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Logic-Sensitivity and Bitstring Semantics in the Square of Opposition

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Abstract

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the precise details of their interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term ‘bitstring semantics’!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective.

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References

  1. Brown, M. (1984). Generalized quantifiers and the square of opposition. Notre Dame Journal of Formal Logic, 25, 303–322.

    Article  Google Scholar 

  2. Christensen, R. (2013). The logic of \(\Delta \). Thought, 2, 350–356.

    Article  Google Scholar 

  3. Cohn, A., Bennett, B., Gooday, J., & Gotts, N. M. (1997). Qualitative spatial representation and reasoning with the region connection calculus. GeoInformatica, 1, 275–316.

    Article  Google Scholar 

  4. Correia, M. (2012). Boethius on the square of opposition. In J.-Y. Béziau & D. Jacquette (Eds.), Around and Beyond the Square of Opposition (pp. 41–52). Basel: Springer.

    Chapter  Google Scholar 

  5. Correia, M. (2017). Aristotle’s squares of opposition. South American Journal of Logic, 3, 313–326.

    Google Scholar 

  6. Correia, M. (2017). Logic in apuleius and boethius. Revista Portuguesa de Filosofia, 73, 1035–1052.

    Article  Google Scholar 

  7. Demey, L. (2012). Structures of oppositions for public announcement logic. In J.-Y. Béziau & D. Jacquette (Eds.), Around and Beyond the Square of Opposition (pp. 313–339). Basel: Springer.

    Chapter  Google Scholar 

  8. Demey, L. (2015). Interactively illustrating the context-sensitivity of Aristotelian diagrams. LNCS 9405In H. Christiansen, I. Stojanovic, & G. Papadopoulos (Eds.), Modeling and Using Context (pp. 331–345). Berlin/Heidelberg: Springer.

    Chapter  Google Scholar 

  9. Demey, L. (2018). Computing the maximal Boolean complexity of families of Aristotelian diagrams. Journal of Logic and Computation, 28, 1323–1339.

    Article  Google Scholar 

  10. Demey, L. (2019). Aristotelian diagrams in the debate on future contingents. Sophia, 58, 321–329.

    Article  Google Scholar 

  11. Demey, L. (2019). Boolean considerations on John Buridan’s octagons of oppositions. History and Philosophy of Logic, 40, 116–134.

    Article  Google Scholar 

  12. Demey, L. (2019). Metalogic, metalanguage and logical geometry. Logique et Analyse, 248, 453–478.

    Google Scholar 

  13. Demey, L. (2021). Logic-sensitivity of Aristotelian diagrams in non-normal modal logics. Axioms, 10(128), 1–25.

    Google Scholar 

  14. Demey, L., & Smessaert, H. (2017). Logical and geometrical distance in polyhedral Aristotelian diagrams in knowledge representation. Symmetry, 9(10), 204.

    Article  Google Scholar 

  15. Demey, L., & Smessaert, H. (2018). Combinatorial bitstring semantics for arbitrary logical fragments. Journal of Philosophical Logic, 47, 325–363.

    Article  Google Scholar 

  16. Falcão, P. (2022). Visualizing polymorphisms and counter-polymorphisms in S5 modal logic. LNCS 13462In V. Giardino, S. Linker, R. Burns, F. Bellucci, J.-M. Boucheix, & P. Viana (Eds.), Diagrammatic Representation and Inference (pp. 296–311). Berlin/Heidelberg: Springer.

    Chapter  Google Scholar 

  17. García-Cruz, J. D. (2017). Aristotelian relations in PDL: The hypercube of dynamic oppositions. South American Journal of Logic, 3, 389–414.

    Google Scholar 

  18. Geudens, C., & Demey, L. (2021). On the Aristotelian roots of the modal square of opposition. Logique et Analyse, 255, 313–348.

    Google Scholar 

  19. Givant, S., & Halmos, P. (2009). Introduction to Boolean Algebras. New York, NY: Springer.

    Google Scholar 

  20. Gombocz, W. L. (1990). Apuleius is better still: a correction to the square of opposition. Phronesis, 43, 124–131.

    Google Scholar 

  21. Klement, K. (2019). New logic and the seeds of analytic philosophy. Boole, Frege. In J. A. Shand (ed.), A Companion to Nineteenth-Century Philosophy, pp. 454–479. Hoboken, NJ: Wiley-Blackwell.

  22. Lemaire, J. (2017). Is Aristotle the father of the square of opposition? In J.-Y. Béziau & S. Gerogiorgakis (Eds.), New Dimensions of the Square of Opposition (pp. 33–69). Munich: Philosophia Verlag.

    Chapter  Google Scholar 

  23. Lemanski, J., & Schang, F. (2022). A bitstring semantics for calculus CL. In J.-Y. Beziau & I. Vandoulakis (Eds.), The Exoteric Square of Opposition (pp. 171–193). Cham: Springer.

    Chapter  Google Scholar 

  24. Londey, D., & Johanson, C. (1984). Apuleius and the square of opposition. Phronesis, 29, 165–173.

    Article  Google Scholar 

  25. Londey, D., & Johanson, C. (1987). The Logic of Apuleius. Leiden: Brill.

    Book  Google Scholar 

  26. Lutz, C., & Wolter, F. (2006). Modal logics of topological relations. Logical Methods in Computer Science, 2, 1–41.

    Google Scholar 

  27. Moktefi A., & Schang F. (2023). Another side of categorical propositions: The Keynes-Johnson octagon of oppositions. History and Philosophy of Logic, pp. 1–17, forthcoming. https://www.tandfonline.com/doi/full/10.1080/01445340.2022.2143711.

  28. Nelson, E. J. (1932). The square of opposition. The Monist, 42, 269–278.

    Article  Google Scholar 

  29. Orenstein, A. (2015). Geach, Aristotle and predicate logics. Philosophical Investigations, 38, 96–114.

    Article  Google Scholar 

  30. Parsons, T. (2017). The traditional square of opposition. In E. N. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford, CA: CSLI.

    Google Scholar 

  31. Pizzi, C. (2016). Generalization and composition of modal squares of opposition. Logica Universalis, 10, 313–325.

    Article  Google Scholar 

  32. Pizzi, C. (2017). Contingency logics and modal squares of opposition. In J.-Y. Béziau & S. Gerogiorgakis (Eds.), New Dimensions of the Square of Opposition (pp. 201–220). Munich: Philosophia Verlag.

    Chapter  Google Scholar 

  33. Pozzi, L. (1974). Studi di Logica Antica e Medioevale. Padova: Liviana Editrice.

    Google Scholar 

  34. Roelandt, K. (2016). The Meaning of Most. Proportional and Comparative Interpretations in Dutch. Utrecht: LOT Publications.

    Google Scholar 

  35. Simons, P. (2004). Judging correctly: Brentano and the reform of elementary logic. In D. Jacquette (Ed.), The Cambridge Compansion to Brentano (pp. 45–65). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  36. Smessaert, H., & Demey, L. (2014). Logical and geometrical complementarities between Aristotelian diagrams. LNCS 8578In T. Dwyer, H. Purchase, & A. Delaney (Eds.), Diagrammatic Representation and Inference (pp. 246–260). Berlin/Heidelberg: Springer.

    Chapter  Google Scholar 

  37. Smessaert, H., & Demey, L. (2014). Logical geometries and information in the square of opposition. Journal of Logic, Language and Information, 23, 527–565.

    Article  Google Scholar 

  38. Smessaert, H., & Demey, L. (2017). The unreasonable effectiveness of bitstrings in logical geometry. In J.-Y. Béziau and G. Basti (eds.), The Square of Opposition: A Cornerstone of Thought, pp. 197–214. Basel: Springer.

  39. Smessaert, H., & Demey, L. (2022). On the logical geometry of geometric angles. Logica Universalis, 16, 581–601.

    Article  Google Scholar 

  40. Sullivan, M. W. (1967). Apuleian Logic. The Nature, Sources, and Influence of Apuleius’s Peri hermeneias. Amsterdam: North-Holland.

    Google Scholar 

  41. Vignero, L. (2021). Combining and relating Aristotelian diagrams. LNCS 12909In A. Basu, G. Stapleton, S. Linker, C. Legg, E. Manalo, & P. Viana (Eds.), Diagrammatic Representation and Inference (pp. 221–228). Berlin/Heidelberg: Springer.

    Chapter  Google Scholar 

  42. Wolter, F., & Zakharyaschev, M. (2002). Qualitative spatio-temporal representation and reasoning: a computational perspective. In G. Lakemeyer & B. Nebel (Eds.), Exploring Artificial Intelligence in the New Millenium (pp. 175–216). San Francisco, CA: Morgan Kauffman.

    Google Scholar 

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Acknowledgements

This research was funded by the ID-N project BITSHARE: Bitstring Semantics for Human and Artificial Reasoning (IDN-19-009, Internal Funds, KU Leuven). The first author holds a research professorship (BOFZAP) at KU Leuven. Both authors would like to thank Alex De Klerck, Hans Smessaert and two anonymous reviewers for their feedback on an earlier version of this paper.

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Authors and Affiliations

  1. Center for Logic and Philosophy of Science, KU Leuven, Leuven, Belgium

    Lorenz Demey & Stef Frijters

  2. KU Leuven Institute for Artificial Intelligence, Leuven, Belgium

    Lorenz Demey

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  1. Lorenz Demey
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  2. Stef Frijters
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Contributions

Both authors contributed to the conceptualization of this paper. The open problem and the two partial solutions (as reported in Sections 4 – 5) are due to Lorenz Demey, while the two full solutions (as reported in Section 5) are due to Stef Frijters. The first complete draft of the manuscript was written by Lorenz Demey. Stef Frijters delivered extensive feedback on several versions of the manuscript. Both authors read and approved the final manuscript.

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Correspondence to Lorenz Demey.

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Demey, L., Frijters, S. Logic-Sensitivity and Bitstring Semantics in the Square of Opposition. J Philos Logic 52, 1703–1721 (2023). https://doi.org/10.1007/s10992-023-09723-6

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