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
Angell, R. B.: 1962, ‘A Propositional Logic with Subjunctive Conditionals’, Journal of Symbolic Logic 27, 327-343.
Aristotle: 1928, The Works of Aristotle Translated into English, vol. I, Oxford University Press, Oxford.
Astroh, M.: 1999, ‘Connexive Logic’, Nordic Journal of Philosophical Logic 4, 31-71.
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.
Boethius, A. M. T. S.: 1969, De hypotheticis syllogismis, Paideia, Brescia.
Felscher, W.: 1985, ‘Dialogues, Strategies and Intuitionistic Provability’, Annals of Pure and Applied Logic 28, 217-254.
Gabbay, D. M.: 1987, Modal Provability Foundations for Negation by Failure, ESPRIT, Technical Report TI 8, Project 393, ACORD.
Gardner, M.: 1996, The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations, Games, Puzzles and Word Plays, Copernicus (Springer-Verlag), New York.
Grice, H. P.: 1967, Conditionals. Privately Circulated Notes, University of California, Berkeley.
Grice, H. P.: 1989, Studies in the Way of Words, MIT-Press, Cambridge, MA.
Hoepelman, J. P. and A. J. M. van Hoof: 1988, ‘The Success of Failure’, Proceedings of COLING, Budapest, pp. 250-254.
Krabbe, E. C. W.: 1985, ‘Formal Systems of Dialogue Rules’, Synthese 63, 295-328.
Lewy, C.: 1976, Meaning and Modality, Cambridge University Press, Cambridge, London, New York, Melbourne.
Linneweber-Lammerskitten, H.: 1988, Untersuchungen zur Theorie des hypothetischen Urteils, Nodus Publikationen, Cambridge, London, New York, Melbourne.
Lorenzen, P. and K. Lorenz: 1978, Dialogische Logik,Wissenschaftliche Buchgesellschaft, Darmstadt.
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.
MacColl, H.: 1877b, ‘The Calculus of Equivalent Statements and Integration Limits’, Proceedings of the London Mathematical Society 9, 9-20.
MacColl, H.: 1878, ‘The Calculus of Equivalent Statements (II)’, Proceedings of the London Mathematical Society 9, 177-186.
MacColl, H.: 1880, ‘Symbolical Reasoning (I)’, Mind 5, 45-60.
MacColl, H.: 1906, Symbolic Logic and its Applications, Longmans, Green & Co, London, New York, Bombay.
McCall, S.: 1963, Aristotle's Modal Syllogisms, North-Holland, Amsterdam.
McCall, S.: 1964, ‘A New Variety of Implication’, Journal of Symbolic Logic 29, 151-152.
McCall, S.: 1966, ‘Connexive Implication’, Journal of Symbolic Logic 31, 415-432.
McCall, S.: 1967a, ‘Connexive Implication and the Syllogism’, Mind 76, 346-356.
McCall, S.: 1967b, ‘MacColl’, in P. Edwards (ed.): 1975, Encyclopedia of Philosophy, Macmillan, London. vol. IV, pp. 545-546.
McCall, S.: 1990, ‘Connexive Implication’, in A. R. Anderson and N. D. Belnap, Entailment I, Princeton University Press, Princeton, NJ, pp. 432-441.
Pizzi, C. and T. Williamson: 1997, ‘Strong Boethius' Thesis and Consequential Implication’, Journal of Philosophical Logic 26, 569-588.
Rahman, S.: 1993, Ñber Dialoge, protologische Kategorien und andere Seltenheiten, Peter Lang, Frankfurt a. M., Berlin, New York, Paris, Wien.
Rahman, S.: 1997, Die Logik der zusammenhängenden Behauptungen im frühen Werk von Hugh MacColl, “Habilitationsschrift”, to appear in Birkhäuser.
Rahman, S.: 1998, Redundanz und Wahrheitswertbestimmung bei Hugh MacColl, FR 5.1 Philosophie, Universität des Saarlandes, Memo Nr. 23.
Rahman, S.: 1999a, ‘Ways of Understanding Hugh MacColl's Concept of Symbolic Existence’, to appear in Nordic Journal of Philosophical Logic.
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.
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.
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.
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.
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.
Rahman, S. and H. Rückert: 1998, Dialogische Logik und Relevanz, FR 5.1 Philosophie, Universität des Saarlandes, Memo No. 27.
Rahman, S. and H. Rückert: 1998-99, ‘Die pragmatischen Sinn-und Geltungskriterien der Dialogischen Logik beim Beweis des Adjunktionssatzes’, Philosophia Scientiae 3, 145-170.
Rahman, S. and H. Rückert: 1999, ‘Dialogische Modallogik (für T, B, S4 und S5)’, to appear in Logique et Analyse.
Rahman, S., H. Rückert and M. Fischmann: 1999, ‘On Dialogues and Ontology. The Dialogical Approach to Free Logic’, to appear in Logique et Analyse.
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.
Read, S.: 1994, Thinking About Logic, Oxford University Press, Oxford, New York.
Routley, R. and H. Montgomery.: 1968, ‘On Systems Containing Aristotle's Thesis’, The Journal of Symbolic Logic 3, 82-96.
Rückert, H.: 1999, ‘Why Dialogical Logic?’, in H. Wansing (ed.), Essays on Non-Classical Logic, King's College University Press, London, to appear.
Smullyan, R.: 1968, First Order Logic, Springer, Heidelberg.
Venn, J.: 1881, Symbolic Logic, Chelsea Publishing Company, New York.
Weingartner, P.: 1997, ‘Reasons for Filtering Classical Logic’, in D. Batens (ed.), Proceedings of the First World Congress on Paraconsistency, in print.
Weingartner, P. and G. Schurz: 1986, ‘Paradoxes Solved by Simple Relevance Criteria’, Logique et Analyse 113, 3-40.
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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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DOI: https://doi.org/10.1023/A:1010351931769