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<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?
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).
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.
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.
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<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.
Ah, right, good!
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.
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.
Ah, so there are already revisions… (I didn’t have a (4) in the version I read).