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Algebraic proof theory for LE-logics

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arxiv 1808.04642 v2 pith:XPJWWOMA submitted 2018-08-14 math.LO

Giuseppe Greco , Peter Jipsen , Fei Liang , Alessandra Palmigiano , Apostolos Tzimoulis This is my paper · ORCID
classification math.LO
keywords le-logicsnormalextensionsalgebraicanalyticaxiomaticcalculuscertain
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In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties.

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Cited by 3 Pith papers

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  1. Vector spaces as Kripke frames

    cs.LO 2019-08 accept novelty 7.0 of 10

    Vector spaces equipped with a bilinear product are shown to form Kripke-style frames whose subspace lattices are complete residuated lattices, yielding a complete vector space semantics for the modal non-associative L...

  2. Modelling socio-political competition

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    A many-valued, multi-type modal logic for socio-political competition is axiomatized and proven complete with respect to graph-based semantics over enriched reflexive graphs.

  3. The logic of vague categories

    math.LO 2019-08 conditional novelty 4.0 of 10

    The basic normal lattice-based modal logic is sound and complete with respect to many-valued enriched formal contexts, with an illustrative proposal for analyzing multi-market competition.

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