Logic for Computer Scientists/Modal Logic/Kripke Semantics

Definition 1
A Kripke frame is a pair $$\langle W, R \rangle$$, where $$W$$ is a non-empty set (of possible worlds) and $$R$$ a binary relation on $$W$$. We write $$wRw'$$ iff $$(w,w') \in R$$ and we say that world $$w'$$ is accessible from world $$w$$, or that $$w'$$ is reachable from $$w$$, or that $$w'$$ is a successor of $$w$$.

A Kripke model is a triple $$\langle W, R, V \rangle$$, with $$W$$ and $$R$$ as above and $$V$$ is a mapping $$\mathcal{P} \mapsto 2^W$$, where $$\mathcal{P}$$ is the set of propositional variables. $$V(p)$$ is intended to be the set of worlds at which $$p$$ is true under the valuation $$V$$.

Given a model $$\langle W, R, V \rangle$$ and a world $$w\in W$$, we define the satisfaction relation $$\models $$ by:

$$\begin{matrix} w \models p & \mbox{iff }  & w \in V(p) \\

w \models \lnot A &  \mbox{iff } & w \not\models  A \\

w \models A \land B & \mbox{iff } & w \models A \mbox{ and } w \models B \\

w \models A \lor B & \mbox{iff } & w \models A \mbox{ or } w \models B \\

w \models A \to B & \mbox{iff } & w \not\models A \mbox{ or } w \models B \\

w \models \Box A & \mbox{iff }& \mbox{for all } v \in W, (w,v)\not\in R \mbox{ or } v\models A \\

w \models \diamond A   & \mbox{iff }& \mbox{there exists some } v \in W, \mbox{ with } (w,v)\in R \mbox{ and } v\models A \\ \end{matrix}$$

We say that $$w$$ satisfies $$A$$ iff $$w \models A$$ (without mentioning the valuation $$V$$). A formula $$A$$ is called satisfiable in a model $$\langle W, R, V \rangle$$, iff there exists some $$w\in W$$, such that $$w \models A$$. A formula $$A$$ is called satisfiable in a frame $$\langle W, R \rangle$$, iff there exists some valuation $$V$$ and some world $$w\in W$$, such that $$w \models A$$. A formula $$A$$ is called valid in a model $$\langle W, R, V \rangle$$, written as $$\langle W, R, V \rangle\models A$$ iff it is true at every world in $$W$$. A formula $$A$$ is valid in a frame $$\langle W, R \rangle$$, written as   $$\langle W, R \rangle \models A $$  iff it is valid in all models  $$\langle W, R, V \rangle$$.

Lemma 1
The operators $$\diamond$$ and $$\square$$ are dual, i.e. for all formulae $$A$$ and all frames $$\langle W, R \rangle$$, the equivalence $$\diamond A \leftrightarrow \lnot \square \lnot A$$ holds.