Possible World

Let $\mathcal{F} = (W, R)$ be a Kripke frame.

Definition. A possible world (or simply world) is an element $w \in W$. Each world determines a truth assignment: given a valuation $V \colon \mathrm{Prop} \to \mathcal{P}(W)$, the set $\{p : w \in V(p)\}$ is the collection of atomic propositions true at $w$.

In the semantics of modal logic, truth is relative to a world. The modal operators quantify over worlds linked by the accessibility relation $R$: necessity $\Box\varphi$ holds at $w$ iff $\varphi$ holds at every $v$ with $wRv$; possibility $\Diamond\varphi$ holds at $w$ iff $\varphi$ holds at some such $v$.

Proposition. Each world $w \in W$ induces a Boolean algebra homomorphism $\mathrm{ev}_w \colon \mathcal{P}(W) \to \{0, 1\}$ defined by $\mathrm{ev}_w(S) = 1$ iff $w \in S$. These homomorphisms separate points: $w \neq v$ iff there exists $S \subseteq W$ with $\mathrm{ev}_w(S) \neq \mathrm{ev}_v(S)$.

Proof sketch. $\mathrm{ev}_w$ is a characteristic function and hence preserves $\cap$, $\cup$, and complementation. For separation, take $S = \{w\}$. $\square$

Proposition (Algebraic counterpart). In the algebraic semantics of modal logic, worlds correspond to ultrafilters (equivalently, prime filters) of the modal algebra. For a Heyting algebra $H$, the prime filters of $H$ form the points of its spectrum, and truth at a world $w$ corresponds to membership in the prime filter associated with $w$.

Examples.

• In epistemic logic, a world $w$ represents a state of knowledge; the set $\{v : wRv\}$ contains all states the agent considers compatible with $w$.
• In temporal logic, each world is a moment in time, and $wRv$ means $v$ is in the future of $w$.
• In a finite Kripke frame $(W, R)$ with $|W| = n$, there are $2^n$ possible valuations on a single propositional variable, giving $2^n$ distinct Kripke models for one atom.