In reply to “why a real number question”, we might want to weaken the theory to the point were only equalities and inequalities can be stated. There are two weaker desiderata one might hold. Let (A|X) be the plausibility of A given X.
Transitivity:
if (A|X) > (B|X) and (B|X) > (C|X), then (A|X) > (C|X)
Universal Comparability:
one of the following must hold
(A|X) > (B|X)
(A|X) = (B|X)
(A|X) < (B|X)
If you keep both, you might as well use a real number—doing so will capture all of the desired behavior. If you throw out Transitivity, I have a series of wagers I’d like to make with you. If you throw out Universal Comparability, then you get lattice theories in which propositions are vertexes and permitted comparisons are edges.
On the other hand, you might find just a single real number too restrictive, so you use more than one. Then you get something like Dempster-Shafer theory.
In short, there are alternatives.
As for why should the plausibility of the negation of a statement depend only on the plausibility of the statement, the answer (I believe) is that we are considering only the sorts of propositions in which the Law of the Excluded Middle holds. So if we are using only a single real number to capture plausibility, we need f{(A|X),(!A|X)} = (truth|X) = constant, and we have no freedom to let (!A|X) depend on the details of A.
Gray Area,
In reply to “why a real number question”, we might want to weaken the theory to the point were only equalities and inequalities can be stated. There are two weaker desiderata one might hold. Let (A|X) be the plausibility of A given X.
Transitivity: if (A|X) > (B|X) and (B|X) > (C|X), then (A|X) > (C|X)
Universal Comparability: one of the following must hold (A|X) > (B|X) (A|X) = (B|X) (A|X) < (B|X)
If you keep both, you might as well use a real number—doing so will capture all of the desired behavior. If you throw out Transitivity, I have a series of wagers I’d like to make with you. If you throw out Universal Comparability, then you get lattice theories in which propositions are vertexes and permitted comparisons are edges.
On the other hand, you might find just a single real number too restrictive, so you use more than one. Then you get something like Dempster-Shafer theory.
In short, there are alternatives.
As for why should the plausibility of the negation of a statement depend only on the plausibility of the statement, the answer (I believe) is that we are considering only the sorts of propositions in which the Law of the Excluded Middle holds. So if we are using only a single real number to capture plausibility, we need f{(A|X),(!A|X)} = (truth|X) = constant, and we have no freedom to let (!A|X) depend on the details of A.