Awareness Logic: Kripke Lattices as a Middle Ground between Syntactic and Semantic Models

TL;DR

This paper introduces a new awareness logic model that combines the intuitive lattice structure of HMS with the formal language-based approach of FH, explicitly distinguishing uncertainty from unawareness.

Contribution

It develops a Kripke lattice model of awareness that unifies syntax-free and syntax-based frameworks, preserving their properties and equivalences.

Findings

Model is equivalent to HMS and FH models

Provides transformations preserving formula satisfaction

Establishes completeness and equivalence with FH model

Abstract

The literature on awareness modeling includes both syntax-free and syntax-based frameworks. Heifetz, Meier \& Schipper (HMS) propose a lattice model of awareness that is syntax-free. While their lattice approach is elegant and intuitive, it precludes the simple option of relying on formal language to induce lattices, and 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 our model 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 where originally proposed.