We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and non-monotonic logics) and for a general, semantics-independent classification of standard connectives via equations on consequence relations (these include Girard's "multiplicatives" and "additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as good representations in each case (especially from the implementation point of view) are explained. We end by briefly outlining (with examples) some methods for developing non-uniform, but still efficient, representations of consequence relations.

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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

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.

In this paper we dene cut-free hypersequent calculi for some intermediate logics semantically characterized by bounded Kripke models. In particular we consider the logics characterized by Kripke models of bounded width Bwk , by Kripke models of bounded cardinality Bck and by linearly ordered Kripke models of bounded cardinality Gk . The latter family of logics coincides with nite-valued Godel logics. Our calculi turn out to be very simple and natural. Indeed, for each family of logics (respectively, Bwk , Bck and Gk ), they are dened by adding just one structural rule to a common system, namely the hypersequent calculus for Intuitionistic Logic. This structural rule reects in a natural way the characteristic semantical features of the corresponding logic. 1 Introduction Kripke models provide a suitable semantical characterization of propositional intermediate logics (see [8]), that is, logics including the Intuitionistic one and included in Classical Logic. In this paper we inv...

. Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. Keywords: Partial functions, many-valued logic, sorted logic, tableau. 1 Introduction Many practical applications of deduction systems in mathematics and computer science rely on the correct and efficient treatment of partial functions. For this purpose...

A sequent calculus S3 for ̷Lukasiewicz’s logic L3 is presented. The completeness theorem is proved relatively to a bivalent semantics equivalent to the non truthfunctional bivalent semantics for L3 proposed by Suszko in 1975. A distinguishing property of the approach proposed here is that we are keeping the format of the classical sequent calculus as much as possible. Mathematics Subject Classification: 03B50, 03F03 1.

We suggest a general formalism of four-valued reasoning, called biconsequence relations, intended to serve as a logical framework for reasoning with incomplete and inconsistent data. The formalism is based on a four-valued semantics suggested by Belnap [3]. As for the classical sequent calculus, any four-valued connective can be dened in biconsequence relations using suitable introduction and elimination rules. In addition, various three-valued and partial logics are shown to be special cases of this formalism obtained by imposing appropriate additional logical rules. We show also that such rules are instances of a single logical principle called coherence. The latter can be considered as a general requirement securing that the information we can infer in this framework will be classically coherent. Keywords: biconsequence relations, four-valued logics, inconsistency, incompleteness, partial logics, three-valued logics. 1 1

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finitevalued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight. Keywords: finite-valued logic, labeled calculus, signed formula, sets-as-signs

This paper tries to identify the basic problems encountered in automated theorem proving in manyvalued logics and demonstrates to which extent they can be currently solved. To this end a number of recently developed techniques are reviewed. We list the avenues of research in many-valued theorem proving that are in our eyes the most promising. 1 Introduction The purpose of this note is to review a number of techniques that lead to a computationally adequate representation of the search space of many-valued logics and to identify the avenues of research in many-valued theorem proving that are in our eyes the most promising. We do not mention the large number of possible applications of many-valued theorem proving, but refer to [15] for an extensive list of applications and to [18] for a case study. If one is doing many-valued deduction, typically a number of problems that are not as much prominent in classical deduction have to be addressed: 1. The number of case distinctions is much la...