Abstract
We study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.
Highlights
Neighbourhood models are a well-established tool to study generalizations and variants of relational semantics and non-normal modal logics.1 They have been successfully applied to i.a. the dynamics of evidence and beliefs [40], the logic of ability [8, 32], conflict-tolerant deontic logic [17], the logical relations between obligationsF
The generalization from relational semantics to neighbourhood semantics allows one to invalidate certain schemata that are problematic for a given interpretation of the modal operator, and to include other schemata that would trivialize any normal modal logic
The semantics of the dynamic operator RG is provided by the following clause: M, w RGφ iff MG, w φ, In what follows, we show how various types of resolution and, more generally, informational dynamics across agents, can be defined and distinguished in the context of neighbourhood semantics
Summary
Neighbourhood models are a well-established tool to study generalizations and variants of relational semantics and non-normal modal logics. They have been successfully applied to i.a. the dynamics of evidence and beliefs [40], the logic of ability [8, 32], conflict-tolerant deontic logic [17], the logical relations between obligations. Neighbourhood models are a well-established tool to study generalizations and variants of relational semantics and non-normal modal logics.. In [20, 21], operations on monotonic neighbourhood models are studied from an abstract, algebraic viewpoint, giving rise to highly generic completeness results.6 Notwithstanding these important achievements, the counterpart of intersections of accessibility relations for neighbourhood semantics is largely unknown. Given any function M that specifies, for each neighbourhood function Ni in the original model, how many members of Ni(w) should go in the intersection for the world w, we can define a unique new neighbourhood function NM This new neighbourhood function can be used to interpret a corresponding classical modal operator M. Definition of pointwise intersection as an operation on neighbourhood models, and introduces various formal languages that feature pooling modalities.
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