NOL Seminar with Thomas Ågotnes
published: 2023-08-15
event date:
2023-08-28
The Nordic Online Logic Seminar
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Date: Monday, 28 August 2023 at 16:00 CEST (UTC+2) on Zoom
Speaker: Thomas Ågotnes, Professor of Information Science, University of
Bergen (Norway) and Professor of Logic, Shanxi University (China)
Title: Somebody Knows and Weak Conjunctive Closure in Modal Logic
Abstract
Normal modal logics are closed under conjunctive closure. There are, however,
interesting non-normal logics that are not, but which nevertheless satisfy a
weak form of conjunctive closure. One example is a notion of group knowledge in
epistemic logic: somebody-knows. While something is general knowledge if it is
known by everyone, this notion holds if it is known by someone.
Somebody-knows is thus weaker than general knowledge but stronger than
distributed knowledge. We introduce a modality for somebody-knows in the style
of standard group knowledge modalities, and study its properties. Unlike most
other group knowledge modalities, somebody-knows is not a normal modality; in
particular it lacks the conjunctive closure property. We provide an equivalent
neighbourhood semantics for the language with a single somebody-knows modality,
together with a completeness result: the somebody-knows modalities are
completely characterised by the modal logic EMN extended with a particular weak
conjunctive closure axiom. The neighbourhood semantics and the completeness and
complexity results also carry over other logics with weak conjunctive closure,
including the logic of so-called local reasoning (Fagin et al., 1995) with
bounded “frames of mind”, correcting an existing completeness result in the
literature (Allen 2005). The talk is based on joint work with Yi N. Wang.