Awareness and knowledge

3^rd December 2013, 16:00 Ashton Lecture Theater
Dr. Hans van Ditmarsch
LORIA Nancy
France

Abstract

Modal logics of knowledge model uncertainty. Logics of awareness model incompleteness (as in vocabulary restriction) - a topic considered of great interest in economics. I have been working on these matters with Tim French (Perth), Fernando Velazquez (Sevilla), and Yi Wang (Bergen). We compare different epistemic notions in the presence of awareness of propositional variables: the logics of implicit knowledge (in which explicit knowledge is definable as implicit knowledge plus awareness), explicit knowledge, and speculative knowledge. Speculative knowledge goes back to the motivation in Levesque's 'A Logic of Implicit and Explicit Belief': one can speculate over variables of which one is unaware, e.g. if you are unaware of p, then p v ~p is still speculatively known by you. A cornerstone of our framework is the notion of awareness bisimulation - this is the proper notion of structural similarity on the structures enriched with awareness of propositional variables proposed by Fagin and Halpern in 'Belief, awareness, and limited reasoning'. A more 'standard' sort of bisimulation is also suitable for these logics. We provide correspondence between bisimulation and modal equivalence on image-finite models for these logics. The logic of speculative knowledge is equally expressive as the logic of explicit knowledge, and the logic of implicit knowledge is more expressive than both. The logics have complete axiomatizations. Dynamics can also be added: any conceivable change of knowledge or awareness can be modelled in this setting. The dynamic versions of all three logics are, surprising, equally expressive.