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Natural necessity is analyzed in terms of “sub-nomic stability” (introduced in Chapter 1). The various species of necessity correspond to the various nonmaximal sets possessing sub-nomic stability. This approach explains what natural necessity has in common with other varieties of necessity by virtue of which they all qualify as varieties of the same thing. Necessities relative to some arbitrary class of facts (merely relative necessities) are thereby distinguished from genuine varieties of necessity (contrary to David Lewis's account of “must”). Different strata of natural law possess different grades of natural necessity. This approach explains what makes one variety of necessity “stronger” than another. Indeed, this approach explains why all varieties of necessity have characteristic places in a single, well-defined ordering from strongest to weakest. It is thus shown how natural laws can be genuinely necessary despite being contingent.

The laws' necessity makes them laws. Their necessity consists of membership in a nonmaximal sub-nomically stable set. Therefore, the laws are laws in virtue of belonging to a nonmaximal sub-nomically stable set. What, then, is responsible for making true the various subjunctive conditionals rendering that set stable? With these subjunctive facts, ontological bedrock is reached. They (along with various sub-nomic facts) are primitive. They are the lawmakers. This proposal reverses the standard picture of laws supporting counterfactuals. It is argued that any view failing to locate subjunctive facts among the lawmakers (such as essentialism) will find it difficult to avoid adhocery in accounting for the laws' relation to counterfactuals. It is also argued that instantaneous rates of change (such as velocity and acceleration in classical physics) should be analyzed in terms of ontologically primitive subjunctive facts. Finally, this account entails that the laws of fundamental physics must be “complete.”

An analysis is provided of different kinds of necessity: logical necessity (or analyticity), a posterior necessity, the necessity of the past, causal ultimacy, everlasting existence, and physical necessity.

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.

The laws of nature are at most physically necessary, and they are not metaphysically necessary. Dispositional Essentialists claim that if natural laws derive from powers, then the laws of nature are metaphysically necessary. But the idea that properties have dispositional essences does not entail necessitarianism for several reasons. There might be no laws of nature. The laws might have exceptions, or be probabilistic. There are non-dispositional properties that could figure in contingent laws. The world might have contained different properties. Finally, even if a property has a dispositional essence, it might have had a slightly different causal profile. Furthermore, the Necessitarian’s views are less revisionary than they initially seem.