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  1. Proof Analysis in Modal Logic.Sara Negri - 2005 - Journal of Philosophical Logic 34 (5-6):507-544.
    A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.
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  • Normalization and Excluded Middle. I.Jonathan P. Seldin - 1989 - Studia Logica 48 (2):193 - 217.
    The usual rule used to obtain natural deduction formulations of classical logic from intuitionistic logic, namely.
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  • A Cut-Free Gentzen-Type System for the Modal Logic S.Masahiko Sato - 1980 - Journal of Symbolic Logic 45 (1):67-84.
  • Normalization Theorems for Full First Order Classical Natural Deduction.Gunnar Stålmarck - 1991 - Journal of Symbolic Logic 56 (1):129-149.
  • A New S4 Classical Modal Logic in Natural Deduction.Maria Da Paz N. Medeiros - 2006 - Journal of Symbolic Logic 71 (3):799 - 809.
    We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.
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  • Gentzen Method in Modal Calculi, II.Masao Ohnishi & Kazuo Matsumoto - 1975 - Journal of Symbolic Logic 40 (3):467-468.
  • A Cut-Free Gentzen Formulation Of The Modal Logic S5.T. Braüner - 2000 - Logic Journal of the IGPL 8 (5):629-643.
    The goal of this paper is to introduce a new Gentzen formulation of the modal logic S5. The history of this problem goes back to the fifties where a counter-example to cut-elimination was given for an otherwise natural and straightforward formulation of S5. Since then, several cut-free Gentzen style formulations of S5 have been given. However, all these systems are technically involved, and furthermore, they differ considerably from Gentzen's original formulation of classical logic. In this paper we give a new (...)
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  • A New S4 Classical Modal Logic in Natural Deduction.Maria Paz N. Medeirodas - 2006 - Journal of Symbolic Logic 71 (3):799-809.
    We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.
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