| Summary: | The paper is devoted to the study of one of the aspects of the so-called Evidence Algorithm programme advanced by Academician V.M.~Glushkov and connected with the problem of automated theorem-proving search in the signature of first-order theories that can use different logics, such as classical, intuitionistic, and modal ones. An approach to the construction of such sequent logics (with or without equality) is described. It exploits the original notions of admissibility and compatibility, which permits to avoid preliminary skolemization being a forbidden operation for a number of non-classical logics in general. Following the approach, the cut-free sequent (modal) calculi avoiding the dependence of inference search on different orders of quantifier rules applications are described. Results on the coextensivity of sequent calculi are given. The research gives a way to the construction of computer-oriented quantifier-rule-free calculi for classical and intuitionistic logics and their modal extensions.
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