Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories Set of sets and Pos of posets: Every functor Set→Pos has a Pos-enriched left Kan extension Pos→Pos. Functors arising in this way are said to have a presentation in discrete arities. In the case that Set→Pos is actually Set-valued, we call the corresponding left Kan extension Pos→Pos its posetification. A Set-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A Pos-functor with a presentation in discrete arities preserves surjections. The inclusion Set→Pos is dense. A functor Pos→Pos has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets'. A functor Pos→Pos is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves Pos-enriched sifted colimits.
A. Balan, A. Kurz, and J. Velebil, “Positive fragments of coalgebraic logics,” Logical Methods in Computer Science, vol. 11, no. 3, Sep. 2015. DOI: 10.2168/LMCS-11(3:18)2015
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