We will start with the simplest case, namely a frame for the basic modal language, $\mathcal{L}_\omega(1)$, (so with oneunary modal operator, denoted $\Diamond$).

Definition

A frame for $\mathcal{L}_\omega(1)$ is a pair $\mathfrak{F} = (W,R)$ with $W$ a non-empty set and $R$ a binary relation on $W$.

(This is also called a basic Kripke frame, after the philosopher and logician Saul Kripke.)

The terminology often used refers to $W$ as the set of possible worlds. Its elements are sometimes called worlds, sometimes states, sometimes points, depending on the context and the whim of the writer. The relation $R$ is called the accessibility relation so $R w v$ says ‘$v$ is accessible from $w$’.

Frames in Multimodal Logics

(N.B. Here we are still restricting to multimodal logics in which the modal operators are unary. The generalisation to allowing more general $n$-ary modalities will be considered later.)

The generalisation is not difficult. In $\mathcal{L}_\omega(1)$, the single unary modality $\Diamond$ is modeled by one binary relation. In $\mathcal{L}_\omega(n)$, there are $n$-unary modalities so the frames have $n$-binary relations. Explicitly we have

Definition

A frame for $\mathcal{L}_\omega(n)$ is a $(n+1)$-tuple $\mathfrak{F} = (W,\{R_i\}_{\{i=1,\ldots, n\}})$ with $W$ a non-empty set and for each $i=1,\ldots, n$, $R_i$ a binary relation on $W$.

(More to go here … frames with unary and more general relations.)