Somebody Knows

Abstract

Several different notions of group knowledge have been extensively studied in the epistemic and doxastic logic literature, including common knowledge, general knowledge (everybody-knows) and distributed knowledge. In this paper we study a natural notion of group knowledge between general and distributed knowledge: somebody-knows. While something is general knowledge if and only if it is known by everyone, this notion holds if and only if it is known by someone. This is stronger than distributed knowledge, which is the knowledge that follows from the total knowledge in the group. We introduce a modality for somebody-knows in the style of standard group knowledge modalities, and study its properties. Unlike the other mentioned 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. We also show that the satisfiability problem for this logic is PSPACE-complete. The neighbourhood semantics and the completeness and complexity results also carry over to logics for so-called local reasoning (Fagin et al. 1995) with bounded ``frames of mind'', correcting an existing completeness result in the literature (Allen 2005).