Journal of Philosophical Logic

, Volume 46, Issue 3, pp 333–353

Cut-Elimination for Quantified Conditional Logic

Article
  • 577 Downloads

Abstract

A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.

Keywords

Cut-elimination Quantified conditional logics Classical higher-order logic Semantic embedding Cut-simulation 

References

  1. 1.
    Andrews, P. (2014). Church’s type theory. In Zalta, E.N. (Ed.) The Stanford Encyclopedia of Philosophy, Spring 2014 edn.: Stanford University.Google Scholar
  2. 2.
    Andrews, P.B. (1971). Resolution in type theory. Journal of Symbolic Logic, 36(3), 414–432.CrossRefGoogle Scholar
  3. 3.
    Andrews, P.B. (1972). General models and extensionality. Journal of Symbolic Logic, 37(2), 395–397.CrossRefGoogle Scholar
  4. 4.
    Andrews, P.B. (1972). General models, descriptions, and choice in type theory. Journal of Symbolic Logic, 37(2), 385–394.CrossRefGoogle Scholar
  5. 5.
    Backes, J., & Brown, C.E. (2011). Analytic tableaux for higher-order logic with choice. Journal of Automated Reasoning, 47(4), 451–479.CrossRefGoogle Scholar
  6. 6.
    Benzmüller, C. (1999). Equality and extensionality in automated higher-order theorem proving. Ph.D. thesis, Saarland University.Google Scholar
  7. 7.
    Benzmüller, C. (1999). Extensional higher-order paramodulation and RUE-resolution. In Ganzinger, H. (Ed.) Automated Deduction - CADE-16, 16th International Conference on Automated Deduction, Trento, Italy, July 7-10, 1999, Proceedings, no. 1632 in LNCS, pp. 399–413. : Springer. doi:10.1007/3-540-48660-7_39
  8. 8.
    Benzmüller, C. (2009). Automating access control logic in simple type theory with LEO-II. In Gritzalis, D., & López, J. (Eds.) Emerging Challenges for Security, Privacy and Trust, 24th IFIP TC 11 International Information Security Conference, SEC 2009, Pafos, Cyprus, May 18-20, 2009. Proceedings, IFIP, vol. 297, pp. 387–398. doi:10.1007/978-3-642-01244-0_34: Springer.
  9. 9.
    Benzmüller, C. (2011). Combining and automating classical and non-classical logics in classical higher-order logic. Annals of Mathematics and Artificial Intelligence (Special issue Computational logics in Multi-agent Systems (CLIMA XI)), 62(1-2), 103–128. doi:10.1007/s10472-011-9249-7.Google Scholar
  10. 10.
    Benzmüller, C. (2013). Automating quantified conditional logics in HOL. In Rossi, F. (Ed.) 23rd International Joint Conference on Artificial Intelligence (IJCAI-13), pp. 746-753, Beijing, China.Google Scholar
  11. 11.
    Benzmüller, C. (2013). Cut-free calculi for challenge logics in a lazy way. In Clint van Alten Petr Cintula, C.N. (Ed.) Proceedings of the International Workshop on Algebraic Logic in Computer Science.Google Scholar
  12. 12.
    Benzmüller, C. (2013). A top-down approach to combining logics. In Proc. of the 5th International Conference on Agents and Artificial Intelligence (ICAART), pp. 346–351. SciTePress Digital Library, Barcelona, Spain. doi:10.5220/0004324803460351.
  13. 13.
    Benzmüller, C. (2015). Higher-order automated theorem provers. In Delahaye, D., & Woltzenlogel Paleo, B. (Eds.) All about Proofs, Proofs for All, Mathematical Logic and Foundations, pp. 171-214, College Publications.Google Scholar
  14. 14.
    Benzmüller, C., Brown, C., & Kohlhase, M. (2004). Higher-order semantics and extensionality. Journal of Symbolic Logic, 69(4), 1027–1088. doi:10.2178/jsl/1102022211.CrossRefGoogle Scholar
  15. 15.
    Benzmüller, C., Brown, C., & Kohlhase, M. (2008). Cut elimination with xi-functionality. In Benzmüller, C., Brown, c., Siekmann, J., & Statman, R. (Eds.) Reasoning in Simple Type Theory — Festschrift in Honor of Peter B. Andrews on His 70th Birthday, Studies in Logic, Mathematical Logic and Foundations, pp. 84–100. College Publications.Google Scholar
  16. 16.
    Benzmüller, C., Brown, C., & Kohlhase, M. (2009). Cut-simulation and impredicativity. Logical Methods in Computer Science, 5 (1:6), 1–21. doi:10.2168/LMCS-5(1:6)2009.Google Scholar
  17. 17.
    Benzmüller, C., Gabbay, D., Genovese, V., & Rispoli, D. (2012). Embedding and automating conditional logics in classical higher-order logic. Annals of Mathematics and Artificial Intelligence, 66(1-4), 257–271. doi:10.1007/s10472-012-9320-z.CrossRefGoogle Scholar
  18. 18.
    Benzmüller, C., & Genovese, V. Quantified conditional logics are fragments of HOL. In The International Conference on Non-classical Modal and Predicate Logics (NCMPL). Guangzhou (Canton), China (2011). The conference had no published proceedings; the paper is available as. arXiv:1204.5920v1.
  19. 19.
    Benzmüller, C., & Paulson, L. (2008). Exploring properties of normal multimodal logics in simple type theory with LEO-II. In Benzmüller, C., Brown, C., Siekmann, J., & Statman, R. (Eds.) Reasoning in Simple Type Theory — Festschrift in Honor of Peter B. Andrews on His 70th Birthday, Studies in Logic, Mathematical Logic and Foundations, pp. 386–406. College Publications.Google Scholar
  20. 20.
    Benzmüller, C., & Miller, D. (2014). Automation of Higher-Order Logic. Chapter. In Gabbay, D.M., Siekmann, J.H., & Woods, J. (Eds.) Handbook of the History of Logic, Volume 9 — Computational Logic, North Holland, Elsevier, pp. 215–254.Google Scholar
  21. 21.
    Benzmüller, C., & Paulson, L. (2010). Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL, 18(6), 881–892. doi:10.1093/jigpal/jzp080.CrossRefGoogle Scholar
  22. 22.
    Benzmüller, C., & Paulson, L. (2013). Quantified multimodal logics in simple type theory. Logica Universalis (Special Issue on Multimodal Logics), 7(1), 7–20. doi:10.1007/s11787-012-0052-y.CrossRefGoogle Scholar
  23. 23.
    Benzmüller, C., Paulson, L.C., Sultana, N., & Theiß, F. (2015). The higher-order prover LEO-II. Journal of Automated Reasoning, 55(4), 389–404. doi:10.1007/s10817-015-9348-y.CrossRefGoogle Scholar
  24. 24.
    Benzmüller, C., & Sultana, N. (2013). LEO-II Version 1.5. In Blanchette , J.C., & Urban, J. (Eds.) PxTP 2013, EPiC Series, vol. 14, pp. 2–10. EasyChair.Google Scholar
  25. 25.
    Benzmüller, C., & Woltzenlogel Paleo, B. (2014). Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers. In Schaub, T., Friedrich, G., & O’Sullivan, B. (Eds.) ECAI 2014, Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 93–98. IOS Press. doi:10.3233/978-1-61499-419-0-93.
  26. 26.
    Brown, C.E. (2004). Set comprehension in church’s type theory. Ph.D. thesis, Department of Mathematical Sciences Carnegie Mellon University. See also Chad E. Brown (2007), Automated Reasoning in Higher-Order Logic, College Publications.Google Scholar
  27. 27.
    Brown, C.E. (2005). Reasoning in extensional type theory with equality. In Nieuwenhuis, R. (Ed.) Proc. of CADE-20, LNCS, vol. 3632, pp. 23–37. Springer.Google Scholar
  28. 28.
    Brown, C.E. (2012). Satallax: An automatic higher-order prover. In Gramlich, B., Miller, D., & Sattler, U. (Eds.) Automated Reasoning - 6th International Joint Conference, IJCAR 2012, Manchester, UK, June 26-29, 2012. Proceedings, LNCS, vol. 7364, pp. 111–117. Springer. doi:10.1007/978-3-642-31365-3_11.
  29. 29.
    Brown, C.E., & Smolka, G. (2010). Analytic tableaux for simple type theory and its first-order fragment. Logical Methods in Computer Science, 6(2).Google Scholar
  30. 30.
    Chellas, B. (1975). Basic conditional logic. Journal of Philosophical Logic, 4 (2), 133–153.CrossRefGoogle Scholar
  31. 31.
    Chellas, B. (1980). Modal logic: an introduction. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  32. 32.
    Church, A. (1932). A set of postulates for the foundation of logic. Annals of Mathematics, 33(2), 346–366.CrossRefGoogle Scholar
  33. 33.
    Church, A. (1936). An unsolvable problem of elementary number theory. American Journal of Mathematics, 58(2), 354–363.CrossRefGoogle Scholar
  34. 34.
    Church, A. (1940). A formulation of the simple theory of types. Journal of Symbolic Logic, 5(2), 56–68.CrossRefGoogle Scholar
  35. 35.
    Chwistek, L. (1948). The limits of science: Outline of logic and of the methodology of the exact sciences. London: Routledge and Kegan Paul.Google Scholar
  36. 36.
    Delgrande, J. (1987). A first-order conditional logic for prototypical properties. Artificial Intelligence, 33(1), 105–130.CrossRefGoogle Scholar
  37. 37.
    Delgrande, J. (1998). On first-order conditional logics. Artificial Intelligence, 105(1–2), 105–137.CrossRefGoogle Scholar
  38. 38.
    Fitting, M. (2002). Interpolation for first order S5. Journal of Symbolic Logic, 67(2), 621–634.CrossRefGoogle Scholar
  39. 39.
    Frege, G. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle (1879). Translated in [43].Google Scholar
  40. 40.
    Friedman, N., Halpern, J., & Koller, D. (2000). First-order conditional logic for default reasoning revisited. ACM Transactions on Computational Logic, 1(2), 175–207.CrossRefGoogle Scholar
  41. 41.
    Girard, J.Y. (1971). Une extension de l’interpretation de Gödel à l’analyse, et son application à l’élimination des coupures dans l’analyse et la théorie des types. In Fenstad, J.E. (Ed.) 2nd scandinavian logic symposium, pp. 63–92, North-Holland, Amsterdam.Google Scholar
  42. 42.
    Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte der Mathematischen Physik, 38, 173–198.CrossRefGoogle Scholar
  43. 43.
    van Heijenoort, J. (1967). From Frege to Gödel: A source Book in Mathematics, 1879–1931, 3rd printing, 1997 edn. Source books in the history of the sciences series. Harvard University Press, Cambridge, MA.Google Scholar
  44. 44.
    Henkin, L. (1950). Completeness in the theory of types. Journal of Symbolic Logic, 15(2), 81–91.CrossRefGoogle Scholar
  45. 45.
    Huet, G.P. (1972). Constrained resolution: a complete method for higher order logic. Ph.D. thesis, Case Western Reserve University.Google Scholar
  46. 46.
    Huet, G.P. (1973). A mechanization of type theory. In Proceedings of the 3rd International Joint Conference on Artificial Intelligence, pp. 139–146.Google Scholar
  47. 47.
    Kohlhase, M. (1994). A mechanization of sorted higher-order logic based on the resolution principle. Ph.D. thesis, Saarland University.Google Scholar
  48. 48.
    McDowell, R., & Miller, D. (2002). Reasoning with higher-order abstract syntax in a logical framework. ACM Transactions on Computational Logic, 3(1), 80–136.CrossRefGoogle Scholar
  49. 49.
    Mints, G. (1999). Cut-elimination for simple type theory with an axiom of choice. Journal of Symbolic Logic, 64(2), 479–485.CrossRefGoogle Scholar
  50. 50.
    Muskens, R. (2007). Intensional models for the theory of types. Journal of Symbolic Logic, 72(1), 98–118.CrossRefGoogle Scholar
  51. 51.
    Nute, D. (1980). Topics in conditional logic. Dordrecht: Reidel.CrossRefGoogle Scholar
  52. 52.
    Olivetti, N., Pozzato, G., & Schwind, C. (2007). A sequent calculus and a theorem prover for standard conditional logics. ACM Transactions on Computational Logic, 8(4).Google Scholar
  53. 53.
    Paleo, B.W. (2014). An embedding of neighbourhood-based modal logics in hol. Unpublished draft; available at https://github.com/Paradoxika/ModalLogic.
  54. 54.
    Pattinson, D., & Schröder, L. (2011). Generic modal cut elimination applied to conditional logics. Logical Methods in Computer Science, 7(1). doi:10.2168/LMCS-7(1:4)2011.
  55. 55.
    Prawitz, D. (1968). Hauptsatz for higher order logic. Journal of Symbolic Logic, 33(3), 452–457.CrossRefGoogle Scholar
  56. 56.
    Ramsey, F.P. (1926). The foundations of mathematics. In Proceedings of the London Mathematical Society, 2, vol. 25, pp. 338–384.Google Scholar
  57. 57.
    Rasga, J. (2007). Sufficient conditions for cut elimination with complexity analysis. Annals of Pure and Applied Logic, 149(1–3), 81–99. doi:10.1016/j.apal.2007.08.001.CrossRefGoogle Scholar
  58. 58.
    Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262.CrossRefGoogle Scholar
  59. 59.
    Schütte, K. (1960). Semantical and syntactical properties of simple type theory. Journal of Symbolic Logic, 25(4), 305–326.CrossRefGoogle Scholar
  60. 60.
    Smullyan, R.M. (1963). A unifying principle for quantification theory. Proc Nat Acad Sciences, 49, 828–832.CrossRefGoogle Scholar
  61. 61.
    Stalnaker, R. (1968). A theory of conditionals. In Studies in Logical Theory , pp. 98–112. Blackwell.Google Scholar
  62. 62.
    Sutcliffe, G. (2009). The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning, 43(4), 337–362.CrossRefGoogle Scholar
  63. 63.
    Sutcliffe, G., & Benzmüller, C. (2010). Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure. Journal of Formalized Reasoning, 73(1), 1–27.Google Scholar
  64. 64.
    Tait, W.W. (1966). A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic. Bulletin of the American Mathematical Society, 72(6), 980–983.CrossRefGoogle Scholar
  65. 65.
    Takahashi, M. (1967). A proof of cut-elimination theorem in simple type theory. Journal of the Mathematical Society of Japan, 19, 399–410.CrossRefGoogle Scholar
  66. 66.
    Takeuti, G. (1954). On a generalized logic calculus. Japanese. Journal of Mathematics 23, 39–96 (1953). Errata: ibid, 24, 149–156.Google Scholar
  67. 67.
    Takeuti, G. (1960). An example on the fundamental conjecture of GLC. Journal of the Mathematical Society of Japan, 12, 238–242.CrossRefGoogle Scholar
  68. 68.
    Takeuti, G. (1975). Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 81. Elsevier.Google Scholar
  69. 69.
    Wisniewski, M., & Steen, A. (2015). Embedding of First-Order Nominal Logic into Higher-Order Logic. In Beziau, J. Y. et al. Handbook of the 5th world congress and school on universal logic (UNILOG’15), Istanbul, Turkey (pp. 337–339).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  2. 2.CSLI/Cordura HallStanford UniversityStanfordUSA

Personalised recommendations