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abramdemski comments on Reflection in Probabilistic Logic - Less Wrong
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Comments (171)
The informal idea seems very interesting, but I'm hesitant about the formalisation. The one-directional => in the reflection principle worries me. This "sort of thing" already seems possible with existing theories of truth. We can split the Tarski biconditional A<=>T("A") into the quotation principle A=>T("A") and the dequotation principle T("A")=>A. Many theories of truth keep one or the other. A quotational theory may be called a "glut theory": the sentences reguarded as true are a strict superset of those which are true, which means we have T("A") and T("~A") for some A. Disquotational theories, on the other hand, will be "gap theories": the sentences asserted true are a strict subset of those which are true, so that we have ~T("A") and ~T("~A") for some A. This classification oversimplifies the situation (it makes some unfounded assumptions), but...
The point is, the single-direction arrow makes this reflection principle look like a quotational ("glut") theory.
Usually, for a glut theory, we want a restricted disquotational principle to hold. The quotational principle already holds, so we might judge the success of the theory by how wide a class of sentences can be included in a disquotational principle. (Similarly, we might judge gap theories by how wide a class of sentences can be included in a quotational principle.) The paper doesn't establish anything like this, though, and that's a big problem.
The reflection principle a<P("A")<b => P('a<P("A")<b')=1 gives us very little, unless I'm missing something... it seems to me you need a corresponding disquotational principle to ensure that P('a<P("A")<b')=1 actually means something.
Is there anything blocking you from setting P("a<P('A')<b")=1 for all cases, and therefore satisfying the stated reflection principle trivially?
Yes, the theory is logically coherent, so it can't have P("a < P('A')") = 1 and P("a > P('A')") = 1.
For example, the following disquotational principle follows from the reflection principle (by taking contrapositives):
P( x <= P(A) <= y) > 0 ----> x <= P(A) <= y
The unsatisfying thing is that one direction has "<=" while the other has "<". But this corresponds to a situation where you can make arbitrarily precise statements about P, you just can't make exact statements. So you can say "P is within 0.00001 of 70%" but you can't say "P is exactly 70%."
I would prefer be able to make exact statements, but I'm pretty happy to accept this limitation, which seems modest in the scheme of things---after all, when I'm writing code I never count on exact comparisons of floats anyway!
This was intended as a preliminary technical report, and we'll include much more discussion of these philosophical issues in future write-ups (also much more mathematics).
Ok, I see!
To put it a different way (which may help people who had the same confusion I did):
We can't set all statements about probabilities to 1, because P is encoded as a function symbol within the language, so we can't make inconsistent statements about what value it takes on. ("Making a statement" simply means setting a value of P to 1.)
I'm very pleased with the simplicity of the paper; short is good in this case.
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!
The negated statements become 'or', so we get x <= P(A) or P(A) <= y, right?
To me, the strangest thing about this is the >0 condition... if the probability of this type of statement is above 0, it is true!
I agree that the derivation of (4) from (3) in the paper is unclear. The negation of a<b<c is not a>=b>=c.
Ah, so there are already revisions... (I didn't have a (4) in the version I read).