nLab modal similarity type


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


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

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


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. See the history of this page for a list of all contributions to it.