Let Ω be a non-empty set and F be a field of subsets (i.e., set of subsets closed under finite unions and complements). A Popper function on F is a real-valued function P defined for pairs of members of F such that:
- if P(Ω−Y|Y)<1, then P(−|Y) is a finitely additive probability on F
- if P(X|Y)=P(Y|X)=1, then P(Z|X)=P(Z|Y).
On the other hand, a comparative probability function on F (Paul Bartha has talked about things like this) is a function from pairs of members of F to [0,∞] such that:
- C(X,Y)C(Y,Z)=C(X,Z) provided the left-hand side is defined (0 times ∞ and ∞ times zero are the undefined cases)
- C(−,Y) is a finitely-additive measure if Y is non-empty.
Alan Hajek, I think, has suggested that one can define a comparative probability function in terms of a Popper function. We can do it as follows. If P is a normal Popper function, then let CP(X,Y)=P(X|X∪Y)/P(Y|X∪Y). And given a comparative probability function, we can define a normal Popper function by PC(X|Y)=CP(X∩Y|Y).
I haven't written out the details, but it looks like we then have:
Proposition. If P is a normal Popper function, then CP is a comparative probability function, and if C is a comparative probability function, then PC is a normal Popper function. Moreover, if P is a normal Popper function and C=CP, then PC=P, while if C is a comparative probability function and P=PC, then CP=C.
Thus, there is a nice one-to-one correspondence between comparative probabilities and normal Popper functions.
Personally, I find comparative probabilities to be easier to prove theorems about than Popper functions, because I find (6) much easier to remember than (3).