Much of the usefulness of probability derives from its rich logical and mathematical structure. That structure comprises the probability calculus. The classical probability calculus is familiar and well understood, but it will turn out that the calculus of nomic probabilities differs from the classical probability calculus in some interesting and important respects. The purpose of this chapter is to develop the calculus of nomic probabilities, and at the same time to investigate the logical and mathematical structure of nomic generalizations. The mathematical theory of nomic probability is formulated in terms of possible worlds. Possible worlds can be regarded as maximally specific possible ways things could have been. This notion can be filled out in various ways, but the details are not important for present purposes. I assume that a proposition is necessarily true iff it is true at all possible worlds, and I assume that the modal logic of necessary truth and necessary exemplification is a quantified version of S5. States of affairs are things like Mary’s baking pies, 2 being the square root of 4, Martha’s being smarter than John, and the like. For present purposes, a state of affairs can be identified with the set of all possible worlds at which it obtains. Thus if P is a state of affairs and w is a possible world, P obtains at w iff w∊P. Similarly, we can regard monadic properties as sets of ordered pairs ⧼w,x⧽ of possible worlds and possible objects. For example, the property of being red is the set of all pairs ⧼w,x⧽ such that w is a possible world and x is red at w. More generally, an n-place property will be taken to be a set of (n+l)-tuples ⧼w,x1...,xn⧽. Given any n-place concept α, the corresponding property of exemplifying a is the set of (n + l)-tuples ⧼w,x1,...,xn⧽ such that x1,...,xn exemplify α at the possible world w. States of affairs and properties can be constructed out of one another using logical operators like conjunction, negation, quantification, and so on.