# Logic for Computer Scientists/Modal Logic/Kripke Semantics

## Definition 1

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

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

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

${\displaystyle {\begin{matrix}w\models p&{\mbox{iff }}&w\in V(p)\\w\models \lnot A&{\mbox{iff }}&w\not \models A\\w\models A\land B&{\mbox{iff }}&w\models A{\mbox{ and }}w\models B\\w\models A\lor B&{\mbox{iff }}&w\models A{\mbox{ or }}w\models B\\w\models A\to B&{\mbox{iff }}&w\not \models A{\mbox{ or }}w\models B\\w\models \Box A&{\mbox{iff }}&{\mbox{for all }}v\in W,(w,v)\not \in R{\mbox{ or }}v\models A\\w\models \diamond A&{\mbox{iff }}&{\mbox{there exists some }}v\in W,{\mbox{ with }}(w,v)\in R{\mbox{ and }}v\models A\\\end{matrix}}}$

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

## Lemma 1

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