In 1983 Karl Popper and David Miller published an argument to the effect that probability theory could be used to disprove induction. Popper had long been an opponent of induction. Since probability theory in general, and Bayes in particular is often seen as rescuing induction from the standard objections, the argument is significant.
Well, it wasn't actually an equation. That's why I used the =||= symbol. It was a bientailment. It asserts logical equivalence (in classical logic), and it means something slightly different than an equals symbol. The equation with the plus signs and the logical equivalence shouldn't be confused.
I'm back and there's been no response, so I'll be specific. Starting from
p(A v B|B) – p(A v B) + p(A v ~B|B) – p(A v ~B) = .15
Using p(X v Y) = p(X) + p(Y) - p(XY), we get .15 = p(A|B) + p(B|B) - p(AB|B) - p(A) - p(B) + p(AB) + p(A|B) + p(~B|B) - p(A~B|B) - p(A) - p(~B) + p(A~B) = p(A|B) + 1 - p(A) - p(A) - p(B) + p(AB) + p(A|B) - p(A) - p(~B) + p(A~B) = 2 p(A|B) - 2 p(A)
= twice the thing you started from, which is bad.
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0Manfred
Ah, I'm sorry. Do you agree that the equation I quoted above is incorrect, though? I'm going to have to leave now, but the relevant basic equations of probability are P(X v Y) = P(X) + P(Y) - P(X&Y), and P(X&Y) = P(X) * P(X | Y)
In 1983 Karl Popper and David Miller published an argument to the effect that probability theory could be used to disprove induction. Popper had long been an opponent of induction. Since probability theory in general, and Bayes in particular is often seen as rescuing induction from the standard objections, the argument is significant.
It is being discussed over at the Critical Rationalism site.