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Friday, May 8, 2009

Peirce and the problem of the single case

I just ran into this April 30, 2009 post "Peirce and the Problem of the Single Case" by Jonah Schupbach at the blog Choice & Inference. It's always fun to take a look at what people are saying about Peirce's ideas, and there are a number of comments on the thread there.

The question is that of how probabilities can be meaningful in a single case. Peirce's example is that of a man who must pick one card either (1) from a deck correctly identified to him as containing 25 white cards and 1 black card or (2) from a deck correctly identified to him as containing 25 black cards and 1 white card. The man will go straight to heaven if he picks a white card and straight to hell if he picks a black card, so there will be no repetition by him of the "experiment." Obviously he should pick from the deck with 25 white cards and 1 black card but, Peirce asks, what consolation can be given to the man if he mischances to draw the single black card from the deck with 25 white cards and 1 black card? Peirce says ("The Doctrine of Chances", 1878, CP 2.652, EP 1:142-154), "He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless." Peirce goes on to say:
[...] in the case supposed, which has no parallel as far as this man is concerned, there would be no real fact whose existence could give any truth to the statement that, if he had drawn from the other pack, he might have drawn a black card. Indeed, since the validity of an inference consists in the truth of the hypothetical proposition that if the premisses be true the conclusion will also be true, and since the only real fact which can correspond to such a proposition is that whenever the antecedent is true the consequent is so also, it follows that there can be no sense in reasoning in an isolated case, at all.
This ties into Peirce's theme that logic is rooted in the social principle and that one must identify oneself with a larger community, even though, as Peirce says elsewhere with possibly a pun alluding to himself, "For an individual whose purse is finite, there really is no such thing as the 'long run' of probabilities."

Now, maybe I'm a dumb bunny, but I must confess that I've never seen the sting of the single-case probability problem, aside from the sting of losing suffered by the man who bets wisely in the scenario. In a universe anything like ours, there will be indefinitely many situations involving 25-1 odds. I don't see why the drawing by a particular individual of a card from a particular deck should be considered a unique instantiation of 25-1 odds even though he cannot repeat the act; and it seems to me that he and anybody else would take all such cases into account, howsoever multiform such cases are, as being governed, in their collective run, by the law of probability; The repetition of exact same actual conditions for a coin flip or a card draw is impossible anyway. One considers idealized cases to which the actual approximates. And if there were not indefinitely many such actual cases, what practical assurance would one have about how the deck is stacked in one way or another in this case? As in Peirce's example, one would need a divine revelation.

So I come to a conclusion, not only like Peirce, that there can be no sense in reasoning about probability in a genuinely isolated case, but also that in a sufficiently isolated case no probability is involved. The conception of a simple, isolated universe consisting of nothing 26 equiprobable options for just one choice ever to be made, implies a degree of determination within that universe which bursts the bounds of that supposed simplicity; the conception merely veils it, or so I figure. So, one needs to take the view not only of an indefinitely large community, but of being ultimately in an indefintitely large universe, which sounds almost like a tautology now that I say it; I mean, one won't find an indefinitely large community in a universe that isn't indefinitely large.

Well, the thought of having actually to defend certain of the ideas that I've stated here makes me feel unsure of my chances. But I'll post it anyway, nobody's posted here for over two weeks!