Abstract
A classical Gentzen-type system is one which employs two-sided sequents, together with structural and logical rules of a certain characteristic form. A decent Gentzen-type system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula property. In this survey we explain the main difficulty in developing classical Gentzen-type systems with these properties for many-valued logics. We then illustrate with numerous examples the various possible ways of overcoming this difficulty, and the strong connection between semantic completeness and cut-elimination in each case. Our examples include practically all 3- valued and 4-valued logics, as well as Godel finite-valued logics and some well-known infinite-valued logics.
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