nLab validity

Idea

In logic, an argument is said to be valid relative to a specified logic if its form in that logic is such that whenever the premises are true then the conclusion must be true.

A logical formula is said to be valid if it true under every interpretation.

Validity in modal logic

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 ….!)