So I was musing about 'one man's modus tollens is another man's modus ponens', and about how you would put that in probabilistic terms.
It seems to be applicable to when you have a probability for P(A v B), update on positive evidence to P(A' v B'), but instead of AB' (or same thing, A>A' and B<B').
I'm just wondering what additional stuff you need to get that; nothing mentally pops out for me as relevant.
The way I see it is this: We consider A likely, but B unlikely. Say P(A)=1-a and P(B)=b, where a and b are small. A and B are currently independent. Then we observe that A implies B (i.e. "¬A v B"). We get probabilities
P(A|A=>B) = P(A=>B|A)P(A)/P(A=>B) = P(B)P(A)/[P(B)P(A)+P(B)P(¬A)+P(¬B)P(¬A)] = [b-ab]/[b+a-ab] or approximately P(A|A=>B) = b/[a+b]
P(B|A=>B) = P(A=>B|B)P(B)/P(A=>B) = 1.P(B)/[P(B)P(A)+P(B)P(¬A)+P(¬B)P(¬A)] = b/[b+a-ab] or approximately P(B|A=>B) = b/[a+b].
So (to first order) we have that observing "A im...
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