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

Neighbourhood structures are the standard semantic tool used to reason about non-normal 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 2-bisimulation, 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 2-bisimulation 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 Hennessy-Milner theorem, and that this is not the case for the other two equivalence notions.

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 (low-dimensional) Euclidean spaces.

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 first-order monotonic modal correspondents, where each first-order 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 first-order 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 counter-factual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given. 1

c t i v it y e p o r t 2009 Table of contents

Abstract. Neighbourhood structures are the standard semantic tool used to reason about non-normal 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 2-bisimilarity 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 back-and-forth style characterisations for 2 2-bisimulations 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 2-bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem’s characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic. 1.

Abstract. Neighbourhood structures are the standard semantic tool used to reason about non-normal 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 2-bisimilarity 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 back-and-forth style characterisations for 2 2-bisimulations 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 2-bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem’s characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic. 1.

hold. In a neighbourhood model, with each state one associates a collection of subsets of the universe (called its neighbourhoods), and a modal formula □ϕ is true at a state s if the truth set of ϕ is a neighbourhood of s. The modal logic of all neighbourhood models is called classical modal logic. Neighbourhood semantics was invented in 1970 by Scott and Montague (independently in [41] and [31]); and Segerberg [42] presents some basic results about neighbourhood models and the classical modal logics that correspond to them. These and other salient results were incorporated by Chellas in his textbook [13]. During the past 15-20 years, non-normal modal logics have emerged in the areas of computer science and social choice theory, where system (or agent) properties are formalised in terms of various notions of ability in strategic games (e.g. [4, 38]). These logics have in common that they are monotonic, meaning they contain the above-mentioned formula □p → □(p ∨ q). The corresponding property of neighbourhood models is that neighbourhood collections are closed under supersets. Nonmonotonic modal logics occur in deontic logic (see e.g. [17]) where monotonicity can lead to paradoxical obligations, and in the modelling of knowledge and related epistemic notions

hold. In a neighbourhood model, with each state one associates a collection of subsets of the universe (called its neighbourhoods), and a modal formula □ϕ is true at a state s if the truth set of ϕ is a neighbourhood of s. The modal logic of all neighbourhood models is called classical modal logic. Neighbourhood semantics was invented in 1970 by Scott and Montague (independently in [41] and [31]); and Segerberg [42] presents some basic results about neighbourhood models and the classical modal logics that correspond to them. These and other salient results were incorporated by Chellas in his textbook [13]. During the past 15-20 years, non-normal modal logics have emerged in the areas of computer science and social choice theory, where system (or agent) properties are formalised in terms of various notions of ability in strategic games (e.g. [4, 38]). These logics have in common that they are monotonic, meaning they contain the above-mentioned formula □p → □(p ∨ q). The corresponding property of neighbourhood models is that neighbourhood collections are closed under supersets. Nonmonotonic modal logics occur in deontic logic (see e.g. [17]) where monotonicity can lead to paradoxical obligations, and in the modelling of knowledge and related epistemic notions