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. Canonical propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules which have the subformula property, introduce exactly one occurrence of a connective in their conclusion, and no other occurrence of any connective is mentioned anywhere else in their formulation. We provide a constructive coherence criterion for the non-triviality of such systems, and show that a system of this kind admits cut elimination i it is coherent. We show also that the semantics of such systems is provided by non-deterministic two-valued matrices (2-Nmatrices). 2Nmatrices form a natural generalization of the classical two-valued matrix, and every coherent canonical system is sound and complete for one of them. Conversely, with any 2-Nmatrix it is possible to associate a coherent canonical Gentzen-type system which has for each connective at most one introduction rule for each side, and is sound and complete for th...

Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1

In this paper we present a method for automated theorem proving in manyvalued logics whose algebra of truth values is a nite distributive lattice with operators. This class of many-valued logics includes many logics that occur in a natural way in applications. The method uses the Priestley dual of the algebra of truth values instead of the algebra itself; this dual is used as a finite set of possible worlds. We first present a procedure that constructs, for every formula in the language of such a logic, a set of signed clauses such that is a theorem if and only if is unsatisfiable. Compared to related approaches, the method presented here leads in many cases to a reduction of the number of clauses that are generated, especially when the set of truth values is not linearly ordered. We then discuss several possibilities for checking the unsatisfiability of , among which a version of signed hyperresolution, and give several examples.

this paper every formula is equivalent to a formula in negation normal form

We apply chaining techniques to automated theorem proving in many-valued logics. In particular, we show that superposition specializes to a refined version of the manyvalued resolution rules introduced by Baaz and Ferm uller, and that ordered chaining can be specialized to a refutationally complete inference system for regular clauses. 1. Introduction A general method for automated theorem proving in finitely-valued logics is the many-valued resolution method by Baaz and Fermuller [1]. Their results have been extended in [7, 8], [10], and [2], where various versions of signed resolution are defined. Signed resolution rules have also been proposed for annotated logics by Kifer and Lozinskii [9] and Lu, Murray and Rosenthal [10]. Hahnle [8] has developed a hyper-resolution method for the so-called regular logics which is directly modeled after classical hyperresolution. The completeness proofs are more or less directly derived from those for classical logic. The calculi in [10] are obt...

We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) many-valued logics --- called projective logics --- characterized by a special format of their semantics. All finite-valued logics as well as infinite-valued Godel logic are projective. As a case-study, sequent of relations calculi for Godel logics are derived. A comparison with some other analytic calculi is provided.

This paper is an overview of a variety of results, all centered around a common theme, namely embedding of non-classical logics into first order logic and resolution theorem proving. We present several classes of non-classical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures. 1

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 Gentzentype system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula property. In this tutorial 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. Our examples include practically all 3-valued logics, the most important class of 4-valued logics, as well as central infinite-valued logics (like GodelDummett logic, S5 and some substructural logics). 1

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