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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 dual-context to formulate S4 modal logic. The modal ***μ -calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the ***μ -calculus and the modal *** -calculus. Subject reduction, confluency, and strong normalization of the modal ***μ -calculus are shown. Finally, the computational interpretation of the modal ***μ -calculus, especially the computational meaning of the modal possibility operator, is discussed.