10 research outputs found

    Uniform Interpolation in provability logics

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    We prove the uniform interpolation theorem in modal provability logics GL and Grz by a proof-theoretical method, using analytical and terminating sequent calculi for the logics. The calculus for G\"odel-L\"ob's logic GL is a variant of the standard sequent calculus, in the case of Grzegorczyk's logic Grz, the calculus implements an explicit loop-preventing mechanism inspired by work of Heuerding

    Relation lifting, with an application to the many-valued cover modality

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    We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad" on categories, one is the preservation by T of "exactness" of certain squares. Both characterisations are generalisations of the "classical" results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks. The results presented in this paper enable us to compute predicate liftings of endofunctors of, for example, generalised (ultra)metric spaces. We illustrate this by studying the coalgebraic cover modality in this setting.Comment: 48 pages, accepted for publication in LMC

    Relation Liftings on Preorders and Posets

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    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

    Non-standard modalities in paraconsistent G\"{o}del logic

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    We introduce a paraconsistent expansion of the G\"{o}del logic with a De Morgan negation ¬\neg and modalities \blacksquare and \blacklozenge. We equip it with Kripke semantics on frames with two (possibly fuzzy) relations: R+R^+ and RR^- (interpreted as the degree of trust in affirmations and denials by a given source) and valuations v1v_1 and v2v_2 (positive and negative support) ranging over [0,1][0,1] and connected via ¬\neg. We motivate the semantics of ϕ\blacksquare\phi (resp., ϕ\blacklozenge\phi) as infima (suprema) of both positive and negative supports of ϕ\phi in R+R^+- and RR^--accessible states, respectively. We then prove several instructive semantical properties of the logic. Finally, we devise a tableaux system for branching fragment and establish the complexity of satisfiability and validity.Comment: arXiv admin note: text overlap with arXiv:2303.1416

    Crisp bi-G\"{o}del modal logic and its paraconsistent expansion

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    In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G\"{o}del modal logic \KbiG. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-G\"{o}del algebra on [0,1][0,1]. We also consider a paraconsistent expansion of \KbiG with a De Morgan negation ¬\neg which we dub \KGsquare. We devise a Hilbert-style calculus for this logic and, as a~con\-se\-quence of a~conservative translation from \KbiG to \KGsquare, prove its completeness w.r.t.\ crisp Kripke models with two valuations over [0,1][0,1] connected via ¬\neg. For these two logics, we establish that their decidability and validity are PSPACE\mathsf{PSPACE}-complete. We also study the semantical properties of \KbiG and \KGsquare. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in K\mathbf{K} (the classical modal logic) and the crisp G\"{o}del modal logic \KG^c. We show that, among others, all Sahlqvist formulas and all formulas ϕχ\phi\rightarrow\chi where ϕ\phi and χ\chi are monotone, define the same classes of frames in K\mathbf{K} and \KG^c

    Fuzzy bi-G\"{o}del modal logic and its paraconsistent relatives

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    We present the axiomatisation of the fuzzy bi-G\"{o}del modal logic (formulated in the language containing \triangle and treating the coimplication as a defined connective) and establish its PSpace-completeness. We also consider its paraconsistent relatives defined on fuzzy frames with two valuations e1e_1 and e2e_2 standing for the support of truth and falsity, respectively, and equipped with \emph{two fuzzy relations} R+R^+ and RR^- used to determine supports of truth and falsity of modal formulas. We establish embeddings of these paraconsistent logics into the fuzzy bi-G\"{o}del modal logic and use them to prove their PSpace-completeness and obtain the characterisation of definable frames

    Two-layered logics for paraconsistent probabilities

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    We discuss two two-layered logics formalising reasoning with paraconsistent probabilities that combine the Lukasiewicz [0,1][0,1]-valued logic with Baaz \triangle operator and the Belnap--Dunn logic

    Relation lifting, with an application to the many-valued cover modality

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    We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad" on categories, one is the preservation by T of "exactness" of certain squares. Both characterisations are generalisations of the "classical" results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks. The results presented in this paper enable us to compute predicate liftings of endofunctors of, for example, generalised (ultra)metric spaces. We illustrate this by studying the coalgebraic cover modality in this setting
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