Two Standard and Two Modal Squares of Opposition

Chapter
Part of the Studies in Universal Logic book series (SUL)

Abstract

In this study, we examine modern reading of the Square of Opposition by means of intensional logic. Explicit use of possible world semantics helps us to sharply discriminate between the standard and modal (‘alethic’) readings of categorical statements. We get thus two basic versions of the Square. The Modal Square has not been introduced in the contemporary debate yet and so it is in the centre of interest. Some properties ascribed by medieval logicians to the Square require a shift from its Standard to Modal version. Not inevitably, because for each of the two squares there exists its mate which can be easily confused with it. The discrimination between the initial and modified versions of the Standard and Modal Square enable us to separate findings about properties of the Square into four groups, which makes their proper comparison possible. The disambiguation so achieved leads to the solution of various puzzles often mentioned in recent literature.

Keywords

Modal logic Modal Square of opposition Possible world semantics Square of opposition 

Mathematics Subject Classification

Primary 03B45 ⋅ Secondary 03B60 ⋅ 03B65 ⋅ 03C80 

References

  1. 1.
    J.L. Ackrill (ed.), Aristotle, De Interpretatione (Clarendon Press, Oxford, 1963)Google Scholar
  2. 2.
    J.-Y. Béziau, New light on the square of opposition and its nameless corner. Logic. Invest. 10, 218–232 (2003)Google Scholar
  3. 3.
    J.-Y. Béziau, The new rising of the square of opposition, in Around and Beyond the Square of Opposition, ed. by J.-Y. Béziau, D. Jacquette (Birkhäuser, Basel, 2012), pp. 3–19CrossRefGoogle Scholar
  4. 4.
    J.-Y. Béziau, The power of the hexagon. Log. Univers. 6, 1–43 (2012)CrossRefGoogle Scholar
  5. 5.
    R. Blanché, Quantity, modality, and other kindred systems of categories. Mind 61, 369–375 (1952)CrossRefGoogle Scholar
  6. 6.
    R. Blanché, Structures intellectuelles: essai sur l’organisation systématique des concepts (Vrin, Paris, 1966)Google Scholar
  7. 7.
    M. Brown, Generalized quantifiers and the square of opposition. Notre Dame J. Formal Logic 25, 303–322 (1984)CrossRefGoogle Scholar
  8. 8.
    A. Church, A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)CrossRefGoogle Scholar
  9. 9.
    D. D’Alfonso, The square of opposition and generalized quantifiers, in Around and Beyond the Square of Opposition, ed. by J.-Y. Béziau, D. Jacquette (Birkhäuser, Basel, 2012), pp. 219–227CrossRefGoogle Scholar
  10. 10.
    A.-A. Dufatanye, From the logical square to Blanché’s hexagon: formalization, applicability and the idea of the normative structure of thoughts. Log. Univers. 6, 45–67 (2012)CrossRefGoogle Scholar
  11. 11.
    M. Duží, Squaring the square of opposition with empty concepts. Slides presented at 4th Congress Square of Oppositions (Vatican), May (2014)Google Scholar
  12. 12.
    M. Duží, B. Jespersen, P. Materna, Procedural Semantics for Hyperintensional Logic: Foundations and Applications of Transparent Intensional Logic (Springer, Houten, 2010)Google Scholar
  13. 13.
    W.H. Gottschalk, The theory of quaternality. J. Symb. Log. 18, 193–196 (1953)CrossRefGoogle Scholar
  14. 14.
    J.C. Joerden, Deontological square, hexagon, and decagon: a deontic framework for supererogation. Log. Univers. 6, 201–216 (2012)CrossRefGoogle Scholar
  15. 15.
    S. Knuuttila, Medieval theories of modality, in The Stanford Encyclopedia of Philosophy, Fall 2013 edn., ed. by E.N. Zalta (2013). < http://plato.stanford.edu/archives/fall2013/entries/modality-medieval/ > 
  16. 16.
    A. Moretti, The geometry of logical opposition. PhD Thesis, University of Neuchâtel (2009)Google Scholar
  17. 17.
    A. Moretti, Why the logical hexagon? Log. Univers. 6, 69–107 (2012)CrossRefGoogle Scholar
  18. 18.
    J.O. Nelson, In defense of the traditional interpretation of the square. Philos. Rev. 63, 401–413 (1954)CrossRefGoogle Scholar
  19. 19.
    T. Parsons, Things that are right with the traditional square of opposition. Log. Univers. 2, 3–11 (2008)CrossRefGoogle Scholar
  20. 20.
    T. Parsons, The traditional square of opposition, in The Stanford Encyclopedia of Philosophy, Spring 2014 edn., ed. by E.N. Zalta (2014). < http://plato.stanford.edu/archives/spr2014/entries/square/ > 
  21. 21.
    R. Pellissier, Setting “n-opposition”. Log. Univers. 2, 235–263 (2008)CrossRefGoogle Scholar
  22. 22.
    J. Raclavský, Defining basic kinds of properties, in The World of Language and the World Beyond Language (A Festschrift for Pavel Cmorej) ed. by T. Marvan, M. Zouhar (Filozofický ústav SAV, Bratislava, 2007), pp. 69–107Google Scholar
  23. 23.
    J. Raclavský, Categorical statements from the viewpoint of transparent intensional logic (in Czech), in Odkud a jak brát stále nové příklady? Elektronická databáze příkladů pro výuku logiky na VŠ, ed. by L. Dostálová et al. (Vydavatelství Západočeské univerzity v Plzni, Plzeň, 2009), pp. 85–101Google Scholar
  24. 24.
    J. Raclavský, Generalized quantifiers from the viewpoint of transparent intensional logic (in Czech), in Odkud a jak brát stále nové příklady? Elektronická databáze příkladů pro výuku logiky na VŠ, ed. by L. Dostálová et al. (Vydavatelství Západočeské univerzity v Plzni, Plzeň, 2009), pp. 103–116Google Scholar
  25. 25.
    J. Raclavský, Names and Descriptions: Logic-Semantical Investigations (in Czech). (Nakladatelství Olomouc, Olomouc, 2009)Google Scholar
  26. 26.
    J. Raclavský, On partiality and Tichý’s transparent intensional logic. Hung. Philos. Rev. 54, 120–128 (2010)Google Scholar
  27. 27.
    J. Raclavský, Explicating truth in transparent intensional logic, in Recent Trends in Logic, vol. 41, ed. by R. Ciuni, H. Wansing, C. Willkomen (Springer, Berlin, 2014), pp. 169–179Google Scholar
  28. 28.
    J. Raclavský, The barber paradox: on its paradoxicality and its relationship to Russell’s paradox. Prolegomena 2, 269–278 (2014)Google Scholar
  29. 29.
    D.H. Sanford, Contraries and subcontraries. Noûs 2, 95–96 (1968)CrossRefGoogle Scholar
  30. 30.
    F. Schang, Questions and answers about oppositions, in The Square of Opposition. A General Framework for Cognition, ed. by J.-Y. Béziau, G. Payette (Peter Lang, Bern, 2011), pp. 283–314Google Scholar
  31. 31.
    H. Smessaert, The classical aristotelian hexagon, versus the modern duality hexagon. Log. Univers. 6, 171–199 (2012)CrossRefGoogle Scholar
  32. 32.
    P. Tichý, Introduction to intensional logic. Unpublished MS (1976)Google Scholar
  33. 33.
    P. Tichý, Foundations of partial type theory. Rep. Math. Logic 14, 57–72 (1982)Google Scholar
  34. 34.
    P. Tichý, The Foundations of Frege’s Logic (Walter de Gruyter, Berlin, 1988)CrossRefGoogle Scholar
  35. 35.
    P. Tichý, Pavel Tichý’s Collected Papers in Logic and Philosophy, ed. by V. Svoboda, B. Jespersen, C. Cheyne (University of Otago Publisher/Filosofia, Dunedin/Prague, 2004)Google Scholar
  36. 36.
    S.L. Uckelman, Three 13th-century views of quantified modal logic, in Advances in Modal Logic, vol. 7, ed. by C. Areces, R. Goldblatt (College Publications, London, 2008), pp. 389–406Google Scholar
  37. 37.
    D. Westerståhl, On the Aristotelian square of opposition, in Kapten Nemos Kolumbarium, Philosophical Communications, ed. by F. Larsson (red.) web series No. 33 (2005). < http://www.ipd.gu.se/digitalAssets/1303/1303475\_Westerst\aahl-onthearistoteliansquare.pdf > , retrieved: 4/6/2014Google Scholar
  38. 38.
    D. Westerståhl, Classical vs. modern squares of opposition, and beyond, in The Square of Opposition – A General Framework for Cognition, ed. by J.-Y. Béziau, G. Payette (Peter Lang, Bern, 2012), pp. 195–229Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of PhilosophyMasaryk UniversityBrnoCzech Republic

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