Article ID Journal Published Year Pages File Type
423384 Electronic Notes in Theoretical Computer Science 2006 12 Pages PDF
Abstract

Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [Prawitz, D., “Natural Deduction: A Proof-theoretical Study”, Stockholm Studies in Philosophy 3, Almqvist and Wiksell, Stockholm, 1965] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic just for a language without ∨, ∃ and with a restricted application of reduction ad absurdum. Reduction steps related to ∨, ∃ and classical negation brings about a lot of problems solved only rather recently [Seldin, J., Normalization and Excluded Middle I, Studia Logica 48 (1989), 193–217; Stalmark, G., Normalizations Theorems for Full First Order Classical Natural Deduction, The Journal of Symbolic Logic 56 (1991); Massi, C.D.B., “Provas de Normalização para a Lógica Clássica”, Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalização para a Lógica Clássica, O que nos faz pensar, Cadernos de Filosofia da PUC-RJ 2 (1990), 49–53]. For classical S4/S5 modal logics, Prawitz defined normalizable systems, but for a language without ∨, ∃ and ◊. We can mention cut-free Gentzen systems for S4/S5 [Meré, M.C., and L.C. Pereira. “A New Cut-Free System for Intuitionistic S4”, Encontro Brasileiro de Lógica, Salvador, 1996; Mints, G.E., “Lewis' Systems and System T (1965–1973)”, Selected Papers in Proof Theory, Studies in Proof Threory, North-Holland, 1992, 221–293; Mints, G.E., Cut-Free Calculi of the S5 Type Studies in Constructive Mathematics and Mathematical Logic, Part II, Seminars in Mathematics 8 (1970), 79–82; Bragança, R.C.M., E.H. Haeusler, and L.C. Pereira, A New Cut-Free Sequent calculus for S5, The Bulletin of Symbolic Logic 5 (1999), 497–498; Braüner, T., A Cut-Free Gentzen Formulation of the Modal Logic S5, Logic Journal of the IGPL 8 (2000), 629–643; H. Wansing. Sequent Calculi for Normal Modal Propositional Logics, Journal of Logic and Computation 4 (1994), 125–142], normalizable natural deduction systems for intuitionistic modal logics [Simpson, A.K., “The Proof Theory and Semantics of Intuitionistic Modal Logic”, Ph.D. thesis, University of Edinburgh, 1994; Masini, A., 2-Sequent Calculus: Intuitionism and Natural Deduction, Journal of Logic and Computation 3 (1993), 533–562] and for full classical S4 [Martins, L.R., and A.T. Martins, “Normalizable Natural Deduction Rules for S4 Modal Operators”, World Congress on Universal Logic, Montreux, 2005, 79–80], but not for full classical S5. Here our focus is in the definition of a classical and normalizable natural deduction system for S5, taken not only □ and ◊ as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure will be based on the strategy proposed by [Massi, C.D.B., “Provas de Normalização para a Lógica Clássica”, Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalização para a Lógica Clássica, O que nos faz pensar, Cadernos de Filosofia da PUC-RJ 2 (1990), 49–53] to cope with the combined use of classical negation, ∨ and ∃. We will extend such results to deal with ◊ too. The elimination rule for ◊ will use the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of □. Weak normalization and subformula property is proved for full S5.

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