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Metalogical Decorations of Logical Diagrams

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Abstract

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams (viz. Aristotelian and duality diagrams) and four kinds of metalogical decorations (viz. those based on the opposition, implication, Aristotelian and duality relations).

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References

  1. Béziau J.-Y.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–232 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Béziau, J.-Y.: Bivalent semantics for De Morgan logic (the uselessness of four-valuedness). In: Carnielli, W., Coniglio, M., D’Ottaviano, I. (eds.) The Many Sides of Logic, pp. 391–402. College Publications, London (2009)

  3. Béziau, J.-Y.: Paralogics and the theory of valuation. In: Carnielli, W., Coniglio, M., D’Ottaviano, I. (eds.) Universal Logic: An Anthology—From Paul Hertz to Dov Gabbay, pp. 361–372. Springer, Basel (2012)

  4. Béziau J.-Y.: The power of the hexagon. Log. Univers. 6, 1–43 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Béziau J.-Y.: The metalogical hexagon of opposition. Argumentos 5, 111–122 (2013)

    Google Scholar 

  6. Béziau, J.-Y.: Opposition and order. In: Béziau J.-Y., Gan-Krzywoszynska, K. (eds.) New Dimensions of the Square of Opposition, pp. 321–336. Philosophia Verlag, Munich (2014)

  7. Béziau, J.-Y.: Paraconsistent logic and contradictory viewpoints. Rev. Bras. Filos. (forthcoming)

  8. Béziau, J.-Y., Jacquette, D. (eds.): Around and Beyond the Square of Opposition. Springer, Basel (2012)

  9. Béziau, J.-Y., Payette, G. (eds.): The Square of Opposition. A General Framework for Cognition. Peter Lang, Bern (2012)

  10. Blanché R.: Quantity, modality, and other kindred systems of categories. Mind 61, 369–375 (1952)

    Article  Google Scholar 

  11. Blanché R.: Sur l’Opposition des Concepts. Theoria 19, 89–130 (1953)

    Article  Google Scholar 

  12. Blanché R.: Opposition et Négation. Revue Philosophique de France et de l’Étranger 147, 187–216 (1957)

    Google Scholar 

  13. Blanché, R.: Structures Intellectuelles: Essai sur l’Organisation Systématique des Concepts. Vrin, Paris (1966)

  14. Bochenski, J.: A Precis of Mathematical Logic. Reidel, Dordrecht (1959)

  15. Copenhaver, B., Normore, C., Parsons, T.: Peter of Spain: Summaries of Logic. Text, Translation, Introduction, and Notes. Oxford University Press, Oxford (2014)

  16. Chatti, S.: Logical Oppositions in Arabic logic: avicenna and averroes. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 21–40. Springer, Basel (2012)

  17. Chatti S.: Avicenna on possibility and necessity. Hist. Philos. Log. 35, 332–353 (2014)

    Article  MathSciNet  Google Scholar 

  18. Chellas, B.: Modal Logic. An Introduction. Cambridge University Press, Cambridge (1980)

  19. Cze zowski T.: On certain peculiarities of singular propositions. Mind 64, 392–395 (1955)

    Article  Google Scholar 

  20. D’Alfonso, D.: The square of opposition and generalized quantifiers. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 219–227. Springer, Basel (2012)

  21. Demey, L.: Algebraic aspects of duality diagrams. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrammatic Representation and Inference, LNCS 7352, pp. 300–302. Springer, Berlin (2012)

  22. 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)

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

  24. Demey, L.: Using syllogistics to teach metalogic (submitted, 2015)

  25. Demey, L.: Partitioning logical space. Manuscript (2015)

  26. Demey, L.: Dependence and independence in logical geometry. Manuscript (2016)

  27. Demey, L., Smessaert, H.: Logische geometrie en pragmatiek. In: Van de Velde, F., Smessaert, H., Van Eynde, F., Verbrugge, S. (eds.) Patroon en argument, pp. 553–564. Leuven University Press, Leuven (2014)

  28. Demey, L., Smessaert, H.: The relationship between Aristotelian and Hasse diagrams. In: Dwyer, T., Purchase, H., Delaney, A. (eds.) Diagrammatic Representation and Inference, LNCS, vol. 8578, pp. 213–227. Springer, Berlin (2014)

  29. Demey, L., Smessaert, H.: Duality in logic and language. In: Béziau, J.-Y. (ed.) Encyclopedia of Logic. College Publications, London (2015)

  30. Demey, L., Smessaert, H.: Combinatorial bitstring semantics for arbitrary logical fragments (submitted, 2015)

  31. Demey, L., Smessaert, H.: Generating the logical relations. Manuscript (2015)

  32. Demey, L., Smessaert, H.: The logical geometry of the Aristotelian rhombic dodecahedron. Manuscript (2015)

  33. Diaconescu, R.: The algebra of opposition (and universal logic interpretations). In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic, pp. 127–143. Springer, Basel (2015)

  34. Diestel, R.: Graph Theory, 4th edn. Springer, Berlin (2010)

  35. Fitting, M., Mendelsohn, R.: First-Order Modal Logic. Kluwer, Dordrecht (1998)

  36. Gottschalk W.H.: The theory of quaternality. J. Symb. Log. 18, 193–196 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hacker E.: The octagon of opposition. Notre Dame J. Form. Log. 16, 352–353 (1975)

    Article  MATH  Google Scholar 

  38. Horn, L.: A Natural History of Negation. University of Chicago Press, Chicago, IL (1989)

  39. Horn, L.: Histoire d’*O: Lexical pragmatics and the geometry of opposition. In: Béziau, J.-Y., Payette, G. (eds.) The Square of Opposition. A General Framework for Cognition, pp. 393–426. Peter Lang, Bern (2012)

  40. Hughes, G.: The modal logic of John Buridan. In: Corsi, G., Mangione, C., Mugnai, M. (eds.) Atti del convegno internazionale di storia della logica, le teorie delle modalità, pp. 93–11. CLUEB, Bologna (1987)

  41. Humberstone, L.: The Connectives. MIT Press, Cambridge, MA (2011)

  42. Humberstone L.: Logical relations. Philos. Perspect. 27, 175–230 (2013)

    Article  MathSciNet  Google Scholar 

  43. Jacoby P.: A triangle of opposites for types of propositions in Aristotelian logic. New Scholast. 24, 32–56 (2012)

    Article  Google Scholar 

  44. Jaspers D.: Logic and colour. Log. Univers. 6, 227–248 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Johnson, W.E.: Logic: Part I. Cambridge University Press, London (1921)

  46. Katzir R., Singh R.: Constraints on the lexicalization of logical operators. Linguist. Philos. 36, 1–29 (2013)

    Article  Google Scholar 

  47. Keynes, J.N.: Studies and Exercises in Formal Logic. MacMillan, London (1884)

  48. Khomskii, Y.: William of Sherwood, singular propositions and the hexagon of opposition. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 43–60. Springer, Basel (2012)

  49. Kretzmann, N.: William of Sherwood’s Introduction to Logic. University of Minnesota Press, Minneapolis (1966)

  50. Lenzen, W.: How to square knowledge and belief. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 305–311. Springer, Basel (2012)

  51. Libert, T.: Hypercubes of duality. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 293–301. Springer, Basel (2012)

  52. Löbner, S.: Quantification as a major module of natural language semantics. In: Groenendijk, J., de Jongh, D., Stokhof, M. (eds.) Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers, pp. 53–85. Foris, Dordrecht (1987)

  53. Löbner, S.: Wahr neben Falsch. Duale Operatoren als die Quantoren natürlicher Sprache. Max Niemeyer Verlag, Tübingen (1990)

  54. Löbner, S.: Dual oppositions in lexical meaning. In: Maienborn, C., von Heusinger, K., Portner, P. (eds.) Semantics: An International Handbook of Natural Language Meaning, vol. I, pp. 479–506. De Gruyter Mouton, Berlin (2011)

  55. Luzeaux D., Sallantin J., Dartnell C.: Logical Extensions of Aristotle’s Square. Log. Univers. 2, 167–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. McNamara, J.: In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (Winter 2014 Edition). CSLI, Stanford (2014)

  57. McCall S.: Contrariety. Notre Dame J. Form. Log. 8, 121–132 (1967)

    Article  MATH  Google Scholar 

  58. Mélès, B.: No group of opposition for constructive logics: the intuitionistic and linear cases. In: Béziau J.-Y., Jacquette, D. (eds.) Around and Beyond the Square of Opposition, pp. 201–217. Springer, Basel (2012)

  59. Moretti, A.: The Geometry of Logical Opposition. PhD thesis, University of Neuchâtel (2009)

  60. Moretti A.: Why the logical hexagon? Log. Univers. 6, 69–107 (2012)

    MathSciNet  MATH  Google Scholar 

  61. Moretti, A.: Arrow-Hexagons. In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic, vol. II, pp. 417–487. Springer, Basel (2015)

  62. Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (Winter 2014 Edition). CSLI, Stanford (2012)

  63. Peckhaus, V.: Algebra of logic, quantification theory, and the square of opposition. In: Béziau J.-Y., Payette, G. (eds.) The Square of Opposition. A General Framework for Cognition, pp. 25–41. Peter Lang, Bern (2012)

  64. Pellissier R.: “Setting” n-opposition. Log. Univers. 2, 235–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Oxford University Press, Oxford (2006)

  66. Piaget, J.: Traité de Logique. Essai de Logistique Opératoire. Colin/Dunod, Paris (1949)

  67. 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)

  68. Rini, A., Cresswell, M.: The World-Time Parallel. Tense and Modality in Logic and Metaphyiscs. Cambridge University Press, Cambridge (2012)

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

    Article  Google Scholar 

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

  71. Seuren, P.: The Logic of Language. Language from Within, Volume II. Oxford University Press, Oxford (2010)

  72. Seuren, P.: The metalogical hexagon. Manuscript (2014)

  73. Seuren P., Jaspers D.: Logico-cognitive structure in the lexicon. Language 90, 607–643 (2014)

    Article  Google Scholar 

  74. Smessaert H.: On the 3D visualisation of logical relations. Log. Univers. 3, 303–332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  75. Smessaert, H.: Boolean differences between two hexagonal extensions of the logical square of oppositions. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrammatic Representation and Inference, LNCS vol. 7352, pp. 193–199. Springer, Berlin (2012)

  76. Smessaert, H.: Duality and reversibility: squares versus crosses; Part 1: the logical geometry of set inclusion. Paper read at the Workshop on Logical Geometry and N-Opposition Theory, Leuven (2012)

  77. Smessaert H.: The classical Aristotelian hexagon versus the modern duality hexagon. Log. Univers. 6, 171–199 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

  79. Smessaert H., Demey L.: Logical geometries and information in the square of oppositions. J. Log. Lang. Inf. 23, 527–565 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  80. Smessaert, H., Demey, L.: Béziau’s contributions to the logical geometry of modalities and quantifiers. In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic, vol. I, pp. 475–494. Springer, Basel (2015)

  81. Smessaert, H., Demey, L.: La géométrie logique du dodécaèdre rhombique des oppositions. In: Chatti, S. (ed.) Le Carré et ses Extensions: Approches Théoriques, Pratiques et Historiques, pp. 127–157. Université de Tunis, Tunis (2015)

  82. Smessaert, H., Demey, L.: The unreasonable effectiveness of bitstrings in logical geometry (submitted, 2015)

  83. van Benthem, J.: Linguistic universals in logical semantics. In: Zaefferer, D. (ed.) Semantic Universals and Universal Semantics, pp. 17–36. Foris, Berlin (1991)

  84. Westerståhl, D.: Classical vs. modern squares of opposition, and beyond. In: Béziau, J.-Y., Payette, G. (eds.) The Square of Opposition. A General Framework for Cognition, pp. 195–229. Peter Lang, Bern (2012)

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

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Authors and Affiliations

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

    Lorenz Demey

  2. Department of Linguistics, KU Leuven, Leuven, Belgium

    Hans Smessaert

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  1. Lorenz Demey
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  2. Hans Smessaert
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Correspondence to Lorenz Demey.

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Parts of this paper were presented at CLMPS 15 (Helsinki). We would like to thank the audience of that talk, as well as Jean-Yves Béziau, Alessio Moretti, Frédéric Sart, Fabien Schang, Margaux Smets and three anonymous reviewers for their valuable feedback. The first author holds a postdoctoral scholarship of the Research Foundation—Flanders (FWO).

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Demey, L., Smessaert, H. Metalogical Decorations of Logical Diagrams. Log. Univers. 10, 233–292 (2016). https://doi.org/10.1007/s11787-015-0136-6

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