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Automated support for investigating paraconsistent and other logics
"... Abstract. We automate the construction of analytic sequent calculi and effective semantics for a large class of logics formulated as Hilbert calculi. Our method applies to infinitely many logics, which include the family of paraconsistent Csystems, as well as to other logics for which neither analy ..."
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Abstract. We automate the construction of analytic sequent calculi and effective semantics for a large class of logics formulated as Hilbert calculi. Our method applies to infinitely many logics, which include the family of paraconsistent Csystems, as well as to other logics for which neither analytic calculi nor suitable semantics have so far been available. 1
Maximal and premaximal paraconsistency in the framework of threevalued semantics
 STUDIA LOGICA,
, 2011
"... Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the c ..."
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Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or nondeterministic threevalued matrices. We show that all reasonable paraconsistent logics based on threevalued deterministic matrices are maximal in our strong sense. This applies to practically all threevalued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on threevalued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these nondeterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) threevalued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core ” of maximal paraconsistency of all possible paraconsistent determinizations of a nondeterministic matrix, thus representing what is really essential for their maximal paraconsistency.
Maximally paraconsistent threevalued logics
 Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning (KR’10
, 2010
"... Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper, we introduce the strongest possible notion of maximal paraconsistency, and investigate it in th ..."
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Cited by 3 (1 self)
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Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper, we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or nondeterministic threevalued matrices. We first show that most of the logics that are based on properly nondeterministic threevalued matrices are not maximally paraconsistent. Then we show that in contrast, in the deterministic case all the natural threevalued paraconsistent logics are maximal. This includes wellknown threevalued paraconsistent logics like P1, LP, J3, PAC and SRM3, as well as any extension of them obtained by enriching their languages with extra threevalued connectives.
Cutfree Sequent Calculi for Csystems with Generalized Finitevalued Semantics
"... In [5], a general method was developed for generating cutfree ordinary sequent calculi for logics that can be characterized by finitevalued semantics based on nondeterministic matrices (Nmatrices). In this paper, a substantial step towards automation of paraconsistent reasoning is made by applyin ..."
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In [5], a general method was developed for generating cutfree ordinary sequent calculi for logics that can be characterized by finitevalued semantics based on nondeterministic matrices (Nmatrices). In this paper, a substantial step towards automation of paraconsistent reasoning is made by applying that method to a certain crucial family of thousands of paraconsistent logics, all belonging to the class of Csystems. For that family, the method produces in a modular way uniform Gentzentype rules corresponding to a variety of axioms considered in the literature. 1
ATaming Paraconsistent (and Other) Logics: An Algorithmic Approach
"... We develop a fully algorithmic approach to “taming ” logics expressed Hilbertstyle, i.e., reformulating them in terms of analytic sequent calculi, and useful semantics. Our approach applies to Hilbert calculi extending the positive fragment of propositional classical logic with axioms of a certain ..."
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We develop a fully algorithmic approach to “taming ” logics expressed Hilbertstyle, i.e., reformulating them in terms of analytic sequent calculi, and useful semantics. Our approach applies to Hilbert calculi extending the positive fragment of propositional classical logic with axioms of a certain general form that contain new unary connectives. Our work encompasses various results already obtained for specific logics. It can be applied to new logics, as well as to known logics for which an analytic calculus or a useful semantics has so far not been available. A Prolog implementation of the method is described.
Nondeterministic combination of connectives
, 2011
"... Combined connectives arise when combining logics [12] and are also useful for analyzing the common properties of two connectives within a given logic [11]. A nondeterministic semantics and a Hilbert calculus are proposed for the meetcombination of connectives (and other language constructors) of a ..."
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Combined connectives arise when combining logics [12] and are also useful for analyzing the common properties of two connectives within a given logic [11]. A nondeterministic semantics and a Hilbert calculus are proposed for the meetcombination of connectives (and other language constructors) of any matrix logic endowed with a Hilbert calculus. The logic enriched with such combined connectives is shown to be a conservative extension of the original logic. It is also proved that both soundness and completeness are preserved. Illustrations are provided for classical propositional logic.
Two Many Values: An algorithmic outlook on Suszko’s Thesis
"... Abstract—In spite of the multiplication of truthvalues, a noticeable shade of bivalence lurks behind the canonical notion of entailment that manyvalued logics inherit from the 2valued case. Can this bivalence be somehow used to our advantage? The present note briefly surveys the progress made in ..."
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Abstract—In spite of the multiplication of truthvalues, a noticeable shade of bivalence lurks behind the canonical notion of entailment that manyvalued logics inherit from the 2valued case. Can this bivalence be somehow used to our advantage? The present note briefly surveys the progress made in the last three decades toward making that theme precise from an abstract point of view and extracting some useful procedures from it, harvesting some of its most favorable crops on the domains of semantics and prooftheory.
Ideal Paraconsistent Logics
, 2011
"... We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every threevalued paraconsistent lo ..."
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We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every threevalued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n> 2 there exists an extensive family of ideal nvalued logics, each one of which is not equivalent to any kvalued logic with k < n.
Extending FloydHoare Logic for Partial Pre and
"... Abstract. Traditional (classical) FloydHoare logic is defined for a case of total pre and postconditions while programs can be partial. In the chapter we propose to extend this logic for partial conditions. To do this we first construct and investigate special program algebras of partial predicate ..."
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Abstract. Traditional (classical) FloydHoare logic is defined for a case of total pre and postconditions while programs can be partial. In the chapter we propose to extend this logic for partial conditions. To do this we first construct and investigate special program algebras of partial predicates, functions, and programs. In such algebras program correctness assertions are presented with the help of a special composition called FloydHoare composition. This composition is monotone and continuous. Considering the class of constructed algebras as a semantic base we then define an extended logic – Partial FloydHoare Logic – and investigate its properties. This logic has rather complicated soundness constraints for inference rules, therefore simpler sufficient constraints are proposed. The logic constructed can be used for program verification.