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Order-Dual Relational Semantics for Non-distributive Propositional Logics: A General Framework

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Abstract

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures (frames) and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic (the logic of bounded lattices) and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as (non-distributive) Generalized Galois Logics (GGL’s).

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Acknowledgments

I wish to sincerely thank the anonymous referees for their comments and recommendations that helped improve the clarity and presentation of this article.

While this article was under review, applications and further clarification of the framework were pursued in two sequel papers, currently under review:

While this article was under review, applications and further clarification of the framework was pursued by this author in a sequel paper Kripke-Galois Semantics for Substructural Logics (2016), currently under review, treating a variety of logical systems, from the Full Lambek and Lambek-Grishin calculi to Modal, Linear and Relevance Logic (without distribution). Other than adopting ‘Kripke-Galois semantics’ in place of the ‘order-dual semantics’ used in the present article and in [32], this follow up paper puts the framework to test by successfully applying it to the familiar logical systems mentioned above. These concrete applications will probably be of help in elucidating the approach and the techniques used.

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  1. Computer Science and Engineering, University of Applied Sciences of Thessaly (TEI of Thessaly), Larissa, Greece

    Chrysafis Hartonas

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  1. Chrysafis Hartonas
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Hartonas, C. Order-Dual Relational Semantics for Non-distributive Propositional Logics: A General Framework. J Philos Logic 47, 67–94 (2018). https://doi.org/10.1007/s10992-016-9417-7

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