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On Gödel Algebras of Concepts

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Logic, Language, and Computation (TbiLLC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10148))

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Abstract

Beside algebraic and proof-theoretical studies, a number of different approaches have been pursued in order to provide a complete intuitive semantics for many-valued logics. Our intention is to use the powerful tools offered by formal concept analysis (FCA) to obtain further intuition about the intended semantics of a prominent many-valued logic, namely Gödel, or Gödel-Dummett, logic. In this work, we take a first step in this direction. Gödel logic seems particularly suited to the approach we aim to follow, thanks to the properties of its corresponding algebraic variety, the class of Gödel algebras. Furthermore, Gödel algebras are prelinear Heyting algebras. This makes Gödel logic an ideal contact-point between intuitionistic and many-valued logics.

In the literature one can find several studies on relations between FCA and fuzzy logics. These approaches often amount to equipping both intent and extent of concepts with connectives taken by some many-valued logic. Our approach is different. Since Gödel algebras are (residuated) lattices, we want to understand which type of concepts are expressed by these lattices. To this end, we investigate the concept lattice of the standard context obtained from the lattice reduct of a Gödel algebra. We provide a characterization of Gödel implication between concepts, and of the Gödel negation of a concept. Further, we characterize a Gödel algebra of concepts. Some concluding remarks will show how to associate (equivalence classes of) formulæ of Gödel logic with their corresponding formal concepts.

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Notes

  1. 1.

    The intended semantics of a logical language consists of the collection of models that intuitively the language talks about. In this specific case the intended semantics’ is the informal description of truth as provability given by Brouwer.

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Acknowledgements

We thank Matteo Bianchi for useful discussions about the subject of the paper. We acknowledge the support of our Marie Curie INdAM-COFUND fellowships.

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Authors and Affiliations

  1. Artificial Intelligence Research Institute (IIIA-CSIC), Campus de la UAB, 08193, Bellaterra, Spain

    Pietro Codara

  2. Dipartimento di Informatica, Università degli Studi di Milano, Via Comelico 39, 20135, Milano, Italy

    Diego Valota

Authors
  1. Pietro Codara
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  2. Diego Valota
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Correspondence to Pietro Codara .

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Editors and Affiliations

  1. Faculty of Technology, Policy and Management, Delft University of Technology, Delft, Zuid-Holland, The Netherlands

    Helle Hvid Hansen

  2. Department of Linguistics, Cornell University, Ithaca, New York, USA

    Sarah E. Murray

  3. School of Electronic Engineering and Computer Science, Queen Mary University of London, London, United Kingdom

    Mehrnoosh Sadrzadeh

  4. Heinrich-Heine-Universität, Düsseldorf, Germany

    Henk Zeevat

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Codara, P., Valota, D. (2017). On Gödel Algebras of Concepts. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_14

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  • DOI: https://doi.org/10.1007/978-3-662-54332-0_14

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