Awareness logic: Kripke lattices as a middle ground between syntactic and semantic models

Abstract

The literature on awareness modelling includes both syntax-free and syntax-based frameworks. Heifetz, Meier and Schipper (HMS) propose a lattice model of awareness that is syntax-free. While their lattice approach is elegant and intuitive, it does not explicitly distinguish uncertainty from unawareness. Contra this, the most prominent syntax-based solution, the Fagin–Halpern (FH) model, accounts for this distinction and offers a simple representation of awareness but lacks the intuitiveness of the lattice structure. Here, we combine these two approaches by providing a lattice of Kripke models, induced by atom subset inclusion, in which uncertainty and unawareness are separate. We show that our model is equivalent to both HMS and FH models by defining transformations between them which preserve satisfaction of formulas of a language for explicit knowledge and obtain completeness through our and HMS’ results. Lastly, we prove that the Kripke lattice model can be shown equivalent to the FH model (when awareness is propositionally determined) also with respect to the language of the Logic of General Awareness, for which the FH model was originally proposed.

Implicit Knowledge in Unawareness Structures - Extended Abstract -

“…Like Kripke structures, HMS models can be constructed canonically and three different sound and complete axiomatizations have been presented (Halpern andRêgo, 2008, Heifetz, Meier, andSchipper, 2008). 1 There have already been a by propositionally determined FH models, namely models where an agent is aware of a formula if and only if she is aware of all atomic formula that appear in it. We also introduce a variant of HMS models in which we take the notion of implicit knowledge and a semantic awareness function as the primitive and then derive explicit knowledge.…”