Thinking Outside the Square of Opposition Box

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


The graphic meaning and formal implications of the canonical Aristotelian square of opposition are favorably compared with alternative ways of representing subsyllogistic logical relations among the four categorical AEIO propositions of syllogistic logic. The canonical arrangement of AEIO propositions in the square is justified by an exhaustive survey of graphically nonequivalent alternatives. The conspicuous omission in a modified expanded Aristotelian square of inversions involving complements of subject terms is critically considered. Two of the four inversions are shown to be logically equivalent to two of the AEIO propositions, and are consequently discounted as offering no potential significant revision of the canonical square. The remaining two inversions are logically distinct from any of the AEIO propositions, and the advantages of adding them to an expanded square are explored. It is shown that the two AEIO-distinct inversions are inferentially, and as contraries, subcontraries, subalterns or contradictories, logically isolated from the AEIO propositions. From this it seems reasonable to conclude that no inversion makes a contribution to the subsyllogistic logical relations represented in an expanded version of the square. The investigation leaves the traditional Aristotelian square unchanged, but affords a deeper appreciation of the canonical square’s optimal two-dimensional diagramming of subsyllogistic logical relations among the four fixed AEIO propositions, and of why inversions of the AEIO propositions are appropriately excluded from the square. The limits and disadvantages of subsyllogistic logic are made conspicuous in the use of contemporary classical logic to translate and verify in algebraic terms all potential square of opposition relations, as a formal basis for excluding inversions, where they are otherwise manifestly excluded without explanation from the canonical square.


AEIO propositions AEIO inversions Canonical Aristotelian square of opposition Graphic equivalences Subsyllogistic logic Surveyability 

Mathematics Subject Classification

03A05 03B05 03B10 03B20 03B65 



I am grateful to an anonymous commentator for useful suggestions and criticisms in preparing the final version of this essay. Special thanks are due to my research assistant Sebastian Elliker at Universität Bern and Tina Marie Jacquette for digitalizing my hand-drawn diagrams.


  1. 1.
    Aristotle: The Complete Works of Aristotle. J. Barnes (ed.). Princeton University Press, Princeton (1984) Google Scholar
  2. 2.
    Bird, O.: Syllogistic and Its Extensions. Prentice-Hall, Englewood Cliffs (1964) Google Scholar
  3. 3.
    Boehner, P.: Medieval Logic. Manchester University Press, Manchester (1952) Google Scholar
  4. 4.
    Boethius, A.M.T.S.: Dialectica Aristotelis. Sebastian Gryphius, Lyons (1551) Google Scholar
  5. 5.
    Boethius, A.M.T.S.: Comentarii in librum Aristotelis (Peri Hermeneias [De Interpretatione]). Meiser, K.C. (ed.). B.G. Teubneri, Leipzig (1877) Google Scholar
  6. 6.
    Boole, G.: The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning. George Bell, London (1847) Google Scholar
  7. 7.
    Boole, G.: An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberley, London (1854) Google Scholar
  8. 8.
    Carroll, L.: Symbolic Logic [1896]. Bartley, W.W. III (ed.). Clarkson N. Potter, New York (1977) Google Scholar
  9. 9.
    Euler, L.: Elements of Algebra [1802]. Hewlett, J. (trans.). Springer, New York (1952) Google Scholar
  10. 10.
    Howard, D.T.: Analytical Syllogistics: A Pragmatic Interpretation of the Aristotelian Logic. Northwestern University Press, Evanston (1970) Google Scholar
  11. 11.
    Iwanus, B.: Remarks about syllogistic with negative terms. Stud. Log. 24, 131–137 (1969) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lukasiewicz, J.: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, 2nd edn., enlarged. Clarendon, Oxford (1955) Google Scholar
  13. 13.
    Jevons, W.S.: Elementary Lessons in Logic: Deductive and Inductive with Copious Questions and Examples and a Vocabulary of Logical Terms, 7th edn. Macmillan, London (1879) Google Scholar
  14. 14.
    Johnson, W.E.: Logic, 3 vols. Cambridge University Press, Cambridge (1921–1924) Google Scholar
  15. 15.
    Keynes, J.N.: Studies and Exercises in Formal Logic: Including a Generalisation of Logical Processes in Their Application to Complex Inferences. 4th edn., re-written and enlarged. Macmillan, London (1906) Google Scholar
  16. 16.
    Kneale, W., Kneale, M.: The Development of Logic. Oxford University Press, Oxford (1962) MATHGoogle Scholar
  17. 17.
    Lejewski, C.: Aristotle’s syllogistic and its extensions. Synthese 15, 125–154 (1963) MATHCrossRefGoogle Scholar
  18. 18.
    Marenbon, J.: Boethius. Oxford University Press, New York (2003). For notes: 17–65 CrossRefGoogle Scholar
  19. 19.
    Martin, C.J.: The logical textbooks and their influence. In: Marenbon, J. (ed.) The Cambridge Companion to Boethius, pp. 56–84. Cambridge University Press, Cambridge (2009) CrossRefGoogle Scholar
  20. 20.
    McLaughlin, R.N.: On the Logic of Ordinary Conditionals. SUNY, Albany (1990) MATHGoogle Scholar
  21. 21.
    Peterson, P.L.: Syllogistic Logic and the Grammar of Some English Quantifiers. Indiana University Press, Bloomington (1988) Google Scholar
  22. 22.
    Prior, A.N.: The logic of negative terms in Boethius. Francisc. Stud. 13, 1–6 (1953) Google Scholar
  23. 23.
    Prior, A.N.: Formal Logic, 2nd edn. Clarendon, Oxford (1962) MATHGoogle Scholar
  24. 24.
    Smith, R.: Logic. In: Barnes, J. (ed.) The Cambridge Companion to Aristotle, pp. 27–65. Cambridge University Press, Cambridge (1995) Google Scholar
  25. 25.
    Thom, P.: The Logic of Essentialism: An Interpretation of Aristotle’s Modal Syllogistic. Springer, Berlin (1996) Google Scholar
  26. 26.
    Venn, J.: Symbolic Logic [1894]. 2nd edn. Franklin Philosophy Monographs, New York (1971) MATHGoogle Scholar
  27. 27.
    Woods, J., Irvine, A.: Aristotle’s early logic. In: Gabbay, D., Woods, J. (eds.) Handbook of the History of Logic, vol. 1: Greek, Indian and Arabic Logic, pp. 27–100. Elsevier (North Holland), Amsterdam (2004) CrossRefGoogle Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

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