Tuesday, January 14, 2014

Low-probability explanations

Here is a plausible principle:

  1. If p explains q1 and p explains q2, then p explains the conjunction of q1 and q2.
Now suppose that p says that persons x1,x2,...,x100 were the buyers of tickets to a fair lottery, with each buying one ticket. Let qi be the proposition that person xi did not win. Suppose that in fact x100 won. Then p explains q1 with a perfectly fine 99/100 stochastic explanation. And by the same token p explains q2, and so on up to q99. So by (1), p explains the conjunction of q1,...,q99. But the probability of that conjunction being true given p is only 1/100. So we have a stochastic explanation despite a low probability.

One can even rig cases where one has a stochastic explanation despite zero probability if (1) extends to infinite conjunctions.


William said...

Perhaps this is more true:

1a. If p explains q1 and p explains q2, and q1 is causally independent from q2, then p explains the conjunction of q1 and q2.

Example: If my painting a white chair explains why it is now entirely red and my painting the chair explains why is is now entirely blue, it does not follow the chair can be both entirely red and blue at the same time because of my painting it.

Alexander R Pruss said...

That p explains q entails that both p and q true. But the chair isn't both entirely red and entirely blue.