Carroll Diagrams: Design and Manipulation

  • Amirouche MoktefiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10871)


The use of diagrams in logic is old. Euler and Venn schemes are among the most popular. Carroll diagrams are less known but are occasionally mentioned in recent literature. The objective of this tutorial is to expose the working of Carroll’s diagrams and their significance from a triple perspective: historical, mathematical and philosophical. The diagrams are exposed, worked out and compared to Euler-Venn diagrams. These schemes are used to solve the problem of elimination which was widely addressed by early mathematical logicians: finding the conclusion that is to be drawn from any number of propositions given as premises containing any number of terms. For this purpose, they designed symbolic, visual and sometimes mechanical devices. The significance of Venn and Carroll diagrams is better understood within this historical context. The development of mathematical logic notably created the need for more complex diagrams to represent n terms, rather than merely 3 terms (the number demanded by syllogisms). Several methods to construct diagrams for n terms, with different strategies, are discussed. Finally, the philosophical significance of Carroll diagrams is discussed in relation to the use of rules to transfer information from a diagram to another. This practice is connected to recent philosophical debates on the role of diagrams in mathematical practices.


Carroll diagram Venn diagram Universe of discourse Diagram for n terms Rules Elimination 


  1. 1.
    Abeles, F.F.: Lewis Carroll’s visual logic. Hist. Philos. Log. 28(1), 1–17 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bartley III, W.W. (ed.): Lewis Carroll’s Symbolic Logic. C. N. Potter, New York (1986)Google Scholar
  3. 3.
    Carroll, L.: The Game of Logic. Macmillan, London (1887)Google Scholar
  4. 4.
    Coumet, E.: The game of logic: a game of universes. In: Guiliano, E. (ed.) Lewis Carroll Observed, pp. 181–195. C. N. Potter, New York (1976)Google Scholar
  5. 5.
    Edwards, A.W.F.: Cogwheels of the Mind: The Story of Venn Diagrams. Johns Hopkins University Press, Baltimore (2004)zbMATHGoogle Scholar
  6. 6.
    Euler, L.: Lettres à une princesse d’Allemagne, vol. 2. Imprimerie de l’Académie impériale des sciences, Saint Petersburg (1768)Google Scholar
  7. 7.
    Geach, P.: Reason and Argument. University of California Press, Berkeley (1976)Google Scholar
  8. 8.
    Lewis, C.I.: A Survey of Symbolic Logic. University of California press, Berkeley (1918)Google Scholar
  9. 9.
    Macula, A.J.: Lewis Carroll and the enumeration of minimal covers. Math. Mag. 68(4), 269–274 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moktefi, A.: Lewis Carroll’s logic. In: Gabbay, D.M., Woods, J. (eds.) British Logic in the Nineteenth Century, pp. 457–505. North-Holland, Amsterdam (2008)Google Scholar
  11. 11.
    Moktefi, A.: Beyond syllogisms: Carroll’s (marked) quadriliteral diagram. In: Moktefi, A., Shin, S.-J. (eds.) Visual Reasoning with Diagrams, pp. 55–71. Basel, Birkhäuser (2013)CrossRefGoogle Scholar
  12. 12.
    Moktefi, A.: On the social utility of symbolic logic: Lewis Carroll against ‘The Logicians’. Stud. Metodol. 35, 133–150 (2015)Google Scholar
  13. 13.
    Moktefi, A.: Diagrams as scientific instruments. In: Benedek, A., Veszelszki, A. (eds.) Virtual Reality – Real Visuality, pp. 81–89. Peter Lang, Frankfurt am Main (2017)Google Scholar
  14. 14.
    Moktefi, A., Bellucci, F., Pietarinen, A.-V.: Continuity, connectivity and regularity in spatial diagrams for N terms. In: Burton, J., Choudhury, L. (eds.) DLAC 2013: Diagrams, Logic and Cognition, CEUR Workshop Proceedings, vol. 1132, pp. 31–35 (2014)Google Scholar
  15. 15.
    Moktefi, A., Pietarinen, A.-V.: On the diagrammatic representation of existential statements with Venn diagrams. J. Logic Lang. Inf. 24(4), 361–374 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Moktefi, A., Shin, S.-J.: A history of logic diagrams. In: Gabbay, D.M., Pelletier, F.J., Woods, J. (eds.) Logic: A History of Its Central Concepts, pp. 611–682. North-Holland, Amsterdam (2012)CrossRefGoogle Scholar
  17. 17.
    Venn, J.: On the diagrammatic and mechanical representation of propositions and reasonings. Philos. Mag. 10, 1–18 (1880)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tallinn University of TechnologyTallinnEstonia

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