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Deep Sequent Systems for Modal Logic
 ARCHIVE FOR MATHEMATICAL LOGIC
"... We see a systematic set of cutfree axiomatisations for all the basic normal modal logics formed by some combination the axioms d,t,b,4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the litera ..."
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We see a systematic set of cutfree axiomatisations for all the basic normal modal logics formed by some combination the axioms d,t,b,4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 8 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Cut Elimination in Coalgebraic Logics
"... We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cutrule is admissible in the ensuing s ..."
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Cited by 6 (6 self)
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We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cutrule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics CK and CK + ID.
Kripke semantics for basic sequent systems
 In Proceedings of the 20th international
"... Abstract. We present a general method for providing Kripke semantics for the family of fullystructural multipleconclusion propositional sequent systems. In particular, many wellknown Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obta ..."
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Abstract. We present a general method for providing Kripke semantics for the family of fullystructural multipleconclusion propositional sequent systems. In particular, many wellknown Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obtain semantic characterizations of analytic sequent systems of this type, as well as of those admitting cutadmissibility. These characterizations serve as a uniform basis for semantic proofs of analyticity and cutadmissibility in such systems. 1
D.: Constructing cut free sequent systems with context restrictions based on classical or intuitionistic logic
"... Abstract. We consider a general format for rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring a generic cut elimination for such rule sets. The rule format encompasses e.g. rules for the boole ..."
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Abstract. We consider a general format for rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring a generic cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic VA. 1
Hypersequent and Labelled Calculi for Intermediate Logics ⋆
"... Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cutfree calculi for nonclassical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rule ..."
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Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cutfree calculi for nonclassical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rules. We show that these methods are closely related and we extend them to capture larger classes of intermediate logics. 1
Journal Applied Logic, 5(4):681–689, 2007.
"... This thesis provides an overview of my work on the proof theory for modal fixed point logics. In particular, it summarizes the main results of the following papers. 1. G. Jäger, M. Kretz, and T. Studer. Cutfree common knowledge. ..."
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This thesis provides an overview of my work on the proof theory for modal fixed point logics. In particular, it summarizes the main results of the following papers. 1. G. Jäger, M. Kretz, and T. Studer. Cutfree common knowledge.