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 if it is true at every state in every model that can be built from . More formally:
A formula is said to be valid at a state is a frame (notation ) if is true at in every model , based ion .
A formula, , is valid in a frame (notation ) if it is valid at every state of .
If is a class of frames (all of the same relational signature), then we say is valid in (notation if for all , and is valid, , if it is valid on the class of all frames.
Fixing a given set of atomic formulae and the relational signature that we are considering:
The set of formulae that are valid on a class,, 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 .
Generally this entry is based on
(any mistakes or errors of interpretation are due to ….!)