You have a set of equations for p(X1), p(X2), etc., where
p(X1) = f1(p(X2), p(X3), … p(Xn))
p(X2) = f2(p(X1), p(X3), … p(Xn))
...
Warrigal is saying: This is a system of n equations in n unknowns. Solve it.
But this has nothing to do with whether you’re using fuzzy logic!
If you define the functions f1, f2, … so that each corresponds to something like
f1(p(X2), p(X3) , …) = p(X2 and (X3 or X4) … )
using standard probability theory, then you’re not using fuzzy logic.
If you define them some other way, you’re using fuzzy logic.
The approach described lets us find a consistent assignment of probabilities (or truth-values, if you prefer) either way.
In fuzzy logic, one requires that the real-numbered truth value of a sentence is a function of its constituents. This allows the “solve it” reply.
If we swap that for probability theory, we don’t have that anymore… instead, we’ve got the constraints imposed by probability theory. The real-numbered value of “A & B” is no longer a definite function F(val(A), val(B)).
Maybe this is only a trivial complication… but, I am not sure yet.
I think I’ve figured it out.
You have a set of equations for p(X1), p(X2), etc., where
p(X1) = f1(p(X2), p(X3), … p(Xn))
p(X2) = f2(p(X1), p(X3), … p(Xn))
...
Warrigal is saying: This is a system of n equations in n unknowns. Solve it.
But this has nothing to do with whether you’re using fuzzy logic!
If you define the functions f1, f2, … so that each corresponds to something like
f1(p(X2), p(X3) , …) = p(X2 and (X3 or X4) … )
using standard probability theory, then you’re not using fuzzy logic. If you define them some other way, you’re using fuzzy logic. The approach described lets us find a consistent assignment of probabilities (or truth-values, if you prefer) either way.
Is this really the case?
In fuzzy logic, one requires that the real-numbered truth value of a sentence is a function of its constituents. This allows the “solve it” reply.
If we swap that for probability theory, we don’t have that anymore… instead, we’ve got the constraints imposed by probability theory. The real-numbered value of “A & B” is no longer a definite function F(val(A), val(B)).
Maybe this is only a trivial complication… but, I am not sure yet.