Full classical S5 in natural deduction with weak normalization
Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on...
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| Published in: | Annals of pure and applied logic 2008-03, Vol.152 (1), p.132-147 |
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| Main Authors: | , |
| Format: | Article |
| Language: | eng |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on Mathematics, 2006] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic only for a language without
∨
,
∃
and with a restricted application of
reduction ad absurdum. Reduction steps related to
∨
,
∃
and classical negation bring about many problems solved only rather recently. For classical S5 modal logic, Prawitz defined a normalizable system, but for a language without
∨
,
∃
,
◊
and, for a propositional language without
◊
, Medeiros [M.da P.N. Medeiros, A new S4 classical modal logic in natural deduction, Journal of Symbolic Logic 71 (3) (2006) 799–809] presented a normalizable system for classical S4. We can mention many cut-free Gentzen systems for S4, S5 and K45/K45D, some normalizable natural deduction systems for intuitionistic modal logics and one more for full classical S4, but not for full classical S5. Here our focus is on the definition of a classical and normalizable natural deduction system for S5, taking not only
□
and
◊
as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure is based on the strategy proposed by Massi [C.D.B. Massi, Provas de normalizaçaõ para a lógica clássica, Ph.D. Thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990] and Pereira and Massi [L.C. Pereira, C.D.B. Massi, Normalização para a lógica clássica, in: O que nos faz pensar, Cadernos de Filosofia da PUC-RJ, vol. 2, 1990, pp. 49–53] for first-order classical logic to cope with the combined use of classical negation, disjunction and the existential quantifier. Here we extend such results to deal with
□
and
◊
too. The elimination rule for
◊
uses the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of
□
. Beyond weak normalization, we also prove the subformula property for full S5. |
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| ISSN: | 0168-0072 |