Frames in Modal Logics
# Frames in Modal Logics

## Warning

The term **frame** is used in a different sense in modal logic than in geometric logic where it stands for an abstraction of the algebraic structure corresponding to the lattice of open sets of a topological space, see frame.

## Frames in Monomodal Logics

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

## Models in Modal Logics

To give the standard (geometric) semantics of modal logics one needs the models and these will be discussed in the entry geometric models for modal logics and the companion algebraic models for modal logics.

## References

Generally this entry is based on

- P. Blackburn, M. de Rijke and Y. Venema,
*Modal Logic*, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

(any mistakes or errors of interpretation are due to ….!)