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On Reduction Rules, Meaning-as-use, and Proof-theoretic Semantics

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Abstract

The intention here is that of giving a formal underpinning to the idea of ‘meaning-is-use’ which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett–Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i.e. the so-called reduction rules. For that we suggest an extension to the Curry– Howard interpretation which draws on the idea of labelled deduction, and brings back Frege’s device of variable-abstraction to operate on the labels (i.e., proof-terms) alongside formulas of predicate logic.

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References

  1. Contu, Patrizio, ‘La Filosofia della Logica in Dummett e Prawitz’, Tesi di Laurea, Dipartimento di Filosofia, Università degli Studi di Firenze, 1994. (http://wwwls. informatik.uni-tuebingen.de/contu/).

  2. de Oliveira Anjolina G., de Queiroz. Ruy J.G.B. : ‘A normalization procedure for the equational fragment of labelled natural deduction’. Logic Journal of the Interest Group in Pure and Applied Logics 7(2), 173–215 (1999)

    Google Scholar 

  3. de Oliveira Anjolina G., de Queiroz. Ruy J.G.B.: ‘A new basic set of proof transformations’. In: Artemov, S., Barringer, H., Garcez, A., Lamb, L., Woods, J. (eds) We Will Show Them: Essays in Honour of Dov Gabbay. Vol. 2, pp. 499–528. College Publications, London (2005)

    Google Scholar 

  4. de Queiroz, Ruy J. G. B., ‘Note on Frege’s notions of definition and the relationship proof theory vs. recursion theory (Extended Abstract)’, in Abstracts of the VIIIth International Congress of Logic, Methodology and Philosophy of Science. Vol. 5, Part I, Institute of Philosophy of the Academy of Sciences of the USSR, Moscow, 1987, pp. 69–73.

  5. de Queiroz Ruy J. G. B.: ‘A proof-theoretic account of programming and the rôle of reduction rules’. Dialectica 42(4), 265–282 (1988)

    Article  Google Scholar 

  6. de Queiroz, Ruy J. G. B., ‘The mathematical language and its semantics: to show the consequences of a proposition is to give its meaning’, in Paul Weingartner, and Gerhard Schurz, (eds.), Reports of the Thirteenth International Wittgenstein Symposium 1988, vol. 18 of Schriftenreihe der Wittgenstein-Gesellschaft, Hölder-Pichler- Tempsky, Vienna, 1989, pp. 259–266. Symposium held in Kirchberg/Wechsel, Austria, August 14–21 1988.

  7. de Queiroz, Ruy J. G. B., ‘Meaning, function, purpose, usefulness, consequences – interconnected concepts (abstract)’, in Abstracts of Fourteenth International Wittgenstein Symposium (Centenary Celebration), 1989, p. 20. Symposium held in Kirchberg/ Wechsel, August 13–20 1989. Full paper published in [12].

  8. de Queiroz, Ruy J. G. B., Proof Theory and Computer Programming. The Logical Foundations of Computation, Ph.D. thesis, Department of Computing, Imperial College, University of London, 1990.

  9. de Queiroz Ruy J. G. B.: ‘Meaning as grammar plus consequences’. Dialectica 45(1), 83–86 (1991)

    Article  Google Scholar 

  10. de Queiroz, Ruy J. G. B., ‘Grundgesetze alongside Begriffsschrift (abstract)’, in Abstracts of Fifteenth International Wittgenstein Symposium, 1992, pp. 15–16. Symposium held in Kirchberg/Wechsel, August 16–23 1992.

  11. de Queiroz, Ruy J. G. B., ‘Normalisation and language-games’, Dialectica 48 (1994), 2, 83–123. Early version presented at Logic Colloquium ’88, Padova. Abstract in JSL 55:425, 1990.

  12. de Queiroz Ruy J. G. B.: ‘Meaning, function, purpose, usefulness, consequences – interconnected concepts’. Logic Journal of the Interest Group in Pure and Applied Logics 9(5), 693–734 (2001)

    Google Scholar 

  13. de Queiroz, Ruy J. G. B., and Anjolina Grisi de Oliveira, ‘Natural deduction for equality: The missing entity’, in E. H. Haeusler, L.C. Pereira, and V. de Paiva, (eds.), Advances in Natural Deduction, Trends in Logic, Kluwer, 2008, pp. 244–273. To appear. Invited talk at Natural Deduction - Rio 2001 .

  14. de Queiroz, Ruy J. G. B., and Dov M. Gabbay, ‘Equality in Labelled Deductive Systems and the functional interpretation of propositional equality’, in P. Dekker, and M. Stokhof, (eds.), Proceedings of the 9th Amsterdam Colloquium 1994, pp. 547–546.

  15. de Queiroz, Ruy J. G. B., and Dov M. Gabbay, ‘The functional interpretation of the existential quantifier’, Bulletin of the Interest Group in Pure and Applied Logics, 3 (1995), 2–3, 243–290. Presented at Logic Colloquium ’91 , Uppsala, August 9–16 1991. Abstract JSL 58(2):753–754, 1993.

  16. de Queiroz, Ruy J. G. B., and Dov M. Gabbay, ‘The functional interpretation of modal necessity’, in Maarten de Rijke, (ed.), Advances in Intensional Logic, Applied Logic Series, Kluwer, 1997, pp. 61–91.

  17. de Queiroz, Ruy J. G. B., and Dov M. Gabbay, ‘Labelled natural deduction’, in H.J. Ohlbach, and U. Reyle, (eds.), Logic, Language and Reasoning. Essays in Honor of Dov Gabbay’s 50th Anniversary, Kluwer, 1999, pp. 173–250.

  18. de Queiroz Ruy J. G. B., Maibaum Thomas S. E.: ‘Proof Theory and Computer Programming’. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36, 389–414 (1990)

    Article  Google Scholar 

  19. de Queiroz Ruy J. G. B., Maibaum Thomas S. E.: ‘Abstract Data Types and Type Theory: Theories as Types’. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 37, 149–166 (1991)

    Article  Google Scholar 

  20. Dummett, Michael A. E., Elements of Intuitionism, reprinted (1985) edn., Series Oxford Logic Guides, Clarendon Press, Oxford, 1977. With the assistance of Roberto Minio.

  21. Dummett, Michael A. E., The Logical Basis of Metaphysics, Duckworth, London, xi+355p, 1991. Revised account (1989) of The William James Lectures given at Harvard University in 1976.

  22. Dummett, Michael A. E., ‘Truth from the constructivist standpoint’, Theoria LXIV (1998), 122–138. Special Issue on the Philosophy of D. Prawitz, P. Pagin (ed.).

  23. Gabbay, Dov M., and Ruy J. G. B. de Queiroz, ‘Extending the Curry-Howard- Tait intepretation to linear, relevant and other resource logics (abstract)’, Journal of Symbolic Logic, 56 (1991), 3, 1139–1140. Presented at the Logic Colloquium ’90 , Helsinki, July 15–22, 1990.

  24. Gabbay Dov M., de Queiroz Ruy J.G.B.: ‘Extending the Curry-Howard intepretation to linear, relevant and other resource logics’. Journal of Symbolic Logic 57(4), 1319–1365 (1992)

    Article  Google Scholar 

  25. Gentzen, Gerhard, ‘Untersuchungen über das logische Schliessen’, Mathematische Zeitschrift, 39 (1935), 176–210 and 405–431. English translation ‘Investigations into Logical Deduction’ in [44], pp. 68–131.

  26. Hacking Ian: ‘What is logic?’. Journal of Philosophy LXXVI(6), 285–319 (1979)

    Article  Google Scholar 

  27. Hilmy S. Stephen: The Later Wittgenstein. The Emergence of a New Philosophical Method. Basil Blackwell, Oxford (1987)

    Google Scholar 

  28. Hintikka K. Jaakko: ‘Quantifiers vs. Quantification Theory’. Linguistic Inquiry 5, 153–177 (1974)

    Google Scholar 

  29. Hintikka, K. Jaakko, ‘Game-Theoretical Semantics: Insights and Prospects’, in J. Hintikka, and J. Kulas, (eds.), The Game of Language, vol. 22 of Synthese Language Library, D. Reidel, Dordrecht, 1983, pp. 1–31.

  30. Hintikka Merrill B., Hintikka Jaakko: Investigating Wittgenstein Basil. Blackwell, Oxford (1986)

    Google Scholar 

  31. Lorenzen, Paul, Einführung in die operative Logik und Mathematik, vol. LXXVIII of Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1955.

  32. Lorenzen, Paul, Normative Logic and Ethics, vol. 236* of B.I-Hochschultaschenb ücher. Systematische Philosophie, Bibliographisches Institut, Mannheim/Zürich, 1969.

  33. Martin-Löf, Per, ‘Truth of a Proposition, Evidence of a Judgement, Validity of a Proof’, Synthese 73 (1987), 407–420. Special Issue on Theories of Meaning, Guest Editor: Maria Luisa Dalla Chiara, collecting articles originally presented as contributions to the conference “Theories of Meaning”, organised by the Florence Center for the History and Philosophy of Science, Firenze, Villa di Mondeggi, June 1985.

  34. Moriconi Enrico: ‘Normalization and meaning theory’. Epistemologia XXIII(2), 281–304 (2000) ISSN 0392–9760. Genova

    Google Scholar 

  35. Pears David F.: The False Prison. A Study of the Development of Wittgenstein’s Philosophy. Volume I. Clarendon Press, Oxford (1987)

    Google Scholar 

  36. Pears David F.: ‘Wittgenstein’s holism’. Dialectica 44, 165–173 (1990)

    Google Scholar 

  37. Prawitz Dag: ‘Meaning and proofs: on the conflict between classical and intuitionistic logic’. Theoria XLIII, 2–40 (1977)

    Google Scholar 

  38. Prawitz, Dag, ‘Proofs and the Meaning and Completeness of the Logical Constants’, in J. Hintikka, I. Niiniluoto, and E. Saarinen, (eds.), Essays on Mathematical and Philosophical Logic, vol. 122 of Synthese Library, D. Reidel Publishing Company, Dordrecht, 1978, pp. 25–40. Proceedings of the Fourth Scandinavian Logic Symposium and of the First Soviet-Finnish Logic Conference, Jyväskylä, Finland, June 29–July 6, 1976.

  39. Prawitz Dag: ‘Truth and objectivity from a verificationist point of view’. In: Dales, H. G., Olivieri, G. (eds) Truth in Mathematics., pp. 41–51. Clarendon Press, Oxford (1998)

    Google Scholar 

  40. Prawitz, Dag, ‘Problems of a generalization of a verificationist theory of meaning’, Topoi, 21 (2002), 87–92. Special Issue dedicated to Justification and Meaning, held at the University of Siena, Certosa di Pontignano, on June 15–17, 2000.

  41. Prawitz, Dag, ‘Meaning approached via proofs’, Synthese, 148 (2006), 3, 507–524. Special Issue on Proof-Theoretic Semantics, Reinhard Kahle and Peter Schroeder- Heister (eds.), collecting contributions to a conference with that title held at the University of Tübingen in January 1999.

  42. Schroeder-Heister Peter: ‘A Natural Extension of Natural Deduction’. Journal of Symbolic Logic 49(4), 1284–1300 (1984)

    Article  Google Scholar 

  43. Schroeder-Heister, Peter, ‘Validity concepts in proof-theoretic semantics’, Synthese, 148 (2006), 3, 525–571. Special Issue on Proof-Theoretic Semantics, Reinhard Kahle and Peter Schroeder-Heister, (eds.), collecting contributions to a conference with that title held at the University of Tübingen in January 1999.

  44. Szabo, M.E. (eds): The Collected Papers of Gerhard Gentzen, Series Studies in Logic and The Foundations of Mathematics. North-Holland, Amsterdam (1969)

    Google Scholar 

  45. Tait, William W., ‘Infinitely long terms of transfinite type’, in J. N. Crossley, and M. A. E. Dummett, (eds.), Formal Systems and Recursive Functions, Series Studies in Logic and The Foundations of Mathematics, North-Holland, Amsterdam, 1965, pp. 176–185. Proceedings of the Logic Colloquium ’63, held in Oxford, UK.

  46. von Kutschera F.: ‘Die Vollständigkeit des Operatorensystems {¬, ∧, ∨, ⊃} für intuitionistische Aussagenlogik im Rahmen der Gentzensemantik’. Archiv für mathematische Logik und Grundlagenforschung 11, 3–16 (1968)

    Article  Google Scholar 

  47. von Kutschera F.: ‘Ein verallgemeinerter Widerlegungsbegriff für gentzenkalküle’. Archiv für mathematische Logik und Grundlagenforschung 12, 104–118 (1969)

    Article  Google Scholar 

  48. Wansing Heinrich: ‘The Idea of a Proof-theoretic Semantics and the Meaning of the Logical Operations’. Studia Logica 64, 3–20 (2000)

    Article  Google Scholar 

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de Queiroz, R.J.G.B. On Reduction Rules, Meaning-as-use, and Proof-theoretic Semantics. Stud Logica 90, 211–247 (2008). https://doi.org/10.1007/s11225-008-9150-5

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