On Some Difficulties Concerning the Definition of Group Implicit Knowledge

Abstract

Implicit group knowledge (or distributive knowledge) is the sum of the knowledge in a group. In this paper, two approaches to group implicit knowledge are compared. According to the first approach, group implicit knowledge is defined as a logical consequence of individual agents’ knowledge sets. According to the second approach, group implicit knowledge is characterized by an intersection of agents’ indistinguishability relations in a Kripke model. Two modal operators are considered: the implicit knowledge operator based on the first approach ($I_G$), and the distributive knowledge operator based on the second one ($D_G$). The paper argues that the principle of full communication (it should be possible for the agents of the group to establish information through communication) is compatible only with the compact group implicit knowledge operator because standard non-compact operator potentially involves infinite communication. In this paper, we introduce four modal operators for compact implicit knowledge, which differ in the condition for information extracting: validity in the model, common knowledge, <<everybody in a group knows>>, <<somebody in a group knows>>. It is shown that all types of compact implicit knowledge are equivalent to the distributive knowledge operator in a class of finite and distinguishing models. This demonstrates that the requirement of logical consequence for knowledge extractions is redundant and can be replaced by an epistemic modality <<somebody knows>>.