A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC) . The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S 5, and can be easily extended to other modal logics.