Algebraic Aspects of Duality Diagrams

  • Lorenz Demey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7352)


Duality phenomena are widespread in logic and language; their behavior is visualized using square diagrams. This paper shows how our recent algebraic account of duality can be fruitfully used to study these diagrams. A duality cube is constructed, and it is shown that 14 duality squares can be embedded into this cube (two of which were hitherto unknown). This number is also an upper bound.


duality negation logic linguistics logical geometry 


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  1. 1.
    van Benthem, J.: Linguistic universals in logical semantics. In: Zaefferer, D. (ed.) Semantic Universals and Universal Semantics, pp. 17–36. Foris, Berlin (1991)Google Scholar
  2. 2.
    Călugăreanu, G.: The total number of subgroups of a finite Abelian group. Scientiae Mathematicae Japonicae 60, 157–168 (2004)MathSciNetMATHGoogle Scholar
  3. 3.
    Demey, L.: Structures of oppositions in public announcement logic. In: Béziau, J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition. Springer (2012)Google Scholar
  4. 4.
    Löbner, S.: Wahr neben Falsch. Duale Operatoren als die Quantoren natürlicher Sprache. Max Niemeyer Verlag, Tübingen (1990)Google Scholar
  5. 5.
    Moretti, A.: A cube extending Piaget-Gottschalk’s formal square (ms.)Google Scholar
  6. 6.
    Smessaert, H.: The classical Aristotelian hexagon versus the modern duality hexagon. Logica Universalis (forthcoming), doi:101007/s11787-011-0031-8Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lorenz Demey
    • 1
  1. 1.University of LeuvenBelgium

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