knowledge-without-partition

Synthese (2010) 172:145–155DOI 10.1007/s11229-009-9469-0S5 knowledge without partitionsDov SametReceived: 23 January 2008 / Accepted: 19 November 2008 / Published online: 18 February 2009© Springer Science+Business Media B.V. 2009Abstract We study set algebras with an operator (SAO) that satisfy the axiomsof S5 knowledge. A necessary and sufficient condition is given for such SAOs thatthe 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 outthe partitional SAOs among all SAOs.Keywords Epistemic logic · Modal logic · S5 · Partitions · Boolean algebras withoperators1 IntroductionThe standard structure used in economic theory, game theory, and decision theory to1describetheknowledgeofanagentisasetof states endowedwithapartition . Aninformal justification of this modeling of knowledge uses the notion of a signal.Theagent is said to observe a signal that may depend on the state. The partition of intosetsofstateswiththesamesignalresultsin .Thus,ineachstate ω theagentcannottell which of the states obtains in (ω) —the element of the partition that containsω—becausesheobservesthesamesignalinallthesestates,butshecantellthatallthestatesoutside (ω) donotobtain,asthesignalsobservedinthesestatesaredifferentfrom the one observed in ω. Of course, the signal is a metaphor for what the agentlearns or knows.Using the partition we can formally describe ...