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Several logical operators are defined as dual pairs, in different types oflogics. Such dual pairs of operators also occur in other algebraic theories,such as mathematical morphology. Based on this observation, this paper proposesto define, at the abstract level of institutions, a pair of abstract dual andlogical operators as morphological erosion and dilation. Standard quantifiersand modalities are then derived from these two abstract logical operators.These operators are studied both on sets of states and sets of models. To copewith the lack of explicit set of states in institutions, the proposed abstractlogical dual operators are defined in an extension of institutions, thestratified institutions, which take into account the notion of open sentences,the satisfaction of which is parametrized by sets of states. A hint on thepotential interest of the proposed framework for spatial reasoning is alsoprovided.

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning