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Using simple algebraic methods we give Lukasiewicz-style refutation calculi for the following intermediate logics: finite logics, LC, Yankov's logic, the logic of the weak law of excluded middle, Medvedev's logic, and certain logics without the finite model property.

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Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics.