The following is an excerpt from a highly-respected paper (Angelica Kratzer, Modals and Conditionals: New and Revised Perspectives, chapter 1 "What Must and Can Must and Can Mean") (my emphasis):
In the possible worlds semantics assumed here, propositions are identified with sets of possible worlds. If $W$ is the set of possible worlds, the set of propositions is $P(W)$—the power set of $W$.
Is it valid to speak of the set of something as ill-defined as "possible worlds" (let alone its power set)?
I mean, we can go on with "the set of all titillating nightmares", and "the set of all deceptively simple buffalo-wing recipes", and on, and on, and on, and thereby cloak the most outlandish ideas with an appearance of formal rigor…
Is there anything in standard set theory to prevent this sort of nonsense?
Best Answer
In modal logic we use the term "possible worlds" to describe some set of "vertices" with an accessibility relation defining "edges". Possible worlds are just a term for some set $W$ which we wish to identify as our frame in the context of Kripke semantics. When we define a valuation on that frame we obtain a model which has certain modal formulas being satisfied depending on the structure of the vertices and edges (in the graph theory sense).
Formally, $\mathcal{F} = \langle W,R \rangle$ is a frame, where $R \subseteq W \times W$, and $\mathcal{M} = \langle W,R, \text{Val}\rangle$ is a model where $\text{Val}: \text{Var} \times W \rightarrow \{0,1\}$ is a valuation function which sends propositions in the set $\text{Var}$ at a world $w \in W$ to a truth value (we can also define probabilities that a modal formula is satisfied at a world by considering a valuation function with values which map to the interval $[0,1]$). Depending on the structure of the accessbility relation $R$, we can have different modal axioms satisfied in the model.
For example, consider the modal axiom $B = p \rightarrow \Box \Diamond p$. If we have that $\mathcal{M}_{w} \vDash B$, $\forall w \in W$, which is read as "the model $\mathcal{M}$ makes true $B$ at all possible worlds", we say that $\mathcal{F} \vDash B$, which is that $B$ is satisfied in the frame $\mathcal{F}$. In this case, the satisfaction of $B = p \rightarrow \Box \Diamond p$ in all possible worlds ensures that the accessibility relation $R$ is symmetric, that is, $w R w' \Rightarrow w' R w, \forall w,w' \in W$. We can characterize modal frames by the satisfaction of modal axioms in this way and give an interpretation of the philosophical phrases such as "it is possible that $\varphi$" and "it is necessary that $\psi$."
In summary, $W$ is just the vertex set of a graph and modal logic studies the satisfactions of modal formulas and other properties of frames and models. I should probably mention that the following are the formal definitions of possibility and necessity.
$\mathcal{M}_{w} \vDash \Diamond \varphi \Leftrightarrow \exists (w,w') \in R \; | \; \mathcal{M}_{w'} \vDash \varphi$
$\mathcal{M}_{w} \vDash \Box \varphi \Leftrightarrow \forall (w,w') \in R \; | \; \mathcal{M}_{w'} \vDash \varphi$
We can define other modalities in a similar manner, thus generalizing to temporal logic, epistemic logic, and other interesting types of logic. Here is a pretty picture I made which gives an example of a model with an orientation (a directed graph upwards as this is representing temporal logic).