We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view.Our approach is adopted from Pitts' proof of uniform interpolation in intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It provides an explicit algorithm constructing the interpolants.
The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss's coalgebraic over posets.
This paper considers two logics. The first one, KGinv, is an expansion of the Gödel modal logic KG with the involutive negation ∼i defined as v(∼iφ, w) = 1 − v(φ, w). The second one, KG bl , is the expansion of KGinv with the bi-lattice connectives and modalities. We explore their semantical properties w.r.t. the standard semantics on [0, 1]-valued Kripke frames and define a unified tableaux calculus that allows for the explicit countermodel construction. For this, we use an alternative semantics with the finite model property. Using the tableaux calculus, we construct a decision algorithm and show that satisfiability and validity in KGinv and KG bl are PSpace-complete.
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