# Somebody Knows and Weak Conjunctive Closure in Modal Logic

## Thomas Ågotnes, University of Bergen and Shanxi University

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.