Uniform Interpolation for Monotone Modal Logic

We reconstruct the syntax and semantics of monotone modal logic, in the style of Moss' coalgebraic logic. To that aim, we replace the box and diamond with a modality nabla which takes a finite collection of finite sets of formulas as its argument. The semantics of this modality in monotone neighborhood models is defined in terms of a version of relation lifting that is appropriate for this setting. We prove that the standard modal language and our r -based one are effectively equi-expressive, meaning that there are effective translations in both directions. We prove and discuss some algebraic laws that govern the interaction of nabla with the Boolean operations. These laws enable us to rewrite each formula into a special kind of disjunctive normal form that we call transparent. For such transparent formulas it is relatively easy to define the bisimulation quantifiers that one may associate with our notion of relation lifting. This allows us to prove the main result of the paper, viz., that monotone modal logic enjoys the property of uniform interpolation.

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