# Classical Natural Deduction for S4 Modal Logic

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## Abstract

This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dualcontext to formulate S4 modal logic. The modal λ*μ*-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λ*μ*-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λ*μ*-calculus are shown. Finally, the computational interpretation of the modal λ*μ*-calculus, especially the computational meaning of the modal possibility operator, is discussed.

### Keywords:

Functional Programming Classical S4 Modal Logic Curry-Howard Isomorphism Lambda-Calculus Staged Computation## Preview

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© Ohmsha and Springer Japan jointly hold copyright of the journal. 2011