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Dialogical Connexive Logic

Abstract

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this context was called συναρτησις(connection). This theory gave a justification for the validity of what we todayexpress through the formulae ¬(a → ¬ a) and ¬(¬ a → a). Hugh MacColl and, more recently, Storrs McCall (from 1877 to 1906 and from1963 to 1975 respectively) searched for a formal system in which the validity ofthese formulae could be expressed. Unfortunately neither of the resulting systems is very satisfactory. In this paper we introduce dialogical games with the help of a new connexive If-Then (“→”), the structural rules of which allow the Proponent to develop (formal) winning strategies not only for the above-mentioned connexive theses but also for (a → b) → ¬(a → ¬ b) and (a → b) → ¬(¬ a → b). Further on, we developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form.

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REFERENCES

  1. Angell, R. B.: 1962, ‘A Propositional Logic with Subjunctive Conditionals’, Journal of Symbolic Logic 27, 327-343.

    Google Scholar 

  2. Aristotle: 1928, The Works of Aristotle Translated into English, vol. I, Oxford University Press, Oxford.

    Google Scholar 

  3. Astroh, M.: 1999, ‘Connexive Logic’, Nordic Journal of Philosophical Logic 4, 31-71.

    Google Scholar 

  4. Barth, E. M. and E. C. W. Krabbe: 1982, From Axiom to Dialogue. A Philosophical Study of Logics and Argumentation, de Gruyter, Berlin, New York.

    Google Scholar 

  5. Boethius, A. M. T. S.: 1969, De hypotheticis syllogismis, Paideia, Brescia.

    Google Scholar 

  6. Felscher, W.: 1985, ‘Dialogues, Strategies and Intuitionistic Provability’, Annals of Pure and Applied Logic 28, 217-254.

    Google Scholar 

  7. Gabbay, D. M.: 1987, Modal Provability Foundations for Negation by Failure, ESPRIT, Technical Report TI 8, Project 393, ACORD.

  8. Gardner, M.: 1996, The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations, Games, Puzzles and Word Plays, Copernicus (Springer-Verlag), New York.

    Google Scholar 

  9. Grice, H. P.: 1967, Conditionals. Privately Circulated Notes, University of California, Berkeley.

    Google Scholar 

  10. Grice, H. P.: 1989, Studies in the Way of Words, MIT-Press, Cambridge, MA.

    Google Scholar 

  11. Hoepelman, J. P. and A. J. M. van Hoof: 1988, ‘The Success of Failure’, Proceedings of COLING, Budapest, pp. 250-254.

  12. Krabbe, E. C. W.: 1985, ‘Formal Systems of Dialogue Rules’, Synthese 63, 295-328.

    Google Scholar 

  13. Lewy, C.: 1976, Meaning and Modality, Cambridge University Press, Cambridge, London, New York, Melbourne.

    Google Scholar 

  14. Linneweber-Lammerskitten, H.: 1988, Untersuchungen zur Theorie des hypothetischen Urteils, Nodus Publikationen, Cambridge, London, New York, Melbourne.

    Google Scholar 

  15. Lorenzen, P. and K. Lorenz: 1978, Dialogische Logik,Wissenschaftliche Buchgesellschaft, Darmstadt.

    Google Scholar 

  16. MacColl, H.: 1877a, ‘Symbolical or Abbreviated Language, with an Application to Mathematical Probability’, The Educational Times and Journal of the College of Preceptors 29, 91-92.

    Google Scholar 

  17. MacColl, H.: 1877b, ‘The Calculus of Equivalent Statements and Integration Limits’, Proceedings of the London Mathematical Society 9, 9-20.

    Google Scholar 

  18. MacColl, H.: 1878, ‘The Calculus of Equivalent Statements (II)’, Proceedings of the London Mathematical Society 9, 177-186.

    Google Scholar 

  19. MacColl, H.: 1880, ‘Symbolical Reasoning (I)’, Mind 5, 45-60.

    Google Scholar 

  20. MacColl, H.: 1906, Symbolic Logic and its Applications, Longmans, Green & Co, London, New York, Bombay.

    Google Scholar 

  21. McCall, S.: 1963, Aristotle's Modal Syllogisms, North-Holland, Amsterdam.

    Google Scholar 

  22. McCall, S.: 1964, ‘A New Variety of Implication’, Journal of Symbolic Logic 29, 151-152.

    Google Scholar 

  23. McCall, S.: 1966, ‘Connexive Implication’, Journal of Symbolic Logic 31, 415-432.

    Google Scholar 

  24. McCall, S.: 1967a, ‘Connexive Implication and the Syllogism’, Mind 76, 346-356.

    Google Scholar 

  25. McCall, S.: 1967b, ‘MacColl’, in P. Edwards (ed.): 1975, Encyclopedia of Philosophy, Macmillan, London. vol. IV, pp. 545-546.

    Google Scholar 

  26. McCall, S.: 1990, ‘Connexive Implication’, in A. R. Anderson and N. D. Belnap, Entailment I, Princeton University Press, Princeton, NJ, pp. 432-441.

    Google Scholar 

  27. Pizzi, C. and T. Williamson: 1997, ‘Strong Boethius' Thesis and Consequential Implication’, Journal of Philosophical Logic 26, 569-588.

    Google Scholar 

  28. Rahman, S.: 1993, Ñber Dialoge, protologische Kategorien und andere Seltenheiten, Peter Lang, Frankfurt a. M., Berlin, New York, Paris, Wien.

    Google Scholar 

  29. Rahman, S.: 1997, Die Logik der zusammenhängenden Behauptungen im frühen Werk von Hugh MacColl, “Habilitationsschrift”, to appear in Birkhäuser.

  30. Rahman, S.: 1998, Redundanz und Wahrheitswertbestimmung bei Hugh MacColl, FR 5.1 Philosophie, Universität des Saarlandes, Memo Nr. 23.

  31. Rahman, S.: 1999a, ‘Ways of Understanding Hugh MacColl's Concept of Symbolic Existence’, to appear in Nordic Journal of Philosophical Logic.

  32. Rahman, S.: 1999b, ‘On Frege's Nightmare. A Combination of Intuitionistic, Free and Paraconsistent Logics’, in H. Wansing (ed.), Essays on Non-Classical Logic, King's College University Press, London, to appear.

    Google Scholar 

  33. Rahman, S.: 1999c, ‘Fictions and Contradictions in the Symbolic Universe of Hugh MacColl’, in J. Mittelstraß (ed.), Die Zukunft desWissens, UVK, Konstanz, pp. 614-620.

    Google Scholar 

  34. Rahman, S.: 1999d, ‘Argumentieren mit Widersprüchen und Fiktionen’, in K. Buchholz, S. Rahman and I. Weber (eds.), Wege zur Vernunft–Philosophieren zwischen Tätigkeit und Reflexion, Campus, Frankfurt a. M., pp. 131-145.

    Google Scholar 

  35. Rahman, S. and W. Carnielli: 1998, The Dialogical Approach to Paraconsistency, FR 5.1 Philosophie, Universität des Saarlandes, Memo No. 8. Also to appear in D. Krause (ed.), Essays on Paraconsistent Logic, Kluwer, Dordrecht.

    Google Scholar 

  36. Rahman, S. and J. A. Roetti: 1999, ‘Dual Intuitionistic Paraconsistency without Ontological Commitments’, presented at the International Congress: Analytic Philosophy at the Turn of the Millennium in Santiago de Compostela (Spain), December 1999.

  37. Rahman, S. and H. Rückert: 1998, Dialogische Logik und Relevanz, FR 5.1 Philosophie, Universität des Saarlandes, Memo No. 27.

  38. Rahman, S. and H. Rückert: 1998-99, ‘Die pragmatischen Sinn-und Geltungskriterien der Dialogischen Logik beim Beweis des Adjunktionssatzes’, Philosophia Scientiae 3, 145-170.

    Google Scholar 

  39. Rahman, S. and H. Rückert: 1999, ‘Dialogische Modallogik (für T, B, S4 und S5)’, to appear in Logique et Analyse.

  40. Rahman, S., H. Rückert and M. Fischmann: 1999, ‘On Dialogues and Ontology. The Dialogical Approach to Free Logic’, to appear in Logique et Analyse.

  41. Read, S.: 1993, ‘Formal and Material Consequence, Disjunctive Syllogism and Gamma’, in Jacobi, K. (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, E. J. Brill, Leiden, New York, Köln.

    Google Scholar 

  42. Read, S.: 1994, Thinking About Logic, Oxford University Press, Oxford, New York.

    Google Scholar 

  43. Routley, R. and H. Montgomery.: 1968, ‘On Systems Containing Aristotle's Thesis’, The Journal of Symbolic Logic 3, 82-96.

    Google Scholar 

  44. Rückert, H.: 1999, ‘Why Dialogical Logic?’, in H. Wansing (ed.), Essays on Non-Classical Logic, King's College University Press, London, to appear.

    Google Scholar 

  45. Smullyan, R.: 1968, First Order Logic, Springer, Heidelberg.

    Google Scholar 

  46. Venn, J.: 1881, Symbolic Logic, Chelsea Publishing Company, New York.

    Google Scholar 

  47. Weingartner, P.: 1997, ‘Reasons for Filtering Classical Logic’, in D. Batens (ed.), Proceedings of the First World Congress on Paraconsistency, in print.

  48. Weingartner, P. and G. Schurz: 1986, ‘Paradoxes Solved by Simple Relevance Criteria’, Logique et Analyse 113, 3-40.

    Google Scholar 

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Rahman, S., Rückert, H. Dialogical Connexive Logic. Synthese 127, 105–139 (2001). https://doi.org/10.1023/A:1010351931769

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Keywords

  • Logical Form
  • Formal System
  • Structural Rule
  • Entailment Relation
  • Dialogical Approach