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Aristotelian diagrams for semantic and syntactic consequence

  • Lorenz Demey
Article
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Abstract

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.

Keywords

Metalogic Semantic consequence Syntactic consequence Square of opposition Aristotelian diagram Logical geometry 

Notes

Acknowledgements

Thanks to Hans Smessaert, Margaux Smets and three anonymous reviewers for their valuable feedback on an earlier version of this paper. The author holds a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO). The research reported in this paper was partially carried out during a research stay at the Institut für Philosophie II of the Ruhr-Universität Bochum, which was financially supported by an FWO travel grant.

References

  1. Beall, J. C., Brady, R. T., Hazen, A. P., Priest, G., & Restall, G. (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 587–598.CrossRefGoogle Scholar
  2. Béziau, J.-Y. (2012). The power of the hexagon. Logica Universalis, 6, 1–43.CrossRefGoogle Scholar
  3. Béziau, J.-Y. (2013). The metalogical hexagon of opposition. Argumentos, 5, 111–122.Google Scholar
  4. Béziau, J.-Y. (2016). Disentangling contradiction from contrariety via incompatibility. Logica Universalis, 10, 157–170.CrossRefGoogle Scholar
  5. Béziau, J.-Y., & Payette, G. (2012). Preface. In J.-Y. Béziau & G. Payette (Eds.), The square of opposition. A general framework for cognition (pp. 9–22). Bern: Peter Lang.Google Scholar
  6. Brown, B. (2015). Stipulation and symmetrical consequence. In J.-Y. Béziau, M. Chakraborty, & S. Dutta (Eds.), New directions in paraconsistent logic (pp. 335–352). Berlin: Springer.CrossRefGoogle Scholar
  7. Chisholm, R. (1963). Supererogation and offence: A conceptual scheme for ethics. Ratio, 5, 1–14.Google Scholar
  8. Ciucci, D., Dubois, D., & Prade, H. (2014). The structure of opposition in rough set theory and formal concept analysis. Toward a bridge between the two settings. In C. Beierle & C. Meghini (Eds.), Foundations of information and knowledge systems (FoIKS 2014) (pp. 154–173). Berlin: Springer.CrossRefGoogle Scholar
  9. Ciucci, D., Dubois, D., & Prade, H. (2016). Structures of opposition induced by relations. The boolean and the gradual cases. Annals of Mathematics and Artificial Intelligence, 76, 351–373.CrossRefGoogle Scholar
  10. Demey, L. (2012). Structures of oppositions in public announcement logic. In J.-Y. Béziau & D. Jacquette (Eds.), Around and beyond the square of opposition (pp. 313–339). Berlin: Springer.CrossRefGoogle Scholar
  11. Demey, L. (2015). Interactively illustrating the context-sensitivity of aristotelian diagrams. In H. Christiansen, I. Stojanovic, & G. Papadopoulous (Eds.), Modeling and using context. Lecture Notes in Computer Science (Vol. 9405, pp. 331–345). Berlin: Springer.CrossRefGoogle Scholar
  12. Demey, L. (2017a). Using syllogistics to teach metalogic. Metaphilosophy, 48, 575–590.CrossRefGoogle Scholar
  13. Demey, L. (2017b). The logical geometry of Russell’s theory of definite descriptions. Unpublished manuscript.Google Scholar
  14. Demey, L. (2018a). Aristotelian diagrams in the debate on future contingents. Sophia.  https://doi.org/10.1007/s11841-017-0632-7 Google Scholar
  15. Demey, L. (2018b). Metalogic, metalanguage, and logical geometry. Logique et Analyse.Google Scholar
  16. Demey, L. (2018c). The role of Aristotelian diagrams in scientific communication. Talk delivered at the 8th Visual Learning Conference. Budapest.Google Scholar
  17. Demey, L., & Smessaert, H. (2014). Logische geometrie en pragmatiek. In F. Van de Velde, H. Smessaert, F. Van Eynde, & S. Verbrugge (Eds.), Patroon en argument (pp. 553–564). Leuven: Leuven University Press.CrossRefGoogle Scholar
  18. Demey, L., & Smessaert, H. (2016). Metalogical decorations of logical diagrams. Logica Universalis, 10, 233–292.CrossRefGoogle Scholar
  19. Demey, L., & Smessaert, H. (2018). Combinatorial bitstring semantics for arbitrary logical fragments. Journal of Philosophical Logic, 47, 325–363.CrossRefGoogle Scholar
  20. Diaconescu, R. (2015). The algebra of opposition (and universal logic interpretations). In A. Koslow & A. Buchsbaum (Eds.), The road to universal logic (pp. 127–143). Berlin: Springer.Google Scholar
  21. Dubois, D., Prade, H., & Rico, A. (2015). The cube of opposition. A structure underlying many knowledge representation formalisms. In Q. Yang & M. Wooldridge (Eds.), Proceedings of the twenty-fourth international joint conference on artificial intelligence (IJCAI 2015) (pp. 2933–2939). New York: AAAI Press.Google Scholar
  22. Hare, R. M. (1967). Some alleged differences between imperatives and indicatives. Mind, 76, 309–326.CrossRefGoogle Scholar
  23. Hart, H. L. A. (1982). Essays on Bentham. Jurisprudence and political theory. Oxford: Clarendon Press.CrossRefGoogle Scholar
  24. Hess, E. (2017). The open future square of opposition: A defense. Sophia, 56, 573–587.CrossRefGoogle Scholar
  25. Kenny, A. (1963). Action, emotion and will. London: Routledge & Kegan Paul.Google Scholar
  26. Kienzler, W. (2013). The logical square and the table of oppositions. Five puzzles about the traditional square of opposition solved by taking up a hint from Frege. Logical Analysis and History of Philosophy, 15, 398–413.Google Scholar
  27. Lenzen, W. (2012). How to square knowledge and belief. In J.-Y. Béziau & D. Jacquette (Eds.), Around and beyond the square of opposition (pp. 305–311). Berlin: Springer.CrossRefGoogle Scholar
  28. Lenzen, W. (2016). Leibniz’s logic and the “cube of opposition”. Logica Universalis, 10, 171–189.CrossRefGoogle Scholar
  29. Lenzen, W. (2017). Caramuel’s theory of opposition. South American Journal of Logic, 3, 1–27.Google Scholar
  30. Marcus, R. B. (1966). Iterated deontic modalities. Mind, 75, 580–582.CrossRefGoogle Scholar
  31. Parsons, T. (2017). The traditional square of opposition. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy (Summer 2017 Edition). Stanford, CA: CSLI.Google Scholar
  32. Pfeifer, N., & Sanfilippo, G. (2017). Probabilistic squares and hexagons of opposition under coherence. International Journal of Approximate Reasoning, 88, 282–294.CrossRefGoogle Scholar
  33. Rini, A., & Cresswell, M. (2012). The world-time parallel. Tense and modality in logic and metaphysics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  34. Scott, D. (1971). On engendering an illusion of understanding. Journal of Philosophy, 68, 787–807.CrossRefGoogle Scholar
  35. Scott, D. (1974). Completeness and axiomatizability in many-valued logic. In L. Henkin (Ed.), Proceedings of the Tarski symposium (pp. 411–435). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
  36. Seuren, P. (2014). Metalogical hexagons in natural logic. Unpublished manuscript.Google Scholar
  37. Smessaert, H., & Demey, L. (2014). Logical geometries and information in the square of oppositions. Journal of Logic, Language and Information, 23, 527–565.CrossRefGoogle Scholar
  38. Sosa, E. (1964). The analysis of ‘knowledge that P’. Analysis, 25, 1–8.CrossRefGoogle Scholar
  39. Vranes, E. (2006). The definition of ‘norm conflict’ in international law and legal theory. European Journal of International Law, 17, 395–418.CrossRefGoogle Scholar
  40. Yao, Y. (2013). Duality in rough set theory based on the square of opposition. Fundamenta Informaticae, 127, 49–64.Google Scholar
  41. Ziegeler, D. (2017). On the empty O-corner of the aristotelian square: A view from singapore english. Journal of Pragmatics, 115, 1–20.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of PhilosophyKU LeuvenLeuvenBelgium

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