International Conference on Theory and Application of Diagrams

Diagrams 2014: Diagrammatic Representation and Inference pp 293-307 | Cite as

The Second Venn Diagrammatic System

  • Renata de Freitas
  • Petrucio Viana
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8578)


We present syntax and semantics of a diagrammatic language based on Venn diagrams in which a diagram is not read as a statement about sets, but as a set itself. We prove that our set of rules is sound and complete with respect to the intended semantics. Our system has two slight advantages in relation to the systems we usually encounter in the literature. First, the drawing of diagrams for terms is made inside the system, i.e., by a completely mechanical process based just on the rules of the system. Second, as a consequence, the validity of an inclusion is also verified inside the system and does not depend on any other means than those afforded by our set of rules. These characteristics are absent in the majority of the Venn diagrammatic systems.


diagrammatic systems Venn diagrams soundness completeness 


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  1. 1.
    Burton, J., Stapleton, G., Howse, J.: Completeness proof strategies for Euler diagram logics. In: Chapman, P., Micallef, L. (eds.) Euler Diagrams 2012: Proceedings of the 3rd International Workshop on Euler Diagrams, Canterbury, UK, July 2 (2012)Google Scholar
  2. 2.
    Halmos, P.: Naive Set Theory. Springer, New York (1974)CrossRefGoogle Scholar
  3. 3.
    Hammer, E., Danner, N.: Towards a model theory of Venn diagrams. J. Philos. Logic 25, 463–482 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  5. 5.
    Stewart, I.: The truth about Venn diagrams. Math. Gaz. 70, 47–54 (1976)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Renata de Freitas
    • 1
  • Petrucio Viana
    • 1
  1. 1.Institute of Mathematics and StatisticsUFF: Universidade Federal FluminenseNiteróiBrasil

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