Q&A: A Question in Philosophy and Probability
A Question in Philosophy and Probability
Question
Hello Rabbi. There is a question I saw a very long time ago, and to this day I keep coming back to it—I haven’t found a solution that puts my mind at ease.
Let us define belief in some proposition (let’s call it P for convenience) as assigning a higher probability to P than to not-P (and similarly, belief in not-P would be assigning a higher probability to not-P than to P).
Now, we have proposition P1, to whose being true I assign a probability of 60%; that is, I believe it. And we also have proposition P2, to whose being true I assign a probability of 70%; that is, I also believe it.
It follows from this that I believe P1 and I believe P2. From here it follows deductively that I believe P1 and P2. However, belief in the state of affairs of P1 and P2 gives me a probability of 0.42. That is, I do not believe that P1 and P2. A contradiction, apparently—where is my mistake?
Thank you very much.
Answer
You can easily see the mistake if you calculate the probabilities of all the possibilities, and then you will immediately see that the probabilities of the four possibilities do not add up to 1. From here it is clear that your event space is not well defined. You need to construct a product space. If you normalize everything by the sum you get (1.69), everything will work out. This is of course if the propositions are independent.
By the way, the probability that both propositions are true really is smaller than the probability that each one is true individually. There is nothing at all absurd about that. You should remember that if you deny the proposition that both propositions are true, it is still possible that each one of them is true separately.
Discussion on Answer
Sorry. I made a mistake here (or I didn’t understand).
But I don’t see the problem. It is true that the certainty you have regarding the truth of both propositions is less than 0.5. So what? The probability that you are not mistaken about any proposition you think is true is very small. The claim that both propositions are true involves a greater risk.
I didn’t understand the Rabbi’s answer. Which four possibilities is he referring to? If he means P1 and P2 (0.42), P1 and not P2 (0.18), not P1 and P2 (0.28), and not P1 and not P2 (0.12), then adding up the products of the possibilities gives 1.