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Implicit Knowledge in Unawareness Structures

Gaia Belardinelli, Burkhard C. Schipper

TL;DR

The paper tackles the problem of reconciling Fagin–Halpern and Heifetz–Meier–Schipper models of unawareness by defining implicit knowledge within HMS and by constructing a category of FH models to capture subjective agent views. It develops a chain of transformations (including the $T$-transform and FH$^*$, HMS$^*$ transformations) that map between FH and HMS frameworks, proving semantic equivalence and preserving satisfaction of the Language of Propositional Awareness. A central contribution is the interdefinability of implicit and explicit knowledge in HMS models, enabling soundness and completeness results for LPA across complemented HMS, implicit-knowledge-based HMS, and FH categories. The framework provides a unified view that bridges syntax-based FH semantics with syntax-free HMS semantics, with potential cross-disciplinary applications in economics, computer science, and decision theory, and opens avenues for future work on tacit knowledge and quantified awareness. Overall, it advances a coherent theory of awareness that encompasses implicit knowledge, explicit knowledge, and subjective perspectives within a rigorous, transformable model landscape.

Abstract

Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic awareness correspondence and accessibility relations modeling implicit knowledge. They are a flexible model of unawareness, and best interpreted from a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and Schipper (2006, 2008) (HMS) model awareness by a lattice of state spaces and explicit knowledge via possibility correspondences. Sublattices thereof can be interpreted as subjective views of agents. Open questions include (1) how implicit knowledge can be defined in HMS structures, and (2) in which way FH structures can be extended to model the agents' subjective views. In this paper, we address (1) by defining implicit knowledge such that it is consistent with explicit knowledge in HMS models. We also introduce a variant of HMS models that instead of explicit knowledge, takes implicit knowledge and awareness as primitives. Further, we address (2) by introducing a category of FH models that are modally equivalent relative to sublanguages and can be interpreted as agents' subjective views depending on their awareness. These constructions allow us to show an equivalence between HMS and FH models. As a corollary, we obtain soundness and completeness of HMS models with respect to the Logic of Propositional Awareness, based on a language featuring both implicit and explicit knowledge.

Implicit Knowledge in Unawareness Structures

TL;DR

The paper tackles the problem of reconciling Fagin–Halpern and Heifetz–Meier–Schipper models of unawareness by defining implicit knowledge within HMS and by constructing a category of FH models to capture subjective agent views. It develops a chain of transformations (including the -transform and FH, HMS transformations) that map between FH and HMS frameworks, proving semantic equivalence and preserving satisfaction of the Language of Propositional Awareness. A central contribution is the interdefinability of implicit and explicit knowledge in HMS models, enabling soundness and completeness results for LPA across complemented HMS, implicit-knowledge-based HMS, and FH categories. The framework provides a unified view that bridges syntax-based FH semantics with syntax-free HMS semantics, with potential cross-disciplinary applications in economics, computer science, and decision theory, and opens avenues for future work on tacit knowledge and quantified awareness. Overall, it advances a coherent theory of awareness that encompasses implicit knowledge, explicit knowledge, and subjective perspectives within a rigorous, transformable model landscape.

Abstract

Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic awareness correspondence and accessibility relations modeling implicit knowledge. They are a flexible model of unawareness, and best interpreted from a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and Schipper (2006, 2008) (HMS) model awareness by a lattice of state spaces and explicit knowledge via possibility correspondences. Sublattices thereof can be interpreted as subjective views of agents. Open questions include (1) how implicit knowledge can be defined in HMS structures, and (2) in which way FH structures can be extended to model the agents' subjective views. In this paper, we address (1) by defining implicit knowledge such that it is consistent with explicit knowledge in HMS models. We also introduce a variant of HMS models that instead of explicit knowledge, takes implicit knowledge and awareness as primitives. Further, we address (2) by introducing a category of FH models that are modally equivalent relative to sublanguages and can be interpreted as agents' subjective views depending on their awareness. These constructions allow us to show an equivalence between HMS and FH models. As a corollary, we obtain soundness and completeness of HMS models with respect to the Logic of Propositional Awareness, based on a language featuring both implicit and explicit knowledge.
Paper Structure (14 sections, 32 theorems, 11 equations, 2 figures, 1 table)

This paper contains 14 sections, 32 theorems, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. HMS Models
  3. From Explicit to Implicit Knowledge
  4. From Implicit to Explicit Knowledge
  5. Category of FH Models
  6. Transformations
  7. From FH Models to Complemented HMS Models
  8. From Complemented HMS Models to FH Models
  9. Equivalence of Complemented HMS and FH Models
  10. From FH to Implicit Knowledge-based HMS Models
  11. From Implicit Knowledge-based HMS to FH Models
  12. Equivalence of Implicit Knowledge-based HMS and FH Models
  13. Logic of Propositional Awareness
  14. Discussion

Key Result

Lemma 1

For every individual $i \in I$ and event $E \in \Sigma$, both $K_i(E)$ and $A_i(E)$ are $S(E)$-based events.

Figures (2)

  • Figure 1: Two Single-Agent Examples of Complemented HMS Models
  • Figure 2: Relations between Approaches to Awareness

Theorems & Definitions (54)

  • Definition 1: HMS Model
  • Definition 2: Knowledge Operator
  • Definition 3: Awareness Operator
  • Lemma 1: Heifetz, Meier, and Schipper, 2006
  • Proposition 1: Heifetz, Meier, and Schipper, 2006
  • Proposition 2: Heifetz, Meier, and Schipper, 2006
  • Lemma 2
  • Definition 4: Implicit Possibility Correspondence
  • Definition 5: Complemented HMS Model
  • Remark 1: Strong Confinement
  • ...and 44 more