Now I have seen some interesting papers that make expanded probability theories that include 0 and 1 as logical falsehood and truth respectively. But that still does not include a special value for contradictions.
Except, contradictions really are the only way you can get to logical truth or falsehood; anything other than that necessarily relies on inductive reasoning at some point. So any probability theory employing those must use contradictions as a means for arriving at these values in the first place.
I do think that there’s not much room for contradictions in probability theories trying to actually work in the real world, in the sense that any argument of the form A->(B & ~B) also has to rely on induction at some point; but it’s still helpful to have an anchor where you can say that, if a certain relationship does exist, then a certain proposition is definitely true.
(This is not like saying that a proposition can have a probability of 0 or 1, because it must rely, at least somewhere down the line, on another proposition with a probability different from 0 and 1).
Except, contradictions really are the only way you can get to logical truth or falsehood; anything other than that necessarily relies on inductive reasoning at some point. So any probability theory employing those must use contradictions as a means for arriving at these values in the first place.
I do think that there’s not much room for contradictions in probability theories trying to actually work in the real world, in the sense that any argument of the form A->(B & ~B) also has to rely on induction at some point; but it’s still helpful to have an anchor where you can say that, if a certain relationship does exist, then a certain proposition is definitely true.
(This is not like saying that a proposition can have a probability of 0 or 1, because it must rely, at least somewhere down the line, on another proposition with a probability different from 0 and 1).