144 research outputs found
Nested sequent calculi and theorem proving for normal conditional logics: The theorem prover NESCOND
Intuitionistic Non-Normal Modal Logics: A general framework
We define a family of intuitionistic non-normal modal logics; they can bee
seen as intuitionistic counterparts of classical ones. We first consider
monomodal logics, which contain only one between Necessity and Possibility. We
then consider the more important case of bimodal logics, which contain both
modal operators. In this case we define several interactions between Necessity
and Possibility of increasing strength, although weaker than duality. For all
logics we provide both a Hilbert axiomatisation and a cut-free sequent
calculus, on its basis we also prove their decidability. We then give a
semantic characterisation of our logics in terms of neighbourhood models. Our
semantic framework captures modularly not only our systems but also already
known intuitionistic non-normal modal logics such as Constructive K (CK) and
the propositional fragment of Wijesekera's Constructive Concurrent Dynamic
Logic.Comment: Preprin
Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics
International audienceThe preferential conditional logic PCL, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalise Lewis' sphere models for counterfactual logics, is proposed. Soundness and completeness of PCL and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The cal-culi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut, that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for PCL and its extensions. Finally, the semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered
Intuitionistic non-normal modal logics: A general framework
International audienceWe define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic
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