Groups, Not Squares: Exorcizing a Fetish

  • Walter CarnielliEmail author
Part of the Studies in Universal Logic book series (SUL)


I argue that the celebrated Square of Opposition is just a shadow of a much deeper relationship on duality, complementarity, opposition and quaternality expressed by algebraic means, and that any serious attempt to make sense of squares and cubes of opposition must take into account the theory of finite groups. By defining a group as triadic if all its elements, other than the identity, have order 3, I show that a natural notion of triality group acting on three-valued structures emerges, generalizing the intuitions of duality and quaternality.


Boolean groups Group theory Square of opposition Triadic groups 

Mathematics Subject Classification (2000)

Primary 03A05 Secondary 05C25 



I acknowledge support from FAPESP Thematic Project LogCons 2010/51038-0, Brazil and from the National Council for Scientific and Technological Development (CNPq), Brazil. I am thankful to the five anonymous referees who have read, commented and criticized this paper.


  1. 1.
    J.C. Agudelo, W.A. Carnielli, Polynomial ring calculus for modal logics: a new semantics and proof method for modalities. Rev. Symb. Log. 11 (4), 150–170 (2011). Pre-print available from CLE e-Prints 9(4), 2009.,n_4,2009.html
  2. 2.
    W.A. Carnielli, The problem of quantificational completeness and the characterization of all perfect quantifiers in 3-valued logics. Zeitschr. f. math. Logik und Grundlagen d. Math. 33, 19–29 (1987)CrossRefGoogle Scholar
  3. 3.
    W.A. Carnielli, Polynomial ring calculus for many-valued logics, in Proceedings of the 35th International Symposium on Multiple-Valued Logic, ed. by B. Werner (IEEE Computer Society, Los Alamitos, 2005), pp. 20–25. Preprint available at CLE e-Prints vol. 5(3),n_3,2005.html
  4. 4.
    W.A. Carnielli, M.C.C. Gracio, Modulated logics and flexible reasoning. Log. Log. Philos. 17 (3), 211–249 (2008)Google Scholar
  5. 5.
    W.A. Carnielli, M. Matulovic, Non-deterministic semantics in polynomial format. Electron. Notes Theor. Comput. Sci. 305, 19–34 (2014). Proceedings of the 8th Workshop on Logical and Semantic Frameworks (LSFA). Open access:
  6. 6.
    W.A. Carnielli, M. Matulovic, The method of polynomial ring calculus and its potentialities. Theor. Comput. Sci. 606 (C), 42–56 (2015)CrossRefGoogle Scholar
  7. 7.
    D. Dubois, H. Prade, De l’organisation hexagonale des concepts de Blanché à l’analyse formelle de concepts et à la théorie des possibilités, in Journées d’Intelligence Artificielle Fondamentale, Lyon, 08/06/2011–10/06/2011 [French] (2011), pp. 113–129. Manuscript. Available at
  8. 8.
    R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (1963).
  9. 9.
    J. Gallian, Contemporary Abstract Algebra, 7th edn. (Brooks Cole, Pacific Grove, CA, 2009)Google Scholar
  10. 10.
    W.H. Gottschalk, The theory of quaternality. J. Symb. Log. 18 (3), 193–196 (1953)CrossRefGoogle Scholar
  11. 11.
    P. Hage, F. Harary, Arapesh sexual symbolism, primitive thought and Boolean groups. L’Homme 23 (2), 57–77 (1983)CrossRefGoogle Scholar
  12. 12.
    P.R. Halmos, S.R. Givant, Logic as Algebra (The Mathematical Association of America, Washington, DC, 1998)Google Scholar
  13. 13.
    F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969)Google Scholar
  14. 14.
    S. Knuuttila, Medieval theories of modality, in The Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta, Fall 2013 Edition (2013).>
  15. 15.
    C. Lévi-Strauss, Anthropologie Structurale (Plon, Paris, 1958). Reprinted in 2012Google Scholar
  16. 16.
    D. McDermott, Artificial intelligence meets natural stupidity, in Mind Design, ed. by J. Haugeland, pp. 143–60 (MIT, Cambridge, MA, 1981)Google Scholar
  17. 17.
    A. Moretti, A Cube Extending Piaget’s and Gottschalk’s Formal Square, ed. by J.-Y. Béziau, K. Gan-Krzywoszyńska. Handbook of the Second World Congress on the Square of Opposition (2010).
  18. 18.
    D. Robinson, A Course in the Theory of Groups (Springer, Berlin, 2012)Google Scholar
  19. 19.
    F. Schang, Oppositions and opposites, in Around and Beyond the Square of Opposition, ed. by J.-Y. Beziau, D. Jacquette (Birkhäuser, Basel, 2012), pp. 147–173CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Centre for Logic, Epistemology and the History of Science, CLE and Department of PhilosophyState University of Campinas - UnicampSPBrazil

Personalised recommendations