respect that principle deserve more points that those that don't. But the violence done to the intuitive thought is nonetheless a cost of these views which Hawthorne glosses over entirely, as far as I can tell.

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