To judge whether a modal logic gives a good encoding of the properties of a particular relational structure, we need to put aside the valuations and to concentrate on the frames. A formula will be valid on a frame $\mathfrak{F}$ if it is true at every state in every model that can be built from $\mathfrak{F}$. More formally:

Definition

A formula $\phi$ is said to be valid at a state$w$ is a frame $\mathfrak{F}= (W,R_1,\ldots, R_n)$ (notation $\mathfrak{F},w \models \phi$) if $\phi$ is true at $w$ in every model $\mathfrak{M} = (\mathfrak{F},V)$, based ion $\mathfrak{F}$.

A formula, $\phi$, is valid in a frame$\mathfrak{F}$ (notation $\mathfrak{F}\models \phi$) if it is valid at every state of $\mathfrak{F}$.

If $\mathbb{F}$ is a class of frames (all of the same relational signature), then we say $\phi$ is valid in $\mathbb{F}$ (notation $\mathbb{F}\models \phi$ if $\mathfrak{F}\models \phi$ for all $\mathfrak{F}\in \mathbb{F}$, and is valid, $\models\phi$, if it is valid on the class of all frames.

Fixing a given set $P$ of atomic formulae and the relational signature that we are considering:

Proposition

The set of formulae that are valid on a class,$\mathbf{F}$, of frames of that signature forms a logic.

(We restrict to Kripke frames (i.e. to binary relations) for simplicity of exposition, and refer to the sources, such as Blackburn et al, for a fuller discussion.)

We denote the logic thus specified by $\Lambda_\mathbf{F}$.

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 ….!)

Revised on December 24, 2010 07:44:47
by Toby Bartels
(75.88.75.53)