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abramdemski comments on Reflection in Probabilistic Logic - Less Wrong
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Actually, we can use coherence to derive a much more symmetric disquotation principle:
P(x>P(A)>y)=1 => x>P(A)>y.
Suppose P(x>P(A)>y)=1. For contradiction, suppose P(A) is outside this range. Then we would also have P(w>P(A)>z)=1 for some (w,z) mutually exclusive with (x,y), contradicting coherence.
Right?
Not quite---if P(A) = x or P(A) = y, then they aren't in any interval (w, z) which is non-overlapping (x, y).
We can obtain P(x > P(A) > y) =1 ---> x >= P(A) >= y by this argument. We can also obtain P(x >= P(A) >= y) > 0 ---> x >= P(A) >= y.
So, herein lies the "glut" of the theory: we will have more > statements than are strictly true. > will behave as >= should: if we see > as a conclusion in the system, we have to think >= with respect to the "true" P.
A "gap" theory of similar kind would instead report too few inequalities...
Yes, there is an infinitesimal glut/gap; similarly, the system reports fewer >= statements than are true. This seems like another way at looking at the trick that makes it work---if you have too many 'True' statements on both sides you have contradictions, if you have too few you have gaps, but if you have too many > statements and too few >= statements they can fit together right.
Ah, right, good!