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40
Simple Consequence Relations
 Information and Computation
, 1991
"... 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 (incl ..."
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Cited by 98 (18 self)
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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 nonmonotonic logics) and for a general, semanticsindependent 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 nonuniform, but still efficient, representations of consequence relations.
Natural 3valued Logics  Characterization and Proof Theory
 Journal of Symbolic Logic
, 1991
"... Introduction Manyvalued logics in general and 3valued logic in particular is an old subject which had its beginning in the work of Lukasiewicz [Luk]. Recently there is a revived interest in this topic, both for its own sake (see, e.g. [Ho]), and also because of its potential applications in sever ..."
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Cited by 43 (14 self)
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Introduction Manyvalued logics in general and 3valued logic in particular is an old subject which had its beginning in the work of Lukasiewicz [Luk]. Recently there is a revived interest in this topic, both for its own sake (see, e.g. [Ho]), and also because of its potential applications in several areas of computer science, like: proving correctness of programs ([Jo]), knowledge bases ([CP]) and Artificial Intelligence ([Tu]). There are, however, a huge number of 3valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear and their proof theory is frequently not well developed. This state of affairs makes both the use of 3valued logics and doing fruitful research on them rather difficult. Our first goal in this work is, accordingly, to identify and characterize a class of 3valued logics which might be called natural. For this we use the general framework for characterizing and inve
A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... 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 threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 28 (11 self)
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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 threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional 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 resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
Formal Inconsistency and Evolutionary Databases
 LOGIC AND LOGICAL PHILOSOPHY
, 2000
"... This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct soun ..."
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Cited by 28 (10 self)
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This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their firstorder extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent firstorder logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.
Connectionbased Theorem Proving in Classical and Nonclassical Logics
 Journal of Universal Computer Science
, 1999
"... Abstract: We present a uniform procedure for proof search in classical logic, intuitionistic logic, various modal logics, and fragments of linear logic. It is based on matrix characterizations of validity in these logics and extends Bibel’s connection method, originally developed for classical logic ..."
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Cited by 22 (14 self)
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Abstract: We present a uniform procedure for proof search in classical logic, intuitionistic logic, various modal logics, and fragments of linear logic. It is based on matrix characterizations of validity in these logics and extends Bibel’s connection method, originally developed for classical logic, accordingly. Besides combining a variety of different logics it can also be used to guide the development of proofs in interactive proof assistants and shows how to integrate automated and interactive theorem proving. 1
Modulated Fibring and the Collapsing Problem
, 2001
"... Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse ..."
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Cited by 20 (12 self)
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Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1
Two’s company: “The humbug of many logical values
 In Logica Universalis
, 2005
"... How was it possible that the humbug of many logical values persisted over the last fifty years? —Roman Suszko, 1976. Abstract. The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of manyvaluedness. According to him, as he would often repeat, “there ..."
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Cited by 16 (12 self)
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How was it possible that the humbug of many logical values persisted over the last fifty years? —Roman Suszko, 1976. Abstract. The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of manyvaluedness. According to him, as he would often repeat, “there are but two logical values, true and false. ” As a matter of fact, a result by WójcickiLindenbaum shows that any tarskian logic has a manyvalued semantics, and results by Suszkoda CostaScott show that any manyvalued semantics can be reduced to a twovalued one. So, why should one even consider using logics with more than two values? Because, we argue, one has to decide how to deal with bivalence and settle down the tradeoff between logical 2valuedness and truthfunctionality, from a pragmatical standpoint. This paper will illustrate the ups and downs of a twovalued reduction of logic. Suszko’s reductive result is quite nonconstructive. We will exhibit here a way of effectively constructing the twovalued semantics of any logic that has a truthfunctional finitevalued semantics and a sufficiently expressive language. From there, as we will indicate, one can easily go on to provide those logics with adequate canonical systems of sequents or tableaux. The algorithmic methods developed here can be generalized so as to apply to many nonfinitely valued logics as well —or at least to those that admit of computable quasi tabular twovalued semantics, the socalled dyadic semantics.
Towards an efficient Tableau Proof Procedure for MultipleValued Logics
 In Proceedings, Workshop on Computer Science Logic
, 1990
"... One of the obstacles against the use of tableaubased theorem provers for nonstandard logics is the inefficiency of tableau systems in practical applications, though they are highly intuitive and extremely flexible from a proof theoretical point of view. We present a method for increasing the effic ..."
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Cited by 15 (5 self)
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One of the obstacles against the use of tableaubased theorem provers for nonstandard logics is the inefficiency of tableau systems in practical applications, though they are highly intuitive and extremely flexible from a proof theoretical point of view. We present a method for increasing the efficiency of tableau systems in the case of multiplevalued logics by introducing a generalized notion of signed formulas and give sound and complete tableau systems for arbitrary propositional finitevalued logics. Introduction One of the main advantages of the method of semantic tableaux [Smullyan, 1968, Beth, 1986] is that it yields analytic proof theories for a wide variety of standard and nonstandard logics within a single framework. With relatively minor modifications tableau proof systems can be designed for such different logics as temporal, intuitionistic and multiplevalued logics [Wolper, 1981, Fitting, 1983, Schmitt, 1989]. In addition, one could easily obtain tableau proof system...
Hypersequent calculi for Gödel logics: a survey
 Journal of Logic and Computation
, 2003
"... 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 Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for ..."
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Cited by 13 (4 self)
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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 Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinitevalued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to firstorder as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1
Are Tableaux an Improvement on TruthTables? CutFree proofs and Bivalence
, 1992
"... We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutf ..."
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Cited by 12 (0 self)
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We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutfree proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableaulike method without affecting its "analytic" nature. 1 Introduction The truthtable method, introduced by Wittgenstein in his Tractatus LogicoPhilosophicus, provides a decision procedure for propositional logic which is immediately implementable on a machine. However this timehonoured method is usually mentioned only to be immediately dismissed because of its incurable inefficiency. The wellknown tableau method (which is closely related to Gentzen's cutfree sequent calculus) is commonly regarded as a "shortcut" in testing the logical validity of complex propositions...