Positive fragments of coalgebraic logics

Abstract

Abstract. 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. ACM CCS: [Theory of computation]:Logic.

Limits in categories of Vietoris coalgebras

“…These coalgebras have significant relevance within the coalgebraic community (e.g. Baldan et al 2014;Balan and Kurz 2011;Balan et al 2013) and we believe that our study can contribute to the topic.…”

Varieties of Positive Modal Algebras and Structural Completeness

“…To name only a few of them, HSC varieties of K 4 -algebras, and of Heyting algebras have been fully described in [14,57]. Moreover, explicit bases for the admissible rules of intuitionistic 1 Observe that passive quasi-equations are vacuously admissible. logic (equiv.…”

“…Hence we have that 0 < a < b < 1 and we can apply condition (ii) in the statement. This condition, together with EDPC for distributive lattices as in (1), implies that there is c ∈ A {0, 1} such that either the pair c, 1 or the pair c, 0 belongs to Cg(a, b). Together with the fact that Qc = 1 and c = 0, this implies that 0, 1 ∈ Cg(a, b).…”

“…The interest of PML and its extensions comes from the fact that, despite their restricted signature, they preserve some mathematically desiderable features typical of normal modal logics, including Sahlqvist theory and a well-behaved Priestley-style duality [10,11,25]. For similar reasons, PML and their generalizations were studied both from the perspective of algebraic logic [30] and from that of coalgebra [1,48].…”

“…K is passively structurally complete (PSC) if every passive quasi-equation is valid. 1 3. K is structurally complete (SC) if every admissible quasi-equation is valid.…”