Focusing in Linear Meta-logic
It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics.
KeywordsClassical Logic Proof System Atomic Formula Intuitionistic Logic Linear Logic
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- [Gen69]Gentzen, G.: Investigations into logical deductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
- [Gir06]Girard, J.-Y.: Le Point Aveugle: Cours de logique: Tome 1, Vers la perfection. Hermann (2006)Google Scholar
- [MP04]Miller, D., Pimentel, E.: Linear logic as a framework for specifying sequent calculus. In: van Eijck, J., van Oostrom, V., Visser, A. (eds.) Logic Colloquium 1999: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, pp. 111–135. A. K. Peters Ltd (2004)Google Scholar
- [NM08]Nigam, V., Miller, D.: Focusing in linear meta-logic: Extended report, http://hal.inria.fr/inria-00281631
- [Par92]Parigot, M.: Free deduction: An analysis of “computations” in classical logic. In: Proceedings of the First Russian Conference on Logic Programming, London, UK, pp. 361–380. Springer, Heidelberg (1992)Google Scholar
- [Pfe89]Pfenning, F.: Elf: A language for logic definition and verified metaprogramming. In: Fourth Annual Symposium on Logic in Computer Science, Monterey, CA, June 1989, pp. 313–321 (1989)Google Scholar
- [Pim01]Pimentel, E.G.: Lógica linear e a especificação de sistemas computacionais. PhD thesis, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil, Written in English (December 2001)Google Scholar
- [Pra65]Prawitz, D.: Natural Deduction. Almqvist & Wiksell, Uppsala (1965)Google Scholar