Nomological Possibility and Necessity

An event E is nomologically possible in history h at time t if the initial segment of that history up to t admits at least one continuation in Ω that lies in E; and E is nomologically necessary in h at t if every continuation of the history’s initial segment up to t lies in E.

More formally, we say that one history, h’, is accessible from another, h, at time t if the initial segments of h and h’ up to time t coincide, i.e., h_t = h_t‘. We then write h’R_th. The binary relation R_t on possible histories is in fact an equivalence relation (reflexive, symmetric, and transitive). Now, an event E ⊆ Ω is nomologically possible in history h at time t if some history h’ in Ω that is accessible from h at t is contained in E. Similarly, an event E ⊆ Ω is nomologically necessary in history h at time t if every history h’ in Ω that is accessible from h at t is contained in E.

In this way, we can define two modal operators, ♦_t and ¤_t, to express possibility and necessity at time t. We define each of them as a mapping from events to events. For any event E ⊆ Ω,

♦_t E = {h ∈ Ω : for some h’ ∈ Ω with h’R_th, we have h’ ∈ E},

¤_t E = {h ∈ Ω : for all h’ ∈ Ω with h’R_th, we have h’ ∈ E}.

So, ♦_t E is the set of all histories in which E is possible at time t, and ¤_t E is the set of all histories in which E is necessary at time t. Accordingly, we say that “ ♦_t E” holds in history h if h is an element of ♦_t E, and “ ¤_t E” holds in h if h is an element of ¤_t E. As one would expect, the two modal operators are duals of each other: for any event E ⊆ Ω, we have ¤_t E = ~ ♦_t ~E and ♦_t E = ~ ¤_t ~E.

Although we have here defined nomological possibility and necessity, we can analogously define logical possibility and necessity. To do this, we must simply replace every occurrence of the set Ω of nomologically possible histories in our definitions with the set H of logically possible histories. Second, by defining the operators ♦_t and ¤_t as functions from events to events, we have adopted a semantic definition of these modal notions. However, we could also define them syntactically, by introducing an explicit modal logic. For each point in time t, the logic corresponding to the operators ♦_t and ¤_t would then be an instance of a standard S5 modal logic.

The analysis shows how nomological possibility and necessity depend on the dynamics of the system. In particular, as time progresses, the notion of possibility becomes more demanding: fewer events remain possible at each time. And the notion of necessity becomes less demanding: more events become necessary at each time, for instance due to having been “settled” in the past. Formally, for any t and t’ in T with t < t’ and any event E ⊆ Ω,

if ♦_t’ E then ♦_t E,

if ¤_t E then ¤_t’ E.

Furthermore, in a deterministic system, for every event E and any time t, we have ♦_t E = ¤_t E. In other words, an event is possible in any history h at time t if and only if it is necessary in h at t. In an indeterministic system, by contrast, necessity and possibility come apart.

Let us say that one history, h’, is accessible from another, h, relative to a set T’ of time points, if the restrictions of h and h’ to T’ coincide, i.e., h’_T’ = h_T’. We then write h’R_T’h. Accessibility at time t is the special case where T’ is the set of points in time up to time t. We can define nomological possibility and necessity relative to T’ as follows. For any event E ⊆ Ω,

♦_T’ E = {h ∈ Ω : for some h’ ∈ Ω with h’R_T’h, we have h’ ∈ E},

¤_T’ E = {h ∈ Ω : for all h’ ∈ Ω with h’R_T’h, we have h’ ∈ E}.

Although these modal notions are much less familiar than the standard ones (possibility and necessity at time t), they are useful for some purposes. In particular, they allow us to express the fact that the states of a system during a particular period of time, T’ ⊆ T, render some events E possible or necessary.

Finally, our definitions of possibility and necessity relative to some general subset T’ of T also allow us to define completely “atemporal” notions of possibility and necessity. If we take T’ to be the empty set, then the accessibility relation RT’ becomes the universal relation, under which every history is related to every other. An event E is possible in this atemporal sense (i.e., ♦_∅E) iff E is a non-empty subset of Ω, and it is necessary in this atemporal sense (i.e., ¤_∅E) if E coincides with all of Ω. These notions might be viewed as possibility and necessity from the perspective of some observer who has no temporal or historical location within the system and looks at it from the outside.

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