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.
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.