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Implicational (semilinear) logics II: additional connectives and characterizations of semilinearity

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Abstract

This is the continuation of the paper (Cintula and Noguera in Arch Math Log 49(4):417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property.

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07, Prague, Czech Republic

    Petr Cintula

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague, Czech Republic

    Carles Noguera

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  1. Petr Cintula
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  2. Carles Noguera
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Correspondence to Carles Noguera.

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Cintula, P., Noguera, C. Implicational (semilinear) logics II: additional connectives and characterizations of semilinearity. Arch. Math. Logic 55, 353–372 (2016). https://doi.org/10.1007/s00153-015-0452-9

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  • DOI: https://doi.org/10.1007/s00153-015-0452-9

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