Non-iterative Modal Logics are Coalgebraic

TL;DR

This paper extends the coalgebraic semantics framework to non-iterative modal logics, establishing soundness and strong completeness, and illustrating with deontic logics that include factual detachment.

Contribution

It generalizes the coalgebraic semantics from rank-1 to non-iterative modal logics using copointed functors, ensuring soundness and strong completeness.

Findings

Non-iterative modal logics can be given canonical coalgebraic semantics.

The semantics are equivalent to neighbourhood semantics with frame conditions.

Deontic logics with factual detachment are included as examples.

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.