I have four propositions $A, B, C, D$ each of which is either true or false. These statements are completely arbitrary and have no relation to each other. Is it logically sound to make the statement, $P(A=T)=\frac{1}{2}$, that is, the probability of A being true is one half? - I realised my initial question had this as a presupposition so I thought it best to check this too.
My actual question is this:
Given the same $A, B, C, D$ and then add in these relations
$TWO(A, B, C)$ (two of A, B and C are true, one is false)
$A \rightarrow D$
$B \rightarrow D$
Can we conlude $P(D=T)=\frac{2}{3}$ or is there a technical reason why applying probabilities to logic is fallacious in some sense? (I'm new to formal logic and appreciate I may have misused a technical term somewhere)
Thanks in advance, Ben
It depends on what the statements are, but most likely not.
You can only do this if you know that $A,B,C$ are all equally likely to occur. This is obviously a major point in probability -- all outcomes must be equally likely. A more subtle note is that they are actually probabilistic. This means they can't be statements whose truthiness or falseness is determined, for instance a statement asserting that the twin prime conjecture is true. Even though this is unproven, it's still either true or untrue; there's no chances involved.