We study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics, establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

We investigate bisimulations for instantial neighbourhood logic and an \-indexed collection of its fragments. For each of these logics we give a Hennessy-Milner theorem and a Van Benthem-style characterisation theorem.