An Analysis of First-Order Logics of Probability

We consider two approaches to giving semantics to first order logics of probability. The first approach puts a probability on the domain, and is appropriate for giving semantics to formulas involving statistical information such as "The probability that a (typical) bird flies is greater than .9." The second approach puts a probability on possible worlds, and is appropriate for giving semantics to formulas describing degrees of belief, such as "The probability that Tweety (a particular bird) flies is greater than .9." We show that the two approaches can be easily combined, allowing us to reason in a straightforward way about statistical information and degrees of belief. We then consider axiornatizing these logics. In general, it can be shown that no complete axiomatization is possible. We provide axiom systems that are sound and complete in cases where a complete axiomatization is possible, showing that they do allow us capture a great deal of interesting reasoning about probability. 1 Introduction Consider the two statements "I he probability that a bird chosen at random will fly is greater than .9" and "The probability that Tweety (a particular bird) flies is greater than .9." It is quite straightforward to capture the sec ond statement by using a possible-world semantics along the lines of that used in [FH88, FHM88, Nil8f>]. Namely, we can imagine a number of possible worlds such that the predicate Flies has a different extension in each one. Thus, Flies(Tweety) would hold in some possible worlds, and not in others. We then put a probability distribution on this set of possible worlds, and check if the set of possible worlds where Flics(Tweety) holds has probability greater than .9. However, as pointed out by Bacchus [Bac88b, Bac88a], this particular possible worlds approach runs into difficulties when trying to represent the first statement, which we may believe as a result of statistical information of the form "More than 90% of all birds fly." What is the formula that should hold at a set of worlds whose probability is greater than .9? The most obvious candidate is perhaps Vx(Btrd(x) => Flies(x)). However, it might very well be the case that in each of the worlds we consider possible, there is at least one bird that doesn't fly. Hence, the statement Wx(Bird(x) => Fltes(x)) holds in none of the worlds (and so has probability 0). Thus it cannot be used to represent the statistical information. As …