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Sequent Systems for Modal Logics

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Handbook of Philosophical Logic

Part of the book series: Handbook of Philosophical Logic ((HALO,volume 8))

Abstract

This chapter surveys the application of various kinds of sequent systems to modal and temporal logic, also called tense logic. The starting point are ordinary Gentzen sequents and their limitations both technically and philosophically. The rest of the chapter is devoted to generalizations of the ordinary notion of sequent. These considerations are restricted to formalisms that do not make explicit use of semantic parameters like possible worlds or truth values, thereby excluding, for instance, Gabbay’s labelled deductive systems, indexed tableau calculi, and Kanger-style proof systems from being dealt with. Readers interested in these types of proof systems are referred to [Gabbay, 1996], [Goré, 1999] and [Pliuškeviene, 1998]. Also Orlowska’s [1988; 1996] Rasiowa-Sikorski-style relational proof systems for normal modal logics will not be considered in the present chapter. In relational proof systems the logical object language is associated with a language of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate finite sequences of relational formulas constructed from relational terms and relational operations. An overview of ordinary sequent systems for non-classical logics is given in [Ono, 1998], and for a general background on proof theory the reader may consult [Troelstra and Schwichtenberg, 2000].

[T]he framework of ordinary sequents is not capable of handling all interesting logics. There are logics with nice, simple semantics and obvious interest for which no decent, cut-free formulation seems to exist . . .. Larger, but still satisfactory frameworks should, therefore, be sought. A. Avron [1996, p. 3]

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Bibliography

  1. M. D’Agostino and M. Mondadori, The Taming of the Cut. Classical Refutations with Analytic Cut, Journal of Logic and Computation 4 (1994), 285–319.

    Google Scholar 

  2. H. Andréka, I. Németi and J. van Benthem, Modal Languages and Bounded Fragments of Predicate Logic, Journal of Philosophical Logic 27 (1998),f 217–274.

    Google Scholar 

  3. A. Avron, On modal systems having arithmetical interpretations, Journal of Symbolic Logic 49 (1984), 935–942.

    Google Scholar 

  4. A. Avron, Using Hypersequents in Proof Systems for Non-classical Logics, Annals of Mathematics and Artificial Intelligence 4 (1991), 225–248.

    Google Scholar 

  5. A. Avron, Natural 3-valued Logics-Characterization and Proof Theory, Journal of Symbolic Logic, (56) 1991, 276–294.

    Google Scholar 

  6. A. Avron, The Method of Hypersequents in Proof Theory of Propositional Non-Classical Logics, in: W. Hodges et al. (eds.), Logic: From Foundations to Applications, Oxford University Press, Oxford, 1996, 1–32.

    Google Scholar 

  7. N.D. Belnap, Display Logic, Journal of Philosophical Logic 11 (1982), 375–417. Reprinted with minor changes as §62 of A.R. Anderson, N.D. Belnap, and J.M. Dunn, Entailment: the logic of relevance and necessity. Vol. 2, Princeton University Press, Princeton, 1992.

    Google Scholar 

  8. N.D. Belnap, Linear Logic Displayed, Notre Dame Journal of Formal Logic 31 (1990), 14–25.

    Google Scholar 

  9. N.D. Belnap, The Display Problem, in: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 1996, 79–92.

    Google Scholar 

  10. J. van Benthem, Essays in Logical Semantics, Kluwer Academic Publishers, Dordrecht, 1986.

    Google Scholar 

  11. J. van Benthem, Language in Action. North-Holland, Amsterdam, 1991.

    Google Scholar 

  12. J. van Benthem, Exploring Logical Dynamics, CSLI Publications, Stanford, 1996.

    Google Scholar 

  13. S. Blamney and L. Humberstone, A Perspective on Modal Sequent Logic, Publications of the Research Institute for Mathematical Sciences, Kyoto University 27 (1991), 763–782.

    Google Scholar 

  14. G. Boolos, Don’t eliminate cut, Journal of Philosophical Logic 13 (1984), 373–378.

    Google Scholar 

  15. T. Borghuis, Interpreting modal natural deduction in type theory, in: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 1993, 67–102.

    Google Scholar 

  16. T. Borghuis, Coming to Terms with Modal Logic: On the interpretation of modalities in typed X-calculus, PhD thesis, Department of Computer Science, University of Eindhoven, 1994.

    Google Scholar 

  17. T. , Modal Pure Type Systems, Journal of Logic, Language and Information 7 (1998), 265–296.

    Google Scholar 

  18. M. Bošic and K. Došen, Models for normal intuitionistic modal logics, Studia Logica 43 (1984), 217–245.

    Google Scholar 

  19. T. Braüner, A Cut-Free Gentzen Formulation of the Modal Logic S5, Logic Journal of the IGPL 8 (2000), 629–643.

    Google Scholar 

  20. R. Bull and K. Segerberg, Basic Modal Logic. In: D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol. II, Extensions of Classical Logic, Reidel, Dordrecht, 1984, 1–88.

    Google Scholar 

  21. C. Cerrato, Modal sequents for normal modal logics, Mathematical Logic Quarterly 39 (1993), 231–240.

    Google Scholar 

  22. C. Cerrato, Modal sequents, in: H. Wansing (ed), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 1996, 141–166.

    Google Scholar 

  23. B. Chellas, Modal Logic: An Introduction, Cambridge University Press, Cambridge, 1980.

    Google Scholar 

  24. R. Davies and F. Pfenning, A Modal Analysis of Staged Computation, 2000, to appear in: Journal of the ACM.

    Google Scholar 

  25. S. Demri and R. Gore, Cut-free display calculi for nominal tense logics. In Proc Tableaux ‘99, pp. 155–170. Lecture Notes in AI, Springer-Verlag, Berlin, 1999.

    Google Scholar 

  26. S. Demri and R. Goré, Display calculi for logics with relative accessibility relations, Journal of Logic, Language and Information 9 (2000), 213–236.

    Google Scholar 

  27. K. Došen, Sequent-systems for modal logic, Journal of Symbolic Logic 50 (1985), 149–159.

    Google Scholar 

  28. K. Došen, Sequent systems and groupoid models I, Studia Logica 47 (1988), 353–389.

    Google Scholar 

  29. A. Dragalin, Mathematical Intuitionism. Introduction to Proof Theory, American Mathematical Society, Providence, 1988.

    Google Scholar 

  30. J. M. Dunn, Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operations, in: J. van Eijk (ed.), Logics in AI, Proc. European Workshop ELIA 1990, Lecture Notesin Computer Science 478, Berlin, 1990, 31–51.

    Google Scholar 

  31. J.M. Dunn, Partial-Gaggles Applied to Logics with Restricted Structural Rules, in: P. Schroeder-Heister and K. Došen (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, 63–108.

    Google Scholar 

  32. J.M. Dunn, Gaggle Theory Applied to Modal, Intuitionistic, and Relevance Logics, in: I. Max and W. Stelzner (eds.), Logik und Mathematik: Frege-Kolloquium 1993, de Gruyter, Berlin, 1995, 335–368.

    Google Scholar 

  33. M. Fitting, Basic Modal Logic, in: D. Gabbay et al. (eds), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Logical Foundations, Oxford UP, Oxford, 1993, 365–448.

    Google Scholar 

  34. H. Friedman, Equality between functionals, in: R. Parikh (ed.), Logic Colloquium Boston 1972–73, Springer Lecture Notes in Mathematics Vol. 453, Springer-Verlag, Berlin, 1975, 22–37.

    Google Scholar 

  35. D. Gabbay, Labelled Deductive Systems: Volume 1. Foundations, Oxford University Press, Oxford, 1996.

    Google Scholar 

  36. D. Gabbay and R. de Quieroz, Extending the Curry Howard interpretation to linear, relevant and other resource logics, Journal of Symbolic Logic 57 (1992), 1319–1365.

    Google Scholar 

  37. G. Gentzen. Investigations into Logical Deduction, in: M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North Holland, Amsterdam, 1969, 68–131.

    Google Scholar 

  38. English translation of: Untersuchungen iüber das logische Schlieißen, Mathematische Zeitschrift 39 (1934), I 176–210, II 405–431.

    Google Scholar 

  39. J.-Y. Girard, Y. Lafont, and P. Taylor, Proofs and Types, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  40. L. Goble, Gentzen systems for modal logics, Notre Dame Journal of Formal Logic 15 (1974), 455–461.

    Google Scholar 

  41. K. Gödel, Eine Interpretation des intuitionistischen Aussagenkalkiüls, Ergebnisse eines mathematischen Kolloquiums 4 (1933), 39–40.

    Google Scholar 

  42. Reprinted and translated in: S. Feferman et al. (eds), Kurt Godel. Collected Works. Vol. 1, Oxford University Press, Oxford, 1986, 300–303.

    Google Scholar 

  43. R. Goldblatt, Logics of Time and Computation, CSLI Lecture Notes 7, Stanford, CSLI Publications,2nd revised and expanded edition, 1992.

    Google Scholar 

  44. R. Goré., Cut-Free Sequent and Tableau Systems for Propositional Normal Modal Logics, PhD thesis, University of Cambridge Computer Laboratory, Technical Report No. 257, 1992.

    Google Scholar 

  45. R. Gore, Intuitionistic Logic Redisplayed, Technical Report TR-ARP-1- 1995, Australian National University, 1995.

    Google Scholar 

  46. R. Gore, Substructural Logics on Display, Logic Journal of the I GPL 6 (1998), 451–504.

    Google Scholar 

  47. R. Gore, Gaggles, Gentzen and Galois: How to disply your favourite susbstructurla logic, Logic Journal of the IGPL 6 (1998), 669–694.

    Google Scholar 

  48. R. Goré, Tableau Methods for Modal and Temporal Logics, in: M. D’Agostino, D. Gabbay, R. Hahnle, and J. Posegga (eds), Handbook of Tableau Methods, Kluwer Academic Publishers, Dordrecht, 1999, 297–396.

    Google Scholar 

  49. R. Goré, Dual Intuitionistic Logic Revisited, in: R. Dyckhoff (ed.), Proceedings Tableaux 2000, LNAI 1847, Springer-Verlag Berlin, 2000, 252–267.

    Google Scholar 

  50. S. Gottwald, Mehrwertige Logik, Akademie-Verlag, Berlin, 1989.

    Google Scholar 

  51. I. Hacking, What is Logic?, The Journal of Philosophy 76 (1979), 285–319.

    Google Scholar 

  52. G. Helman, Restricted Lambda-abstraction and the Interpretation of Some Non-classical Logics, PhD thesis, Department of Philosophy, University of Pittsburgh, 1977.

    Google Scholar 

  53. J.R. Hindley and J.P. Seldin, Introduction to Combinators and X-Calculus, Cambridge UP, Cambridge, 1986.

    Google Scholar 

  54. W.A. Howard, The formulae-as-types notion of construction, in: J.R. Hindley and J.P. Seldin (eds.), To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, 1980, 479–490.

    Google Scholar 

  55. A. Indrzejczak, Generalised Sequent Calculus for Propositional Modal Logics, Logica Trianguli 1 (1997), 15–31.

    Google Scholar 

  56. A. Indrzejczak, Cut-free Double Sequent Calculus for S5, Logic Journal of the IGPL 6 (1998), 505–516.

    Google Scholar 

  57. R. Kashima, Cut-free sequent calculi for some tense logics, Studia Logica 53 (1994), 119–135.

    Google Scholar 

  58. M. Kracht, Power and Weakness of the Modal Display Calculus, in: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 1996, 93–121.

    Google Scholar 

  59. S. Kripke, Semantical analysis of modal logic I: Normal modal propositional calculi, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 11 (1968), 3–16.

    Google Scholar 

  60. F. von Kutschera, Die Vollständigkeit des Operatorensystems {¬,⋀ ,⋁ ,⊃} für die intuitionistische Aussagenlogik im Rahmen der Gentzensemantik, Archiv für Mathematische Logik und Grundlagenforschung, 11 (1968), 3–16.

    Google Scholar 

  61. R. Lavendhomme and T. Lucas, Sequent calculi and decision procedures for weak modal systems, Studia Logica 65 (2000), 121–145.

    Google Scholar 

  62. D. Leivant, On the proof theory of the modal logic for arithmetic ability, Journal of Symbolic Logic 46 (1981), 531–538.

    Google Scholar 

  63. P. Lukowski, Modal interpretation of Heyting-Brouwer Logic, Bulletin of the Section of Logic 25 (1996), 80–83.

    Google Scholar 

  64. A. Martini and A. Masini, A Computational Interpretation of Modal Proofs, in: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 1996, 213–241.

    Google Scholar 

  65. A. Masini, 2-Sequent calculus: a proof theory of modalities, Annals of Pure and Applied Logic 58 (1992), 229–246, 1992.

    Google Scholar 

  66. G. Mints, Cut-free calculi of the S5 type, Studies in constructive mathematics and mathematical logic. Part II. Seminars in Mathematics 8 (1970), 79–82.

    Google Scholar 

  67. G. Mints, Gentzen-type systems and resolution rules. Part I. Propositional Logic, in: P. Martin-Löf and G. Mints (eds), COLOG-88, Lecture Notes in Computer Science 417, Springer-Verlag, Berlin, 198–231, 1990.

    Google Scholar 

  68. G. Mints, A Short Introduction to Modal Logic, CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992.

    Google Scholar 

  69. G. Mints, Indexed systems of sequents and cut-elimination. Journal of Philosophical Logic 26 (1997), 671–696.

    Google Scholar 

  70. H. Nishimura, A Study of Some Tense Logics by Gentzen’s Sequential Method; Publications of the Research Institute for Mathematical Sciences, Kyoto University 16 (1980), 343–353.

    Google Scholar 

  71. M. Ohnishi and K. Matsumoto, Gentzen Method in Modal Calculi, Osaka Mathematical Journal 9 (1957), 113–130.

    Google Scholar 

  72. M. Ohnishi and K. Matsumoto, Gentzen Method in Modal Calculi, II, Osaka Mathematical Journal 11 (1959), 115–120.

    Google Scholar 

  73. M. Ohnishi, A New Version to Gentzen Decision Procedure for Modal Sentential Calculus S5, Mathematical Seminar Notes 10 (1982), Kobe University, 161–170.

    Google Scholar 

  74. H. Ono, Proof-Theoretic Methods in Nonclassical Logic - an Introduction, MSJ Memoirs 2, Mathematical Society of Japan, 1998, 207–254.

    Google Scholar 

  75. E. Orlowska, Relational interpretation of modal logics, in: H. Andreka, D. Monk and I. Nemeti (eds.), Algebraic Logic. Colloquia Mathematica Societatis Janos Bolyai 54, North Holland, Amsterdam, 443–471, 1988.

    Google Scholar 

  76. E. Orlowska, Relational Proof Systems for Modal Logics, in: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 55–77, 1996.

    Google Scholar 

  77. F. Pfenning and R. Davies, A Judgmental Reconstruction of Modal Logic, Department of Computer Science, Carnegie Mellon University, Pittsburgh, 2000.

    Google Scholar 

  78. A. Pliuškeviċienė, Cut-free Calculus for Modal Logics Containing the Barcan Axiom, in: M. Kracht et al. (eds.), Advances in Modal Logic ’96, CSLI Publications, Stanford, 1998, 157–172.

    Google Scholar 

  79. G. Pottinger, Uniform, cut-free formulations of T, 54 and 55 (Abstract), Journal of Symbolic Logic 48 (1983), 900–901.

    Google Scholar 

  80. R. de Queiroz and D. Gabbay, The functional interpretation of modal necessity, in: M. de Rijke (ed.), Advances in Intensional Logic, Kluwer Academic Publishers, Dordrecht, 1997, 61–91.

    Google Scholar 

  81. R. de Queiroz and D. Gabbay, An introduction to labelled natural deduction, in: H.J. Ohlbach and U. Reyle (eds.), Logic Language and Reasoning. Essays in Honour of Dov Gabbay, Kluwer Academic Publishers, Dordrecht, 1999.

    Google Scholar 

  82. C. Rauszer, An algebraic and Kripke-style approach to a certain extension of intuitionistic logic, Dissertationes Mathematicae, vol. CLXVII, Warsaw, 1980.

    Google Scholar 

  83. G. Restall, Display Logic and Gaggle Theory, Reports on Mathematical Logic 29 (1995), 133–146.

    Google Scholar 

  84. G. Restall, Displaying and Deciding Substructural Logics 1: Logics with Contraposition, Journal of Philosophical Logic 27 (1998), 179–216.

    Google Scholar 

  85. G. Restall, An Introduction to Substructural Logics, Routledge, London, 1999.

    Google Scholar 

  86. D. Roorda, Resource Logics. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 1991.

    Google Scholar 

  87. G. Sambin and S. Valentini, The modal logic of provability. The sequential approach, Journal of Philosophical Logic 11 (1982), 311–342.

    Google Scholar 

  88. K. Sasaki, On intuitionistic modal logic corresponding to extended typed λ-calculus for partial functions, Manuscript, Department of Computer Science, Leipzig University, 1999.

    Google Scholar 

  89. M. Sato. A Study of Kripke-type Models for Some Modal Logics by Gentzen’s Sequential Method. Publications of the Research Institute for Mathematical Sciences, Kyoto University 13 (1977), 381–468.

    Google Scholar 

  90. M. Sato, A cut-free Gentzen-type system for the modal logic 55, Journal of Symbolic Logic 45 (1980), 67–84.

    Google Scholar 

  91. K. Schröter, Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 1 (1955), 214–251.

    Google Scholar 

  92. K. Schütte, Vollständige Systeme modaler und intuitionistischer Logik, Springer-Verlag, Berlin, 1968.

    Google Scholar 

  93. T. Shimura, Cut-Free Systems for the Modal Logic 54.3 and S4.3Grz, Reports on Mathematical Logic 25 (1991), 57–73.

    Google Scholar 

  94. G. Shvarts, style systems for K45 and K45D, in: A. Meyer and M. Taitslin (eds.),Logic at Botik ’89, Lecture Notes in Computer Science 363, Springer-Verlag, Berlin, 1989, 245–256.

    Google Scholar 

  95. R. Smullyan, Analytic cut, Journal of Symolic Logic 33 (1968), 560–564.

    Google Scholar 

  96. M. Takano, Subformula property as a substitute for cut-elimination in modal propositional logics, Mathematica Japonica 37 (1992), 1129–1145.

    Google Scholar 

  97. N. Tennant, The Transmission of Truth and the Transitivity of Deduction, in: D. Gabbay (ed.), What is a Logical System?’, Oxford University Press, Oxford, 1994, 161–178.

    Google Scholar 

  98. A. Troelstra. Lectures on Linear Logic, CSLI Lecture Notes 29, CSLI Publications, Stanford, 1992.

    Google Scholar 

  99. A. Troelstra and H. Schwichtenberg, Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science 43, Second Edition, Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  100. R. de Vrijer, Strong normalization in N - HA%. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 90 (1987), 473–478.

    Google Scholar 

  101. H. Wansing, Formulas-as-types for a hierarchy of sublogics of intuitionistic propositional logic, in: H. Wansing and D. Pearce (eds.), Nonclassical Logics and Information Processing, Springer Lecture Notes in Artificial Intelligence 619, Springer-Verlag, Berlin, 1992, 125–145.

    Google Scholar 

  102. H. Wansing, Sequent systems for normal modal propositional logics, Journal of Logic and Computation 4 (1994), 125–142.

    Google Scholar 

  103. H. Wansing, Displaying Modal Logic, Kluwer Academic Publishers, Dordrecht, 1998.

    Google Scholar 

  104. H. Wansing, Displaying the modal logic of consistency, Journal of Symbolic Logic 64 (1999), 1573–1590.

    Google Scholar 

  105. H. Wansing, Predicate Logics on Display, Studia Logica 62 (1999), 49–75.

    Google Scholar 

  106. H. Wansing, Formulas-as-types for temporal logic, Report, Dresden University of Technology, Institute of Philosophy, 2000.

    Google Scholar 

  107. L. Wittgenstein, Philosophical Investigations, Blackwell, Oxford, 1953.

    Google Scholar 

  108. F. Wolter, On Logics With Coimplication, Journal of Philosophical Logic 27 (1998), 353–387.

    Google Scholar 

  109. F. Wolter and M. Zakharyaschev. Intuitionistic Modal Logic, in: A. Cantini et al. (eds.), Logic and Foundations of Mathematics, Kluwer Academic Publishers, Dordrecht, 1999, 227–238.

    Google Scholar 

  110. R. Zach, Proof Theory of Finite-valued Logics, Diplomarbeit, Institut fur Computersprachen, Technische Universität Wien, 1993.

    Google Scholar 

  111. J.J. Zeman Modal Logic. The Lewis-Modal Systems, Oxford University Press, Oxford, 1973.

    Google Scholar 

  112. J. Zucker and R. Tragesser, The adequacy problem for inferential logic, Journal of Philosophical Logic 7 (1978), 501–516.

    Google Scholar 

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  1. Heinrich Wansing
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  1. King’s College, London, UK

    D. M. Gabbay

  2. Centrum für Informations- und Sprachverarbeitung, Ludwig-Maximilians-Universität München, Germany

    F. Guenthner

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Wansing, H. (2002). Sequent Systems for Modal Logics. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0387-2_2

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