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Mereotopology in 2nd-Order and Modal Extensions of Intuitionistic Propositional Logic

We show how mereotopological notions can be expressed by extending intuitionistic propositional logic with propositional quantification and a strong modal operator. We first prove completeness for the logics wrt Kripke models; then we trace the correspondence between Kripke models and topological spaces that have been enhanced with an explicit notion of expressible region (r-spaces). We show how some qualitative spatial notions can be expressed in topological terms. We use the semantical and topological results in order to show how in some extensions of the logics it is possible to express connectedness, non-emptiness and a set of jointly exhaustive, pairwise disjoint, binary relations that play a significant role in qualitative spatial reasoning (RCC8 relations, [RAN 92]).