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Pure extensions, proof rules and hybrid axiomatics
 Preliminary proceedings of Advances in Modal Logic (AiML 2004
, 2004
"... We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language ..."
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We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language
Bisimulation for Neighbourhood Structures
, 2007
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notio ..."
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Cited by 5 (0 self)
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notions of 2 2bisimulation, behavioural equivalence and neighbourhood bisimulation (a notion based on pushouts), with the aim of finding the logically correct notion of equivalence on neighbourhood structures. Our results include relational characterisations for 2 2bisimulation and neighbourhood bisimulation, and an analogue of Van Benthem’s characterisation theorem for all three equivalence notions. We also show that behavioural equivalence gives rise to a HennessyMilner theorem, and that this is not the case for the other two equivalence notions.
Topology, connectedness, and modal logic
 ADVANCES IN MODAL LOGIC
, 2008
"... This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of ..."
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Cited by 5 (3 self)
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This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (lowdimensional) Euclidean spaces.
NEIGHBOURHOOD STRUCTURES: BISIMILARITY AND BASIC MODEL THEORY
, 2009
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denote ..."
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Cited by 4 (1 self)
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give backandforth style characterisations for 2 2bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2 2bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and imagefiniteness. We prove a HennessyMilner theorem for modally saturated and for imagefinite neighbourhood models. Our main results are an analogue of Van Benthem’s characterisation theorem and a modeltheoretic proof of Craig interpolation for classical modal logic.
AGM Belief Revision in Monotone Modal Logics
"... Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several firstorder monotonic modal correspondents, where each firstorder correspondence ..."
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Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several firstorder monotonic modal correspondents, where each firstorder correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for monotone modal logic. A revision problem expressed in a monotone modal system is translated into firstorder logic, the revision is performed, and the new belief set is translated back to the original modal system. An example is provided for the logic of Risky Knowledge that uses modal AGM contraction to construct counterfactual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given. 1
Structures for Epistemic Logic
"... Epistemic modal logic in a narrow sense studies and formalises reasoning about knowledge. In a wider sense, it gives a formal account of the informational attitude that agents may have, and covers notions like knowledge, belief, uncertainty, and hence incomplete or partial information. As is so ofte ..."
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Epistemic modal logic in a narrow sense studies and formalises reasoning about knowledge. In a wider sense, it gives a formal account of the informational attitude that agents may have, and covers notions like knowledge, belief, uncertainty, and hence incomplete or partial information. As is so often the case in modal logic,
Completeness and Definability of a Modal Logic Interpreted over Iterated Strict Partial Orders
"... Any strict partial order R on a nonempty set X defines a function θR which associates to each strict partial order S ⊆ R on X the strict partial order θR(S) = R ◦ S on X. Owing to the strong relationships between Alexandroff TD derivative operators and strict partial orders, this paper firstly call ..."
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Any strict partial order R on a nonempty set X defines a function θR which associates to each strict partial order S ⊆ R on X the strict partial order θR(S) = R ◦ S on X. Owing to the strong relationships between Alexandroff TD derivative operators and strict partial orders, this paper firstly calls forth the links between the CantorBendixson ranks of Alexandroff TD topological spaces and the greatest fixpoints of the θlike functions defined by strict partial orders. It secondly considers a modal logic with modal operators ✷ and ✷ ⋆ respectively interpreted by strict partial orders and the greatest fixpoints of the θlike functions they define. It thirdly addresses the question of the complete axiomatization of this modal logic.