Suppose X is a number uniformly chosen in the interval [0,1] (from 0 to 1, both inclusive). Then the probability that X is not 2 is 1, and so is is the probability that X is not 1/2. But intuitively it is certain that X is not 2, while X might be 1/2.
One solution is to bring in infinitesimals. We then say that the probability that X is not 2 is 1, but the probability that X is not 1/2 is 1−a, for an infinitesimal a. Unfortunately, this leads to paradoxes of nonconglomerability.
Here is an alternative. Introduce certainty operator C(p) and C(p|q) operators that work in parallel with probabilities, subject to axioms like:
- If C(p), then P(p)=1.
- If p is a tautology, then C(p).
- If p entails q and C(p), then C(q).
- If C(p), then ~C(~p).
- If C(p|q) and P(q)>0, then P(p|q)=1.
- If C(p1 or p2 or ...) and C(q|pi) for all i, then C(q).
- If p entails q and ~C(~p), then C(q|p).
- If p entails q and C(p|r), then C(q|r).
- If p entails q and ~C(~q) and C(r|q), then C(r|p).
- If C(p1 or p2 or ...|r) and C(q|pi and r) for all i, then C(q|r).
Then, we can say that while the probability that X isn't 2 and the probability that X isn't 1/2 are the same—both are one—the former is certain while the latter isn't.
7 comments:
You conclude that we could then say that the former is certain, and yet you began by saying that very thing. Why not just say that the solution is that probabilities of 1 do not always mean certainty? That is, I fail to see what an axiomatization would add.
enigMan:
"You conclude that we could then say that the former is certain, and yet you began by saying that very thing. Why not just say that the solution is that probabilities of 1 do not always mean certainty? . . ." When you make it that simple it just isn't any fun. :-)
You could still axiomatize certainty for fun. Math is fun! It is just Pruss's preamble that makes me feel qualmy logically. And, I doubt that there is any other good reason to axiomatize.
It's worth remembering that certainty here is not psychological certainty but some other notion, maybe certainty by a perfect epistemic agent?
But, should we axiomatise probability? Where would we get the axioms from, if not from conceptions that are able to judge axioms? We may need to clarify our conceptions of probability, but do axioms always add clarity? Borel's law of thought was that very unlikely things do not happen, and if so then a probabiilty of 1 is always certainty.
That may be counter-intuitive, but there is intuitive support for Borel's law. Suppose I put four red balls and one blue ball into an empty box, and you blindly put your hand into the box and pick out a ball. You would be surprised to find that the ball was green, but then, I did not say that I only put five balls into the box. So that might happen. What will certainly not happen is that you will pick out a tiny ballroom in which there is a ball. And yet that is not logically impossible, only very, very unlikely, given all we have seen.
Once we have good axioms that we agree on, we will have theorems that we agree on.
Hmm... I found mathematics to be a counter-example to that!
(It was by disagreeing with proofs of theorems of functional analysis that I was led to doubt, not so much the axioms of the real number field, or those of arithmetic, but some formulations of the set-theoretic axiom of infinity, or rather, some of the assumptions underlying those formulations.)
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