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Assertions, Hypotheses, Conjectures, Expectations: Rough-Sets Semantics and Proof Theory

  • Gianluigi  BellinEmail author
Chapter
Part of the Trends in Logic book series (TREN, volume 39)

Abstract

In this chapter bi-intuitionism is interpreted as an intensional logic which is about the justification conditions of assertions and hypotheses, extending C. Dalla Pozza and C. Garola’s pragmatic interpretation [18] of intuitionism, seen as a logic of assertions according to a suggestion by M. Dummett. Revising our previous work on this matter [5], we consider two additional illocutionary forces, \((i)\) conjecturing, seen as making the hypothesis that a proposition is epistemically necessary, and \((ii)\) expecting, regarded as asserting that a propostion is epistemically possible; we show that a logic of expectations justifies the double negation law. We formalize our logic in a calculus of sequents and study bimodal Kripke semantics of bi-intuitionism based on translations in S4. We look at rough set semantics following P. Pagliani’s analysis of “intrinsic co-Heyting boundaries” [40] (after Lawvere). A Natural Deduction system for co-intuitionistic logic is given where proofs are represented as upside down Prawitz trees. We give a computational interpretation of co-intuitionism, based on T. Crolard’s notion of coroutine [16] as the programming construction corresponding to subtraction introduction. Our typed calculus of co-routines is dual to the simply typed lambda calculus and shows features of concurrent and distributed computations.

Keywords

Classical Logic Intuitionistic Logic Natural Deduction Sequent Calculus Epistemic Possibility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Angelelli, I. (1970). The techniques of disputation in the history of logic. The Journal of Philosophy, 67(20), 800–815.CrossRefGoogle Scholar
  2. 2.
    Austin, J. L. (1970). Philosophical papers (2nd ed.). USA: Oxford University Press.Google Scholar
  3. 3.
    Bellin, G., & Scott, P. J. (1994). On the Pi-calculus and linear logic. Theoretical Computer Science, 135, 11–65.CrossRefGoogle Scholar
  4. 4.
    Bellin, G. (2003). Chu’s Construction: A Proof-theoretic Approach. In Ruy J.G.B. de Queiroz (Ed.), Logic for Concurrency and Synchronisation (Vol. 18, pp. 93–114). Dordrecht: Kluwer Trends in Logic.Google Scholar
  5. 5.
    Bellin, G., & Biasi, C. (2004). Towards a logic for pragmatics. Assertions and conjectures. Journal of Logic and Computation, 14(4), 473–506. Special Issue with the Proceedings of the Workshop on Intuitionistic Modal Logic and Application (IMLA-FLOC 2002), V. de Paiva, R. Goré & M. Mendler (Eds.).Google Scholar
  6. 6.
    Bellin, G., & Ranalter, K. (2003). A Kripke-style semantics for the intuitionistic logic of pragmatics ILP. Journal of Logic and Computation, 13(5), 755–775. Special Issue with the Proceedings of the Dagstuhl Seminar on Semantic Foundations of Proof-search, Schloss Dagstuhl.Google Scholar
  7. 7.
    Bellin, G., & Dalla Pozza, C. (2002). A pragmatic interpretation of substructural logics. Reflection on the Foundations of Mathematics (Stanford, CA, 1998), Essays in honor of Solomon Feferman, W. Sieg, R. Sommer and C. Talcott (Eds.), Association for Symbolic Logic, Urbana, IL, Lecture Notes in Logic, (Vol. 15, pp. 139–163).Google Scholar
  8. 8.
    Bellin, G. (2005). A Term Assignment for Dual Intuitionistic Logic. Conference paper presented at the LICS’05-IMLA’05 Workshop, Chicago, IL, June 30.Google Scholar
  9. 9.
    Bellin, G., Hyland, M., Robinson, E., & Urban, C. (2006). Categorical proof theory of classical propositional calculus. Theoretical Computer Science, 364(2), 146–165.CrossRefGoogle Scholar
  10. 10.
    Biasi, C. (2003). Verso una logica degli operatori prammatici asserzioni e congetture. Tesi di Laurea: Facoltà di Scienze, Università di Verona.Google Scholar
  11. 11.
    Biasi, C., & Aschieri, F. (2008). A term assignment for polarized bi-intuitionistic logic and its strong normalization. Fundamenta Informaticae, 84(2), 185–205. Special issue on Logic for Pragmatics.Google Scholar
  12. 12.
    Bing, J. (1982). Uncertainty, decisions and information systems. In C. Ciampi (Ed.), Artificial Intelligence and Legal Information Systems. Amsterdam: North-Holland.Google Scholar
  13. 13.
    Brewka, G. & Gordon, T. (2010). Carneades and Abstract Dialectical Frameworks: A Reconstruction. In: P. Baroni, M. Giacomin & G. Simari (Ed.), Computational Models of Argument, Proceedings of COMMA 2010, IOS Press.Google Scholar
  14. 14.
    Crolard, T. (1996). Extension de l’isomorphisme de Curry-Howard au traitement des exceptions (application d’une ètude de la dualité en logique intuitionniste) (p. 7). Thèse de Doctorat: Université de Paris.Google Scholar
  15. 15.
    Crolard, T. (2001). Subtractive logic. Theoretical Computer Science, 254(1–2), 151–185.CrossRefGoogle Scholar
  16. 16.
    Crolard, T. (2004). A formulae-as-types interpretation of subtractive logic. Journal of Logic and Computation, 14(4), 529–570. Special Issue with the Proceedings of the Workshop on Intuitionistic Modal Logic and Application (IMLA-FLOC 2002), V. de Paiva, R. Goré and M. Mendler (Eds.).Google Scholar
  17. 17.
    Curien, P. L. (2002). Abstract Machines, Control, and Sequents. In G. Barthe, P. Dybjer, L. Pinto, J. Saraiva (Eds.), Applied Semantics, International Summer School, APPSEM 2000, Caminha, Portugal, September 9–15, Advanced Lectures. Lecture Notes in Computer Science, Vol. 2395, Springer.Google Scholar
  18. 18.
    Dalla Pozza, C., & Garola, C. (1995). A pragmatic interpretation of intuitionistic propositional logic. Erkenntnis, 43(1995), 81–109.CrossRefGoogle Scholar
  19. 19.
    Dalla Pozza, C. (1997). Una logica prammatica per la concezione “espressiva” delle norme. In A. Martino (Ed.), Logica delle Norme. Pisa: Servizio Editoriale Universitario.Google Scholar
  20. 20.
    Danos, V., Joinet, J.-B., & Schellinx, H. (1995). LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. In Proceedings of the workshop on Advances in linear logic (pp. 211–224). New York: Cambridge University Press.Google Scholar
  21. 21.
    Dummett, M. (1991). The Logical Basis of Metaphysics. Cambridge: Cambridge University Press.Google Scholar
  22. 22.
    Girard, J.-Y. (1993). On the unity of logic. Annals of Pure and Applied Logic, 59, 201–217.CrossRefGoogle Scholar
  23. 23.
    Gödel, K. (1933). Eine Interpretation des Intuitionistischen Aussagenkalküls. Ergebnisse eines Mathematischen Kolloquiums IV, pp. 39–40.Google Scholar
  24. 24.
    Gordon, T. F., & Walton, D. (2009). Proof burdens and standards In: I. Rahwan & G. Simari (Eds.), Argumentations in Artificial Intelligence (pp. 239–258). New York: Springer.Google Scholar
  25. 25.
    Goré, R. (2000). Dual intuitionistic logic revisited. In TABLEAUX00: Automated reasoning with analytic tableaux and related methods, LNAI (Vol. 1847, pp. 252–267). Berlin: Springer.Google Scholar
  26. 26.
    Hyland, J. M., & De Paiva, V. C. V. (1993). Full intuitionistic linear logic. Annals of Pure and Applied Logic, 64, 273–291.CrossRefGoogle Scholar
  27. 27.
    Kripke, S. A. (1963). Semantical analysis of modal logic I: Normal modal propostional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 9.Google Scholar
  28. 28.
    Kripke, S. A. (1965). Semantical analysis of intuitionistic logic I. Formal Systems and Recursive Functions. In J. N. Crossley & M. A. E. Dummett (Eds.), Studies in logic and the foundations of mathematics (pp. 92–130). Amsterdam: North-Holland.Google Scholar
  29. 29.
    Krivine, J. L. (1990). Lambda-calcul, types et modèles. Paris: Masson.Google Scholar
  30. 30.
    Lambek, J., & Scott, P. J. (1986). Introduction to higher order categorical logic. Cambridge: Cambridge University Press.Google Scholar
  31. 31.
    Levinson, S. C. (1983). Pragmatics. Cambridge: Cambridge University Press.Google Scholar
  32. 32.
    Lawvere, F. W. (1991). Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In A. Carboni, M. C. Pedicchio, & G. Rosolini (Eds.), Category Theory (Como 1990), Lecture Notes in Mantehmatics (Vol. 1488, pp. 279–297). Verlag: Springer.Google Scholar
  33. 33.
    Makkai, M., & Reyes, G. E. (1995). Completeness results for intuitionistic and modal logic in a categorical setting. Annals od Pure and Applied Logic, 72, 25–101.CrossRefGoogle Scholar
  34. 34.
    Martin-Löf, P. (1985). On the meaning and justification of logical laws. In Bernardi & Pagli (Eds.) Atti degli Incontri di Logica Matematica, vol. II, Universita‘ di Siena.Google Scholar
  35. 35.
    Martin-Löf, P. (1987). Truth of a proposition, evidence of a judgement, validity of a proof. Synthese, 73, 407–420.CrossRefGoogle Scholar
  36. 36.
    McKinsey, J. C. C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic, 13, 1–15.CrossRefGoogle Scholar
  37. 37.
    Miglioli, P., Moscato, U., Ornaghi, M., & Usberti, U. (1989). A constructivism based on classical truth. Notre Dame Journal of Formal Logic, 30, 67–90.CrossRefGoogle Scholar
  38. 38.
    Nelson, D. (1949). Constructible falsity. The Journal of Symbolic Logic, 14, 16–26.CrossRefGoogle Scholar
  39. 39.
    Pagliani, P., & Chakraborty, M. (2008). A geometry of approximation rough set theory: Logic algebra and topology of conceptual patterns. Verlag: Springer.CrossRefGoogle Scholar
  40. 40.
    Pagliani, P. (2009). Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis. In L. Polkowski & A. Skowron (Eds.), RSCTC 1998. Lecture Notes in Computer Science (Vol. 1424, pp. 123–130). Heidelberg: Springer.Google Scholar
  41. 41.
    Pagliani, P. (1997). Information gaps as communication needs: A new semantic foundation for some non-classical logics. Journal of Logic, Language and Information, 6(1), 63–99.CrossRefGoogle Scholar
  42. 42.
    Parigot, M. (1991). Free Deduction: An Analysis of “Computations” in Classical Logic. In A. Voronkov (Ed.), Logic Programming, Proceedings of the First Russian Conference on Logic Programming, Irkutsk, Russia, September 14–18, 1990–Second Russian Conference on Logic Programming, St. Petersburg, Russia, September 11–16, Springer Lecture Notes in Computer Science, ISBN 3-540-55460-2, London: Springer, pp. 361-380.Google Scholar
  43. 43.
    Parigot, M. (1992). Lambda-My-Calculus: An Algorithmic Interpretation of Classical Natural Deduction. In A. Voronkov (Ed.), Logic Programming and Automated Reasoning, Proceedings of the International Conference LPAR’92, St. Petersburg, Russia. Springer Lecture Notes in Computer Science 624, 1992, ISBN 3-540-55727-X, pp. 190-201.Google Scholar
  44. 44.
    Parigot, M., (1993). Strong Normalization for Second Order Classical Natural Deduction. In LICS,. (1993). Proceedings (pp. 39–46). Canada: Eighth Annual IEEE Symposium on Logic in Computer Science Montreal.Google Scholar
  45. 45.
    Polkowski, L. (2002). Rough Sets. Physical-Verlag: Mathematical Foundations. Advances in Soft Computing. Heidelberg.CrossRefGoogle Scholar
  46. 46.
    Prawitz, D. (1965). Natural deduction. A proof-theoretic study. Stockholm: Almquist and Wikksell.Google Scholar
  47. 47.
    Prawitz, D. (1987). Dummett on a theory of meaning and its impact on logic. In B. Taylor (Ed.), Michael Dummett: Contributions to philosophy (pp. 117–165). The Hague: Nijhoff.CrossRefGoogle Scholar
  48. 48.
    Ranalter, K. (2008). A semantic analysis of a logic for pragmatics with assertions, obligations, and causal implication. Fundamenta Informaticae, 84(3–4), 443–470.Google Scholar
  49. 49.
    Ranalter, K. (2008). Reasoning about assertions, obligations and causality: on a categorical semantics for a logics for pragmatics. PhD Thesis, Queen Mary, University of London and University of Verona.Google Scholar
  50. 50.
    Rauszer, C. (1974). Semi-Boolean algebras and their applications to intuitionistic logic with dual operations. Fundamenta Mathematicae, 83, 219–249.Google Scholar
  51. 51.
    Rauszer, C. (1977). Applications of Kripke models to Heyting-Brouwer logic. Studia Logica, 36, 61–71.CrossRefGoogle Scholar
  52. 52.
    Reyes, G., & Zolfaghari, H. (1996). Bi-Heyting algebras, toposes and modalities. Journal of Philosophical Logic, 25, 25–43.CrossRefGoogle Scholar
  53. 53.
    Ryan, M., & Schobbens, P.-Y. (1997). Counterfactuals and updates as inverse modalities. Journal of Logic, Language and Information, 6(2), 123–146.CrossRefGoogle Scholar
  54. 54.
    Selinger, P. (2001). Control categories and duality: On the categorical semantics of the lambda-mu calculus. Mathematical Structures in Computer Science, 11, 207–260.CrossRefGoogle Scholar
  55. 55.
    Shapiro, S. (1985). Epistemic and Intuitionistic Arithmetic. In S. Shapiro (Ed.), Intensional mathematics (pp. 11–46). Amsterdam: North-Holland.CrossRefGoogle Scholar
  56. 56.
    Shapiro, S. (1997). Philosophy of mathematics : structure and ontology. New York: Oxford University Press.Google Scholar
  57. 57.
    Stell, J., & Worboys, M. (1997). The Algebraic Structure of Sets of Regions. In S. Hirtle & A. Frank (Eds.), Spatial Information Theory, COSIT’97 Proceedings Lecture Notes in Computer Science (Vol. 1329, pp. 163–174). Verlag: Springer.Google Scholar
  58. 58.
    Thomason, R. (1969). A semantical study of constructible falsity. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 15, 247–257.CrossRefGoogle Scholar
  59. 59.
    Troelstra, A. S. & Schwichtenberg, H. (1996). Basic proof theory. Cambridge tracts in theoretical computer science (Vol. 43). Cambridge: Cambridge University Press.Google Scholar
  60. 60.
    White, G. (2008). Davidson and Reiter on actions. Fundamenta Informaticae, 84(2), 259–289.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly

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