Monday, February 24, 2014

"If there are so many, then probably there are more"

Suppose the police have found one person involved in the JFK assassination. Then simplicity grounds may give us significant reason to think that that one person is the sole killer. But suppose that they have found 15 people involved. Then while the hypothesis H15 that there were exactly 15 conspirators is simpler than the hypothesis Hn that there were exactly n for n>15, nonetheless barring special evidence that they got them all, we should suspect that there are more conspirators at large. With that large number, it's just not that likely that all were caught.

Why is this? I think it's because even though prior probabilities decrease with complexity, the increment of complexity from H15 to, say, H16 or H17 is much smaller than the increment of complexity from H1 to H2. Maybe P(H2)≈0.2P(H1). But surely we do not have P(H16)≈0.2P(H15). Rather, we have a modest decrease, maybe P(H16)≈0.9P(H15) and P(H17)≈0.9P(H16). If so, then P(H16)+P(H17)≈1.7P(H15). Unless we receive specific evidence that favors H15 over H16 and H17, something like this will be true of the posterior probabilities, and so the disjunction of H16 and H17 will be significantly more likely that H15.

Thus we have a heuristic. If our information is that there are at least n items of some kind, but we have no evidence that there are no more, then when n is small, say 1 or 2 or maybe 3, it may be reasonable to think there are no more items of that kind. But if n is bigger—my intuition is that the switch-around is around 6—then under these conditions it is reasonable to think there are more. If there are so many, then probably there are more. And this just follows from the fact that the increase in complexity from 1 to 2 is great, and from 2 to 3 is significant, but from 6 to 7 or maybe even 4 to 5 it's not very large.

This is all just intuitive, since I do not have any precise way to assign prior probabilities. But staying at this intuitive level, we get some nice intuitive applications:

• If after thorough investigation we have found only one kind of good that could justify God's permitting evil, then we have significant evidence that it's the only such good. And if some evil is no justified by that kind of good, then that gives significant evidence that it's not justified. But suppose we've found six, say. And it's easy to find at least six: (1) exercise of virtues that deal with evils; (2) significant freedom; (3) preservation of laws of nature; (4) opportunities to go beyond justice via forgiveness[note 1]; (5) adding variety to life; (6) punishment; (7) the great goods of the Incarnation and sacrifice of the cross. So we have good reason to think there are more permission-of-evil justifying goods that we have not yet found. (Alston makes this point.)
• Suppose our best definition of knowledge has three clauses. Then we might reasonably suspect that we've got the definition. But it is likely, given Gettier stuff, that one needs at least four clauses. But for any proposed definition with four clauses, we should be much more cautious to think we've got them all.
• Suppose we think we have four fundamental kinds of truths, as Chalmers does (physics, qualia, indexicals and that's all). Then we shouldn't be confident that we've got them all. But once we realize that the list leaves out severel kinds (e.g., morality, mathematics, intentions and intentionality, pace Chalmers), our confidence that we have them all should be low.
• If our best physics says that there are two fundamental laws, we have some reason to think we've got it all. But if it says that there six, we should be dubious.