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Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λCalculi
, 1992
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Proof theory for admissible rules
, 2006
"... The admissible rules of a logic are the rules under which the set of theorems of the logic is closed. In this paper a Gentzenstyle framework is introduced for defining analytic proof systems that derive the admissible rules of various nonclassical logics. Just as Gentzen systems for derivability t ..."
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Cited by 9 (1 self)
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The admissible rules of a logic are the rules under which the set of theorems of the logic is closed. In this paper a Gentzenstyle framework is introduced for defining analytic proof systems that derive the admissible rules of various nonclassical logics. Just as Gentzen systems for derivability treat sequents as basic objects, for admissibility, sequent rules are basic. Proof systems are defined here for the admissible rules of classes of both modal logics, including K4, S4, and GL, and intermediate logics, including Intuitionistic logic IPC, De Morgan (or Jankov) logic KC, and logics BCn (n = 1, 2,...) with bounded cardinality Kripke models. With minor restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the corresponding logics.
Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics
"... Abstract. The admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a prooftheoretic framework where ..."
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Abstract. The admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a prooftheoretic framework where the basic objects of the systems are sequent rules. Here, the framework is extended to cover derivability of the admissible rules of intermediate logics and a wider class of modal logics, in this case, by taking hypersequent rules as the basic objects. 1
Utrecht University
"... Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzenstyle framework is introduced for analytic proof systems that derive admissible rules of nonclassical logics. While Gentzen systems for derivability treat sequents as basic o ..."
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Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzenstyle framework is introduced for analytic proof systems that derive admissible rules of nonclassical logics. While Gentzen systems for derivability treat sequents as basic objects, for admissibility, the basic objects are sequent rules. Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic IPC. With minor restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the logics.
Complexity of Admissible Rules in the ImplicationNegation Fragment of Intuitionistic Logic
"... The goal of this paper is to study the complexity of the set of admissible rules of the implicationnegation fragment of intuitionistic logic IPC. Surprisingly perhaps, although this set strictly contains the set of derivable rules (the fragment is not structurally complete), it is also PSPACEcompl ..."
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The goal of this paper is to study the complexity of the set of admissible rules of the implicationnegation fragment of intuitionistic logic IPC. Surprisingly perhaps, although this set strictly contains the set of derivable rules (the fragment is not structurally complete), it is also PSPACEcomplete. This differs from the situation in the full logic IPC where the admissible rules form a coNEXPcomplete set. 1