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A Nondeterministic View on Nonclassical Negations
 Workshop on Negation in Constructive Logic
, 2004
"... We investigate two large families of logics, diering from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant for \the false") by adding various standard Gentzentype rules for ..."
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We investigate two large families of logics, diering from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant for \the false") by adding various standard Gentzentype rules for negation. The logics in the other family are similarly obtained from LJ, the positive fragment of intuitionistic logic (again, with or without ). For all the systems, we provide simple semantics which is based on nondeterministic fourvalued or threevalued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of nondeterminism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ (with or without ) are shown to be conservative over the underlying logic, and it is determined which of them are not. 1
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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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.
Negation: Two Points Of View
 In Gabbay and Wansing [21
"... this paper we look at negation from two different points of view: a syntactical one and a semantical one. Accordingly, we identify two different types of negation. The same connective of a given logic might be of both types, but this might not always be the case. The syntactical point of view is an ..."
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this paper we look at negation from two different points of view: a syntactical one and a semantical one. Accordingly, we identify two different types of negation. The same connective of a given logic might be of both types, but this might not always be the case. The syntactical point of view is an abstract one. It characterizes connectives according to the internal role they have inside a logic, regardless of any meaning they are intended to have (if any). With regard to negation our main thesis is that the availability of what we call below an internal negation is what makes a logic essentially multipleconclusion.
Kripke Completeness of FirstOrder Constructive Logics with Strong Negation
, 2003
"... This paper considers Kripke completeness of Nelson's constructive predicate logic N and its several variants. N is an extension of intuitionistic predicate logic Int by an contructive negation operator called strong negation. The variants of N in consideration are by the axiom of constant dom ..."
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This paper considers Kripke completeness of Nelson's constructive predicate logic N and its several variants. N is an extension of intuitionistic predicate logic Int by an contructive negation operator called strong negation. The variants of N in consideration are by the axiom of constant domain 8x(A(x)B) ! 8xA(x)B, the axiom (A ! B)(B ! A), omitting the axiom A ! (A ! B) and the axiom ::(AA); the last one we would like to call the axiom of potential omniscience and can be interpreted that we can always verify or falsify a statement, with proper additional information. The proofs
Kripke Completeness of FirstOrder Constructive Logics with Strong Negation
"... This paper considers Kripke completeness of Nelson’s constructive predicate logic N3 and its several variants. N3 is an extension of intuitionistic predicate logic Int by an contructive negation operator ∼ called strong negation. The variants of N3 in consideration are by omitting the axiom A → (∼A ..."
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This paper considers Kripke completeness of Nelson’s constructive predicate logic N3 and its several variants. N3 is an extension of intuitionistic predicate logic Int by an contructive negation operator ∼ called strong negation. The variants of N3 in consideration are by omitting the axiom A → (∼A → B), by adding the axiom of constant domain ∀x(A(x) ∨ B) → ∀xA(x) ∨ B, by adding (A → B) ∨ (B → A), and by adding ¬¬(A ∨ ∼A); the last one we would like to call the axiom of potential omniscience and can be interpreted that we can eventually verify or falsify a statement, with proper additional information. The proofs of completeness are by the widelyapplicable treesequent method; however, for those logics with the axiom of potential omniscience we can hardly go through with a simple application of it. For them we present two different proofs: one is by an embedding of classical logic, and the other by the TSg method, which is an extension of the treesequent method.
A Note on Negation in Categorial
"... A version of strong negation is introduced into Categorial Grammar. The resulting syntactic calculi turn out to be systems of connexive logic. ..."
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A version of strong negation is introduced into Categorial Grammar. The resulting syntactic calculi turn out to be systems of connexive logic.