Two-dimensional Modal Logics with Difference Relations

In this thesis, we explore various computational and axiomatisation problems relating to two-dimensional modal logics that exhibit some modest capacity to count. In particular, we consider modal products in which at least one component is the logic of difference (inequality) relations. These formalisms are connected with finite variable fragments of first-order logic and first-order modal logics, extended with some additional counting quantifiers. The contributions provided herein serve as a case study to better steer investigation into more general principles governing the interactions between modal logics, and into understanding the interactions between first-order quantifiers