logical geometry

Introduction to Logical Geometry

  • ESSLLI 2024: the 35th European Summer School in Logic, Language and Information
  • Lorenz Demey & Hans Smessaert
  • Leuven, Belgium, July 29 - August 9, 2024
  • introductory course
  • abstract (on the ESSLLI website)


  • course slides
    1. 1. Basic Definitions, Bitstring Semantics and (Iso)Morphisms
    2. 2. Abstract-Logical Properties of Aristotelian Diagrams, Part I
    3. 3. Visual-Geometric Properties of Aristotelian Diagrams
    4. 4. Abstract-Logical Properties of Aristotelian Diagrams, Part II
    5. 5. Case Studies and Philosophical Outlook


  • recommended readings (the most important papers are marked with ***)
    1. 1. Basic Definitions, Bitstring Semantics and (Iso)Morphisms
      • Geudens, Christophe & Lorenz Demey (2021).
        'On the Aristotelian Roots of the Modal Square of Opposition'. Logique et Analyse 255, 313 - 348.
        => preprint available in PDF format; paper available in PDF format from Peeters Online Journals
      • Demey, Lorenz (2020).
        'Between Square and Hexagon in Oresme's Livre du Ciel et du Monde'. History and Philosophy of Logic 41/1, 36 - 47.
        => preprint available in PDF format; paper available in PDF format from Taylor & Francis Online
      • *** Demey, Lorenz (2019).
        'Metalogic, Metalanguage and Logical Geometry'. Logique et Analyse 248, 453 - 478.
        => preprint available in PDF format; paper available in PDF format from Peeters Online Journals
        => for the course: focus on sections 1 - 2
      • Demey, Lorenz & Hans Smessaert (2022).
        'From Euler Diagrams to Aristotelian Diagrams'. In: Valeria Giardino, Sven Linker, Richard Burns, Francesco Bellucci, Jean-Michel Boucheix & Petrucio Viana (eds.), Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 13462. Cham: Springer, 279 - 295.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz & Hans Smessaert (2024).
        'Euler Diagrams, Aristotelian Diagrams and Syllogistics'. Forthcoming in: Jens Lemanski, Mikkel Willum Johansen, Emmanuel Manalo, Petrucio Viana, Reetu Bhattacharjee & Richard Burns (eds.), Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI). Cham: Springer.
        => preprint available in PDF format
      • Smessaert, Hans & Lorenz Demey (2017).
        'The Unreasonable Effectiveness of Bitstrings in Logical Geometry'. In: Jean-Yves Beziau & Gianfranco Basti (eds.) The Square of Opposition: A Cornerstone of Thought, Springer, Basel. 197 - 214.
        => preprint available in PDF format; paper available in PDF format from Springer
      • *** Demey, Lorenz & Hans Smessaert (2018).
        'Combinatorial Bitstring Semantics for Arbitrary Logical Fragments'. Journal of Philosophical Logic 47/2, 325 - 363.
        => preprint available in PDF format; paper available in PDF format from Springer
        => for the course: focus on sections 1 - 3, 6

    2. 2. Abstract-Logical Properties of Aristotelian Diagrams, Part I
      • *** Smessaert, Hans & Lorenz Demey (2014).
        'Logical Geometries and Information in the Square of Oppositions'. Journal of Logic, Language and Information 23/4, 527 - 565.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz & Hans Smessaert (2016).
        'Duality in Logic and Language'. In: James Fieser & Bradley Dowden (eds.), Internet Encyclopedia of Philosophy. Martin, TN: University of Tennessee.
        => paper available online
      • Demey, Lorenz & Hans Smessaert (2016).
        'Metalogical Decorations of Logical Diagrams'. Logica Universalis 10/2, 233 - 292.
        => preprint available in PDF format; paper available in PDF format from Springer
        => for the course: focus on sections 4 - 6
      • *** Demey, Lorenz & Hans Smessaert (2018).
        'Aristotelian and Duality Relations Beyond the Square of Opposition'. In: Peter Chapman, Gem Stapleton, Amirouche Moktefi, Sarah Perez-Kriz & Francesco Bellucci (eds.), Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 10871. Heidelberg: Springer, 640 - 656.
        => preprint available in PDF format; paper available in PDF format from Springer
      • *** Demey, Lorenz & Hans Smessaert (2020).
        'Using Multigraphs to Study the Interaction Between Opposition, Implication and Duality Relations in Logical Squares'. In: Ahti-Veikko Pietarinen, Peter Chapman, Leonie Bosveld-de Smet, Valeria Giardino, James Corter & Sven Linker (eds.), Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 12169. Cham: Springer, 385 - 393.
        => preprint available in PDF format; paper available in PDF format from Springer

    3. 3. Visual-Geometric Properties of Aristotelian Diagrams
      • Demey, Lorenz & Hans Smessaert (2016).
        'Shape Heuristics in Aristotelian Diagrams'. In: Oliver Kutz, Stefano Borgo and Mehul Bhatt (eds.) Shapes 2015 - The Shape of Things 2015. CEUR Workshop Proceedings Vol. 1616, 35 - 45.
        => volume available at CEUR-WS website
      • *** Demey, Lorenz & Hans Smessaert (2016).
        'The Interaction between Logic and Geometry in Aristotelian Diagrams'. In: Mateja Jamnik, Yuri Uesaka & Stephanie Elzer Schwartz (eds.) Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 9781. Springer International Publishing, 67 - 82.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz & Hans Smessaert (2014).
        'The Relationship between Aristotelian and Hasse Diagrams'. In: Tim Dwyer, Helen Purchase & Aidan Delaney (eds.) Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 8578. Heidelberg: Springer, 213 - 227.
        => preprint available in PDF format; paper available in PDF format from Springer
      • *** Smessaert, Hans, Atsushi Shimojima & Lorenz Demey (2020).
        'Free Rides in Logical Space Diagrams versus Aristotelian Diagrams'. In: Ahti-Veikko Pietarinen, Peter Chapman, Leonie Bosveld-de Smet, Valeria Giardino, James Corter & Sven Linker (eds.) Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 12169. Cham: Springer, 419 - 435.
        => preprint available in PDF format; paper available in PDF format from Springer
        => best paper award at the Diagrams 2020 conference
      • Smessaert, Hans, Atsushi Shimojima & Lorenz Demey (2021).
        'On the Cognitive Potential of Derivative Meaning in Aristotelian Diagrams'. In: Amrita Basu, Gem Stapleton, Sven Linker, Catherine Legg, Emmanuel Manalo & Petrucio Viana (eds.) Diagrammatic Representation and Inference. Lecture Notes in Artificial Intelligence (LNAI) 12909. Cham: Springer, 495 - 511.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz & Hans Smessaert (2017).
        'Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation'. Symmetry 9/10. 22 pp.
        => paper available in PDF format (open access) from MDPI
        => for the course: focus on sections 3 - 4
      • *** Demey, Lorenz & Hans Smessaert (2018).
        'Geometric and Cognitive Differences between Logical Diagrams for the Boolean Algebra B_4'. Annals of Mathematics and Artificial Intelligence 83/2, 185 - 208.
        => preprint available in PDF format; paper available in PDF format from Springer
        => for the course: focus on sections 3 - 7

    4. 4. Abstract-Logical Properties of Aristotelian Diagrams, Part II
      • *** Demey, Lorenz & Hans Smessaert (2018).
        'Combinatorial Bitstring Semantics for Arbitrary Logical Fragments'. Journal of Philosophical Logic 47/2, 325 - 363.
        => preprint available in PDF format; paper available in PDF format from Springer
        => for the course: focus on sections 4 - 5
      • Demey, Lorenz (2019).
        'Boolean Considerations on John Buridan's Octagons of Opposition'. History and Philosophy of Logic 40/2, 116 - 134.
        => preprint available in PDF format; paper available in PDF format from Taylor & Francis
      • *** Demey, Lorenz & Hans Smessaert (2024).
        'Aristotelian and Boolean Properties of the Keynes-Johnson Octagon of Opposition'. Forthcoming in: Journal of Philosophical Logic.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz & Atahan Erbas (2024).
        'Boolean Subtypes of the U4 Hexagon of Opposition'. Axioms 13/2, 20 pp.
        => paper available in PDF format (open access) from MDPI
      • *** Demey, Lorenz (2018).
        'Computing the Maximal Boolean Complexity of Families of Aristotelian Diagrams'. Journal of Logic and Computation 28/6, 1323 - 1339.
        => preprint available in PDF format; paper available in PDF format from Oxford University Press; Prolog code available in PDF format
      • Demey, Lorenz (2015).
        'Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams'. In: Henning Christiansen, Isidora Stojanovic and George Papadopoulos (eds.) Modeling and Using Context. Lecture Notes in Artificial Intelligence (LNAI) 9405. Heidelberg: Springer, 331-345.
        => preprint available in PDF format; paper available in PDF format from Springer; interactive illustration available here
      • *** Demey, Lorenz (2021).
        'Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics'. Axioms 10/3, 25 pp.
        => paper available in PDF format (open access) from MDPI
      • Demey, Lorenz & Stef Frijters (2023).
        'Logic-Sensitivity and Bitstring Semantics in the Square of Opposition'. Journal of Philosophical Logic 52/6, 1703 - 1721.
        => preprint available in PDF format; paper available in PDF format from Springer

    5. 5. Case Studies and Philosophical Outlook
      • *** Demey, Lorenz (2024).
        'The Logical Geometry of Russell's Theory of Definite Descriptions'. Submitted. 33 pp.
        => preprint available in PDF format
      • Demey, Lorenz (2018).
        'A Hexagon of Opposition for the Theism/Atheism Debate', Philosophia 47/2, 387 - 394.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz (2019).
        'Aristotelian Diagrams in the Debate on Future Contingents', Sophia 58/3, 321 - 329.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Demey, Lorenz (2020).
        'From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry'. In: Jens Lemanski (ed.), Mathematics, Logic and Language in Schopenhauer. Berlin: Springer, 181 - 205.
        => preprint available in PDF format; paper available in PDF format from Springer
      • García-Cruz, José David & Lorenz Demey (2024).
        'Aristotelian Diagrams for the Ancient Discussion on Privative and Infinite Negation'. Forthcoming in: Logique et Analyse.
        => preprint available in PDF format
      • Demey, Lorenz (2021).
        'Aristotelian Diagrams for Semantic and Syntactic Consequence', Synthese 198/1, 187 - 207.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Smessaert, Hans & Lorenz Demey (2022).
        'On the Logical Geometry of Geometrical Angles', Logica Universalis 16/4, 581 - 601.
        => preprint available in PDF format; paper available in PDF format from Springer
      • Smessaert, Hans & Lorenz Demey (2023).
        'Aristotelian Diagrams for the Proportional Quantifier 'Most'', Axioms 12/3, 236. 13 pp.
        => paper available in PDF format (open access) from MDPI
      • *** Demey, Lorenz (2019).
        'The Role of Aristotelian Diagrams in the Scientific Communication'. In: András Benedek and Kristóf Nyíri (eds.) Learning and Technology in Historical Perspective. Perspectives on Visual Learning 2. Budapest: Hungarian Academy of Sciences, 69 - 77.
        => preprint available in PDF format