Group belief

Abstract

Abstract While logical formalizations of group notions of knowledge such as common and distributed knowledge have received considerable attention in the literature, most approaches being based on modal logic, group notions of belief have received much less attention. In this paper we systematically study standard notions of group belief under different assumptions about the properties of belief. In particular, we map out (lack of) preservation of belief properties against different standard definitions of group belief. It turns out that what is called group belief most often is not actually belief, i.e. does not have the properties of belief. In fact, even what is called group knowledge is sometimes not actually knowledge either. For example, under the common assumption that belief has the KD45 properties, neither common belief (does not satisfy the negative introspection axiom 5) nor distributed belief (does not satisfy the consistency axiom D) are not actually belief. There has been some confusion in the literature regarding soundness of proposed axiomatizations of logics with distributed knowledge, related to the mentioned lack of preservation. In this paper we also present detailed completeness proofs of sound and complete axiomatizations of KD45 with distributed belief, both with and without common belief.