Sunday, January 2, 2011

Possibility does not entail any degree of probability

In the previous post I noted that I think Wittgenstein was trying to make the point that the mere raising of a possibility counter to common sense does not offer us a justified reason to put that common sense into real doubt of any degree. Real doubt requires more than mere raising of alternative (logical or otherwise) possibilities or the feelings of uncertainty which may result from it. One example of possibility not entailing any probability is this:

Consider the unit line (length of 1). It contains a continuum of numbers from 0 to 1, inclusive. Choose any arbitrary number on that line, say, .333.... There certainly is a possibility that a random selection of any number from that continuum will be .333...; after all, a number between 0 and 1. But what is the probability that it will be that number? The answer is literally, 0.

So in this case, even though there is a possibility of it landing on that number, there is no probability of it ever doing so. Our confidence in it landing on it, if it is to be rational, to to assign it 0 confidence.

Wittgenstein's examples, I believe, used real life possibilities instead of this mathematical example.