Pragmatic and dialogic interpretations of bi-intuitionism. Part I
Keywordsbi-intuitionism, categorical proof theory, justificationism, meaning-asuse, speech-acts theory
AbstractWe consider a “polarized” version of bi-intuitionistic logic [5, 2, 6, 4] as a logic of assertions and hypotheses and show that it supports a “rich proof theory” and an interesting categorical interpretation, unlike the standard approach of C. Rauszer’s Heyting-Brouwer logic [28, 29], whose categorical models are all partial orders by Crolard’s theorem . We show that P.A. Melliès notion of chirality [21, 22] appears as the right mathematical representation of the mirror symmetry between the intuitionistic and co-intuitionistc sides of polarized bi-intuitionism. Philosophically, we extend Dalla Pozza and Garola’s pragmatic interpretation of intuitionism as a logic of assertions  to bi-intuitionism as a logic of assertions and hypotheses. We focus on the logical role of illocutionary forces and justification conditions in order to provide “intended interpretations” of logical systems that classify inferential uses in natural language and remain acceptable from an intuitionistic point of view. Although Dalla Pozza and Garola originally provide a constructive interpretation of intuitionism in a classical setting, we claim that some conceptual refinements suffice to make their “pragmatic interpretation” a bona fide representation of intuitionism. We sketch a meaning-asuse interpretation of co-intuitionism that seems to fulfil the requirements of Dummett and Prawitz’s justificationist approach. We extend the Brouwer-Heyting-Kolmogorov interpretation to bi-intuitionism by regarding co-intuitionistic formulas as types of the evidence for them: if conclusive evidence is needed to justify assertions, only a scintilla of evidence suffices to justify hypotheses.
Angelelli, I., “The techniques of disputation in the history of logic”, The Journal of Philosophy 67, 20 (1970): 800–815. DOI: 10.2307/2024013
Bellin, G., “Assertions, hypotheses, conjectures, expectations: Rough-sets semantics and proof theory”, pp. 193–241 in Advances in Natural Deduction. A Celebration of Dag Prawitz’s Work, L.C. Pereira, E.H. Haeusler, V. de Paia (eds.), “Trends in Logic”, vol. 39, 2014.
Bellin, G., “Chu’s construction: A proof-theoretic approach”, pp. 93–114 in Logic for Concurrency and Synchronisation, Ruy J.G.B. de Queiroz (ed.), “Trends in Logic”, vol. 18, Kluwer, 2003. DOI: 10.1007/0-306-48088-3_2
Bellin, G., “Categorical proof theory of co-intuitionistic linear logic”, accepted by Logical Methods in Computer Science (2014).
Bellin, G., and C. Biasi, “Towards a logic for pragmatics. Assertions and conjectures”, Journal of Logic and Computation, 14, 4 (2004): 473–506. DOI: 10.1093/logcom/14.4.473
Bellin, G., and A. Menti, “On the π-calculus and co-intuitionistic logic. Notes on logic for concurrency and λP systems”, Fundamenta Informaticae 30, 1 (2014): 21–65.
Brewka, G., and T. Gordon, “Carneades and abstract dialectical frameworks: A reconstruction”, in Computational Models of Argument, P. Baroni, M. Giacomin, and G. Simari (eds.), Proceedings of COMMA 2010, IOS Press, 2010.
Crolard, T., “Subtractive logic”, Theoretical Computer Science, 254, 1–2 (2001) 151–185. DOI: 10.1016/S0304-3975(99)00124-3
Crolard, T., “A formulae-as-types interpretation of subtrActive logic”, Journal of Logic and Computation, 14, 4 (2004): 529–570. DOI: 10.1093/logcom/14.4.529
Dalla Pozza, C., and C. Garola, “A pragmatic interpretation of intuitionistic propositional logic”, Erkenntnis, 43 (1995): 81–109.
Czermak, J., “A remark on Gentzen’s calculus of sequents”, Notre Dame Journal of Formal Logic, 18, 3 (1977): 471–474. DOI: 10.1305/ndjfl/1093888021
Dummett, M., The Logical Basis of Metaphysics, Cambridge, Mass.: Cambridge University Press, 1991.
Girard, J-Y., “Linear logic”, Theoretical Computer Science, 50, 1 (1987). DOI: 10.1016/0304-3975(87)90045-4
Goodman, N., “The logic of contradiction”, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 27 (1981): 119–126.
Gordon, T.F., and D. Walton, “Proof burdens and standards”, pp. 239–258 in Argumentations in Artificial Intelligence, I. Rahwan and G. Simari (eds.), 2009.
Goré, R., “Dual intuitionistic logic revisited”, in Tableaux00: Automated Reasoning with Analytic Tableaux and Related Methods, R. Dyckhoff (ed.), Springer, 2000.
Lawvere, F.W., “Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes”, pp. 279–297 in Category Theory (Como 1990), A. Carboni, M.C. Pedicchio, and G. Rosolini (eds.), “Lecture Notes in Mathematics”, vol. 1488, Springer-Verlag 1991.
Lincoln, P., J. Mitchell, and A. Scedrov, “Stochastic interaction and linear logic”, pp. 147–166 in Advances in Linear Logic, J-Y. Girard, Y. Lafont, and L. Regnier (eds.), “London Mathematical Society Lecture Note Series”, vol. 222, Cambridge University Press, Cambridge, UK, New York, NY, USA, 1995.
López-Escobar, E.G.K., “On intuitionistic sentential connectives I”, Revista Colombiana de Matemàticas, XIX (1985): 117–130.
McKinsey, J.C.C., and A. Tarski, “Some theorems about the sentential calculi of Lewis and Heyting”, Journal of Symbolic Logic, 13 (1948): 1–15. DOI: 10.2307/2268135
Melliès, P-A., “Game semantics in string diagrams”, in Proceedings of the Annual ACM/IEEE Symposium in Logic in Computer Science, LICS 2012.
Melliès, P-A., “A micrological study of negation”, manuscript, available at the author’s web page.
Nelson, D., “Constructible falsity”, The Journal of Symbolic Logic, 14 (1949): 16–26. DOI: 10.2307/2268973
Pinto, L., and T. Uustalu, “Relating sequent calculi for bi-intuitionistic propositional logic”, pp. 57–72 in Proceedings of the Third International Workshop on Classical Logic and Computation, CL&C 2010, Brno, Czech Republic, 21-22 August 2010, S. von Bakel, S. Berardi, and U. Berger (eds.), EPTCS 47, 2010.
Prawitz, D., Natural Deduction. A Proof-Theoretic Study, Almquist and Wikksell, Stockholm, 1965.
Prawitz, D., “Meaning and proofs: On the conflict between classical and intuitionistic logic”, Theoria, 43, 1 (1977): 2–40.
Prawitz, D., “Is there a general notion of proof?”, Proceedings of the XIV Congress of Logic, Methodology and Philosophy of Science, Nancy 2011, forthcoming.
Rauszer, C., “Semi-Boolean algebras and their applications to intuitionistic logic with dual operations”, Fundamenta Mathematicae, 83 (1974): 219–249.
Rauszer, C., “Applications of Kripke models to Heyting-Brouwer logic”, Studia Logica, 36 (1977): 61–71. DOI: 10.1007/BF02121115
Shramko, Y., “Dual intuitionistic logic and a variety of negations. The logic of scientific reseach”, Studia Logica, 80 (2005): 347–367. DOI: 10.1007/s11225-005-8474-7
Tranchini, L., “Natural deduction for dual-intuitionistic logic”, Studia Logica, 100, 2 (2012): 631–648. DOI: 10.1007/s11225-012-9417-8
Urbas, I., “Dual intuitionistic logic”, Nore Dame Journal of Formal Logic, 37 (1996): 440–451. DOI: 10.1305/ndjfl/1039886520
Uustalu, T., “A note on anti-intuitionisticlogic”, abstract presented at the Nordic Workshop of Programmong Theory (NWPT’97), Tallin Estonia, 1997.
How to Cite
Number of views and downloads: 373
Number of citations: 15