Certain Doubts

I was just at Fordham talking about assertion, knowledge, and lotteries, and Bryan Frances and I were talking about a particular case that it might be interesting to get others’ reactions to. I won’t tell you who took which position on the case, but here it is.

Suppose you have a lottery in which each ticket has a probability of winning of one in ten to the billionth power. There is no guarantee of a winner, and presumably there will be significantly fewer tickets bought than could be bought. Suppose, for example, that there are a hundred thousand tickets bought.

The drawing is held today, and tomorrow the newspaper (one as trustworthy as, say, the NYTimes) reports two things. First it reports the winning number, and second it reports that Lucky Louie L’Amour is the holder of the winning ticket.

Should you believe either, or both, of the newspaper reports? Should you believe that LLL won the lottery, and should you believe the report of which number is the winning ticket? Before answering, note the incredible improbability of anyone winning, and note the incredible improbability of any given number being the number of the winning ticket. OK, enough priming of the pump: should one believe the newspaper? And if you do believe the newspaper, do you now know, on the basis of the information you have, who won the lottery and what the winning number is?

This is just to let you all know that Stirling is hosting a conference on Social Epistemology next year, August 31st-September 2nd. The speakers include: Jon Kvanvig, Peter Lipton, Alvin Goldman, Elizabeth Fricker, Miranda Fricker, Scott Sturgeon, Ernest Sosa, Sandy Goldberg, and Alan Millar. The commentators include: Martin Kusch, Peter Graham, Igor Douven, Rene van Woudenberg, Klemens Kappell, Kathleen Lennon, Nenad Miscevic, and Erik Olsson. There will also be a number of invited ‘discussants-at-large’.

For more details, click here.

A detachment rule for an operator tells you conditions under which the operator can be removed from that which it governs. So, to use a straightforward example, the detachment rule for the necessity operator is just the inference rule that allows us to infer p from []p:
[]p |- p.

I’m interested in detachment rules for the probability operator. If the operator P is taken to mean “it is probable that”, then the analogue of the inference rule for the necessity operator is a disaster:
P(p) |- p.
The counterexamples should be obvious, but I’ll give one anyway. In a ten ticket lottery, this rule would allow you to conclude that the winner is one of the first 6 tickets sold.

There is pressure, however, to find some true detachment rule, even if the simple one is obviously false.

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