Synthese (2010) 172:145–155 DOI 10.1007/s11229-009-9469-0 S5 knowledge without partitions Dov Samet Received: 23 January 2008 / Accepted: 19 November 2008 / Published online: 18 February 2009 © Springer Science+Business Media B.V. 2009 Abstract We study set algebras with an operator (SAO) that satisfy the axioms of S5 knowledge. A necessary and sufficient condition is given for such SAOs that the knowledge operator is defined by a partition of the state space. SAOs are con- structed for which the condition fails to hold. We conclude that no logic singles out the partitional SAOs among all SAOs. Keywords Epistemic logic · Modal logic · S5 · Partitions · Boolean algebras with operators 1 Introduction The standard structure used in economic theory, game theory, and decision theory to describe the knowledge of an agent is a set of states  endowed with a partition . 1 An informal justification of this modeling of knowledge uses the notion of a signal. The agent is said to observe a signal that may depend on the state. The partition of  into sets of states with the same signal results in . Thus, in each state ω the agent cannot tell which of the states obtains in (ω)—the element of the partition that contains ω—because she observes the same signal in all these states, but she can tell that all the states outside (ω) do not obtain, as the signals observed in these states are different from the one observed in ω. Of course, the signal is a metaphor for what the agent learns or knows. Using the partition we can formally describe the agent’s knowledge in terms of subsets of states. We say that the agent knows a given subset of states E in state ω if 1 In similar structures that are used for the semantics of modal logic, states are referred to as possible worlds D. Samet (B ) The Faculty of Management, Tel Aviv University, Tel Aviv 69978, Israel e-mail: dovsamet@gmail.com 123