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.

Last revised on March 19, 2015 at 07:20:00.
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