Skip to main content
SpringerLink
Log in

Logical and Geometrical Complementarities between Aristotelian Diagrams

  • Conference paper
Diagrammatic Representation and Inference (Diagrams 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8578))

Included in the following conference series:

Abstract

This paper concerns the Aristotelian relations of contradiction, contrariety, subcontrariety and subalternation between 14 contingent formulae, which can get a 2D or 3D visual representation by means of Aristotelian diagrams. The overall 3D diagram representing these Aristotelian relations is the rhombic dodecahedron (RDH), a polyhedron consisting of 14 vertices and 12 rhombic faces (Section 2). The ultimate aim is to study the various complementarities between Aristotelian diagrams inside the RDH. The crucial notions are therefore those of subdiagram and of nesting or embedding smaller diagrams into bigger ones. Three types of Aristotelian squares are characterised in terms of which types of contradictory diagonals they contain (Section 3). Secondly, any Aristotelian hexagon contains 3 squares (Section 4), and any Aristotelian octagon contains 4 hexagons (Section 5), so that different types of bigger diagrams can be distinguished in terms of which types of subdiagrams they contain. In a final part, the logical complementarities between 6 and 8 formulae are related to the geometrical complementarities between the 3D embeddings of hexagons and octagons inside the RDH (Section 6).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Jacoby, P.: A Triangle of Opposites for Types of Propositions in Aristotelian Logic. The New Scholasticism 24(1), 32–56 (1950)

    Article  Google Scholar 

  2. Sesmat, A.: Logique II. Les Raisonnements. Hermann, Paris (1951)

    Google Scholar 

  3. Blanché, R.: Structures Intellectuelles. Essai sur l’organisation systématique des concepts. Librairie Philosophique J. Vrin, Paris (1969)

    Google Scholar 

  4. Demey, L.: Structures of Oppositions in Public Announcement Logic. In: Béziau, J.Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 313–339. Springer, Basel (2012)

    Chapter  Google Scholar 

  5. Smessaert, H.: On the 3D visualisation of logical relations. Logica Universalis 3(2), 303–332 (2009)

    Article  MathSciNet  Google Scholar 

  6. Smessaert, H.: Boolean differences between two hexagonal extensions of the logical Square of Oppositions. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS (LNAI), vol. 7352, pp. 193–199. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  7. Fish, A., Flower, J.: Euler Diagram Decomposition. In: Stapleton, G., Howse, J., Lee, J. (eds.) Diagrams 2008. LNCS (LNAI), vol. 5223, pp. 28–44. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Flower, J., Stapleton, G., Rodgers, P.: On the drawability of 3D Venn and Euler diagrams. Journal of Visual Languages and Computing (2013)

    Google Scholar 

  9. Urbas, M., Jamnik, M., Stapleton, G., Flower, J.: Speedith: A Diagrammatic Reasoner for Spider Diagrams. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS (LNAI), vol. 7352, pp. 163–177. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Cheng, P.C.-H.: Algebra Diagrams: A HANDi Introduction. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS (LNAI), vol. 7352, pp. 178–192. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  11. Engelhardt, Y.: Objects and Spaces: The Visual Language of Graphics. In: Barker-Plummer, D., Cox, R., Swoboda, N. (eds.) Diagrams 2006. LNCS (LNAI), vol. 4045, pp. 104–108. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  12. Jin, Y., Esser, R., Janneck, J.W.: Describing the Syntax and Semantics of UML Statecharts in a Heterogeneous Modelling Environment. In: Hegarty, M., Meyer, B., Hari Narayanan, N. (eds.) Diagrams 2002. LNCS (LNAI), vol. 2317, pp. 320–334. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  13. Sauriol, P.: Remarques sur la théorie de l’hexagone logique de Blanché. Dialogue 7, 374–390 (1968)

    Article  Google Scholar 

  14. Moretti, A.: The Geometry of Logical Opposition. Ph.D. thesis, University of Neuchâtel (2009)

    Google Scholar 

  15. Dubois, D., Prade, H.: From Blanché’s Hexagonal Organization of Concepts to Formal Concept Analysis and Possibility Theory. Logica Universalis 6, 149–169 (2012)

    Article  MathSciNet  Google Scholar 

  16. Coxeter, H.S.M.: Regular Polytopes. Dover Publications (1973)

    Google Scholar 

  17. Zellweger, S.: Untapped potential in Peirce’s iconic notation for the sixteen binary connectives. In: Hauser, N., Roberts, D.D., Evra, J.V. (eds.) Studies in the Logic of Charles Peirce, pp. 334–386. Indiana University Press (1997)

    Google Scholar 

  18. Demey, L., Smessaert, H.: The relationship between Aristotelian and Hasse diagrams. In: Dwyer, T., Purchase, H.C., Delaney, A. (eds.) Diagrams 2014. LNCS (LNAI), vol. 8578, pp. 215–229. Springer, Heidelberg (2014)

    MATH  Google Scholar 

  19. Smessaert, H., Demey, L.: Logical Geometries and Information in the Square of Oppositions. Submitted research paper (2014)

    Google Scholar 

  20. Khomskii, Y.: William of Sherwood, singular propositions and the hexagon of opposition. In: Béziau, J.Y., Payette, G. (eds.) New Perspectives on the Square of Opposition, pp. 43–60. Peter Lang, Bern (2011)

    Google Scholar 

  21. Kretzmann, N.: William of Sherwood’s Introduction to Logic. Minnesota Archive Editions, Minneapolis (1966)

    Google Scholar 

  22. Czezowski, T.: On certain peculiarities of singular propositions. Mind 64(255), 392–395 (1955)

    Article  Google Scholar 

  23. Pellissier, R.: Setting n-opposition. Logica Universalis 2(2), 235–263 (2008)

    Article  MathSciNet  Google Scholar 

  24. Béziau, J.Y.: New light on the square of oppositions and its nameless corner. Logical Investigations 10, 218–232 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Hughes, G.: The modal logic of John Buridan. In: Atti del Congresso Internazionale di Storia Della Logica: La Teorie Delle Modalitá, pp. 93–111. CLUEB, Bologna (1989)

    Google Scholar 

  26. Read, S.: John Buridan’s Theory of Consequence and his Octagons of Opposition. In: Béziau, J.Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 93–110. Springer, Basel (2012)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Linguistics, KU Leuven, Belgium

    Hans Smessaert

  2. Center for Logic and Analytic Philosophy, KU Leuven, Belgium

    Lorenz Demey

Authors
  1. Hans Smessaert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lorenz Demey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Monash University, PO Box 197, 3145, Caulfield East, VIC, Australia

    Tim Dwyer

  2. University of Glasgow, University Avenue, G12 8QQ, Glasgow, UK

    Helen Purchase

  3. University of Brighton, Lewes Road, BN2 4GJ, Brighton, UK

    Aidan Delaney

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Smessaert, H., Demey, L. (2014). Logical and Geometrical Complementarities between Aristotelian Diagrams. In: Dwyer, T., Purchase, H., Delaney, A. (eds) Diagrammatic Representation and Inference. Diagrams 2014. Lecture Notes in Computer Science(), vol 8578. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44043-8_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44043-8_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44042-1

  • Online ISBN: 978-3-662-44043-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation