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Logics of Formal Inconsistency
 Handbook of Philosophical Logic
"... 1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory ..."
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Cited by 45 (19 self)
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1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory
A Taxonomy of Csystems
, 2002
"... The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictorin ..."
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Cited by 41 (15 self)
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The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called Csystems, as the LFIs that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of NonContradiction, and we also sharply distinguish these two from the Principle of NonTriviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main Csystems based on classical logic, showing how several wellknown logics in the literature can be recast as such a kind of Csystems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully
Minimal Classical Logic and Control Operators
 In ICALP: Annual International Colloquium on Automata, Languages and Programming, volume 2719 of LNCS
, 2003
"... We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction. ..."
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Cited by 30 (5 self)
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We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction.
A Definitional TwoLevel Approach to Reasoning with HigherOrder Abstract Syntax
 Journal of Automated Reasoning
, 2010
"... Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co ..."
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Cited by 14 (3 self)
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Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use it in a multilevel reasoning fashion, similar in spirit to other metalogics such as Linc and Twelf. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of nonstratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuationmachine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly
2005, ‘A ProofTheoretic Foundation of Abortive Continuations (Extended version
"... Abstract. We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural ” implementation of this logic is Parigot’s classical natural deduction. We then move on to the comp ..."
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Cited by 9 (5 self)
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Abstract. We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural ” implementation of this logic is Parigot’s classical natural deduction. We then move on to the computational side and emphasize that Parigot’s λµ corresponds to minimal classical logic. A continuation constant must be added to λµ to get full classical logic. The extended calculus is isomorphic to a syntactical restriction of Felleisen’s theory of control that offers a more expressive reduction semantics. This isomorphic calculus is in correspondence with a refined version of Prawitz’s natural deduction.
Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Cited by 6 (0 self)
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
AntiIntuitionism and Paraconsistency
 URL = http://www.cle.unicamp.br/eprints/vol 3,n 1,2003.html
, 2003
"... This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of antiintuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is s ..."
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Cited by 3 (1 self)
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This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of antiintuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that antiintuitionistic logics are paraconsistent, and in particular we develop a first antiintuitionistic hierarchy starting with Johansson 's dual calculus and ending up with Godel's threevalued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these antiintuitionistic logics with wellknown paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) manyvalued )n## we show that the antiintuitionistic hierarchy (I )n## obtained from (I )n## does coincide with the hierarchy of the manyvalued paraconsistent logics (P )n## . Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of selfduality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multipleconclusion logics are used as an appropriate environment to deal with them.
Light Functional Interpretation
 Lecture Notes in Computer Science, 3634:477 – 492, July 2005. Computer Science Logic: 19th International Workshop, CSL
, 2005
"... an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs ..."
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Cited by 2 (1 self)
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an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs
Circumscribing Embedded Implications (Without Stratifications)
 Journal of Logic Programming
, 1992
"... This paper is a study of circumscription, not in classical logic, as usual, but in intuitionistic logic. We first review the intuitionistic circumscription of Horn clause logic programs, which was discussed in previous work, and we then consider the larger class of embedded implications . The ordina ..."
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Cited by 2 (2 self)
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This paper is a study of circumscription, not in classical logic, as usual, but in intuitionistic logic. We first review the intuitionistic circumscription of Horn clause logic programs, which was discussed in previous work, and we then consider the larger class of embedded implications . The ordinary circumscription axiom turns out to be inappropriate for this class of rules, and we analyze two alternatives: (1) prioritized circumscription, which works for stratified embedded implications; and (2) partial circumscription, which is independent of the stratification. We then show that these two approaches coincide by identifying a single structure that serves as the final Kripke model for both circumscription axioms. This means that prioritized circumscription and partial circumscription entail exactly the same set of implicational queries. Several applications of these ideas are described, including: (1) an interpretation of negationasfailure; (2) a formalization of indefinite reasoni...
Intuitionistic propositional Logic with the Converse Ackermann Property”, Teorema vol XXII/12
, 2003
"... En este artículo extendemos el espectro de las lógicas con la Conversa de la Propiedad Ackermann. Definimos y axiomatizamos lógicas proposicionales intuicionistas positivas, lógicas subintuicionistas y lógicas intuicionistas con la Conversa de la Propiedad Ackermann. Especialmente interesante es la ..."
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Cited by 1 (1 self)
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En este artículo extendemos el espectro de las lógicas con la Conversa de la Propiedad Ackermann. Definimos y axiomatizamos lógicas proposicionales intuicionistas positivas, lógicas subintuicionistas y lógicas intuicionistas con la Conversa de la Propiedad Ackermann. Especialmente interesante es la versión de la negación intuicionista propia de algunos de estos sistemas. Presentamos semánticas de tipo relacional ternario para cada una de las lógicas que estudiamos en este trabajo. The range of propositional logics with the Converse Ackermann Property is proved to be wider than currently assumed. Subintuitionistic, positive intuitionistic and intuitionistic propositional logics with the Converse Ackermann Property are defined and axiomatized. A particular version of intuitionistic negation is involved in some of these systems. Complete ternary relational semantics are offered for all logics studied in the paper. I.