nLab modal similarity type

Idea

In modal logic, the relational signature of the relational structures being described by a given logic are sometimes called a modal similarity type.

Definition

A modal similarity type is given by a pair $\tau = (O,\rho)$ where $O$ is a (usually non-empty) set and $\rho : O \to \mathbb{N}$ is a function. The elements of $O$ are called the (labels for) modal operators and, if $\Delta \in O$, then $\rho(\Delta)$ indicates the arity of the modal operator $\Delta$, i.e. the number of arguments to which $\Delta$ is applied, so that, if $\rho(\Delta)= n$, the label $\Delta$ stands will stand for an $n$-ary relation on a given set.

A modal similarity type thus determines a single sorted relational signature, $\Sigma$ where, in the notation used in the page on signatures in logic, $Rel(\Sigma) = O$ and $ar: Rel(\Sigma) \to S^*$ is just $\rho$.

References

General books on modal logics include

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