Natural Deduction for Full S5 Modal Logic with Weak Normalization

• Authors: Ana Teresa Martins;Lília Ramalho Martins
• Venue: Electronic Notes in Theoretical Computer Science (ENTCS)
• Year: 2006

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 Normalizacao para a Logica Classica'', Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalizacao para a Logica Classica, 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 [Mere, M.C., and L.C. Pereira. ''A New Cut-Free System for Intuitionistic S4'', Encontro Brasileiro de Logica, 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; Braganca, 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; Brauner, 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 Normalizacao para a Logica Classica'', Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalizacao para a Logica Classica, 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.