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From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2021 Association for Computing Machinery.
ISSN
1529-3785
eISSN
1557-945X
DOI
10.1145/3461661
Publisher site
See Article on Publisher Site

Abstract

We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

Journal

ACM Transactions on Computational Logic (TOCL)Association for Computing Machinery

Published: Jul 22, 2021

Keywords: Natural deduction

References