[EMAIL PROTECTED] wrote:
> Consider the following line of reasoning. Let p be the proposition
> "Ronald was born in New York." From p, we can infer q: Ronald was born
> in the United States. From q, we can infer r: It is possible that
> Ronald was born in New Jersey. On the other hand, from p we can infer
> s: It is not possible that Ronald was born in New Jersey. We have
> arrived at a contradiction. What is wrong?
There is no paradox if you use probability theory as extended logic to
analyze the situation. Let X represent our background information
(understanding of spatial relationships and U.S. geography), and let's
analyze the above.
1. "From p, we can infer q..."
That is, our background information X tells us that (p => q) is true.
In symbols,
P(p => q | X) = 1,
and hence P(q | p, X) = 1.
2. "From q, we can infer r: It is possible that Ronald was born in New
Jersey."
That is, P(r1 | q, X) > 0, where r1 is "Ronald was born in New Jersey".
3. "On the other hand, from p we can infer s: It is not possible that
Ronald was born in New Jersey."
That is, P(r1 | p, X) = 0.
4. "We have arrived at a contradiction."
You have, but we Bayesians have not. There is no rule in probability
theory that allows one to conclude P(r1 | p, X) > 0 from P(r1 | q, X) >
0 and P(q | p, X) = 1. You appear to be using some form of modus ponens
in constructing your paradox, but modus ponens doesn't generalize to
probabilities; the closest we can get is
P(r1 | p, X)
= { since (p => q) is known to be true }
P(r1 and q | p, X)
= { product rule }
P(q | p, X) * P(r1 | q, p, X)
= { since (p => q) is known to be true }
P(r1 | q, p, X)
To get the paradox we would have to conclude P(r1 | q, p, X) > 0 from
P(r1 | q, X) > 0, but probability theory, quite sensibly, does not allow
this: it is easy to find cases where new information (p, in this case)
should turn a possibility into an impossibility.
In a sense, what is happening in the proposed paradox is that important
context is being dropped; this does not happen with probability theory
because the product rule ensures that contextual information is retained.
-- Kevin S. Van Horn
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