Non-iterative Modal Logics are Coalgebraic (2006.05396v2)

Abstract: A modal logic is \emph{non-iterative} if it can be defined by axioms that do not nest modal operators, and \emph{rank-1} if additionally all propositional variables in axioms are in scope of a modal operator. It is known that every syntactically defined rank-1 modal logic can be equipped with a canonical coalgebraic semantics, ensuring soundness and strong completeness. In the present work, we extend this result to non-iterative modal logics, showing that every non-iterative modal logic can be equipped with a canonical coalgebraic semantics defined in terms of a copointed functor, again ensuring soundness and strong completeness via a canonical model construction. Like in the rank-1 case, the canonical coalgebraic semantics is equivalent to a neighbourhood semantics with suitable frame conditions, so the known strong completeness of non-iterative modal logics over neighbourhood semantics is implied. As an illustration of these results, we discuss deontic logics with factual detachment, which is captured by axioms that are non-iterative but not rank~1.