Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variations on distributed belief

John Lindqvist, Fernando R. Velázquez-Quesada, Thomas Ågotnes

TL;DR

This work introduces cautious ($\mathop{\mathrm{D}^{\forall}_{}}$) and bold ($\mathop{\mathrm{D}^{\exists}_{}}$) distributed belief as robust alternatives to standard distributed belief ($\mathop{\mathrm{D}}$) by leveraging maximally consistent subgroups. Cautious belief uses a universal quantification over maximally consistent subgroups, yielding a normal, relational modality that preserves some but not all doxastic properties; bold belief uses an existential quantification and adopts neighbourhood semantics, capturing credulous aggregation without a relational basis. The paper establishes definability and expressivity results, shows how these notions relate to ordinary belief, and characterizes when properties like seriality, reflexivity, and transitivity are preserved. The findings illuminate how distributed belief can be made resilient to conflicting information and highlight expressive gaps between the variants, guiding future axiomatizations and applications in distributed epistemic settings.

Abstract

Motivated by the search for forms of distributed belief that do not collapse in the face of conflicting information, this paper introduces the notions of cautious and bold distributed belief. Both notions rely on maximally consistent subgroups of agents, with cautious quantifying universally and bold quantifying existentially. As a result, while the cautious distributed belief of a group is inconsistent only when all group members are individually inconsistent, the bold distributed belief of a group is never inconsistent. The paper discusses these two notions, presenting their respective modalities and semantic interpretations, discussing some of their basic properties, studying whether they preserve doxastic properties from the members of the group, and comparing them not only with standard distributed belief but also with one another, both at the level of modalities and at the level of language expressivity.

Variations on distributed belief

TL;DR

This work introduces cautious () and bold () distributed belief as robust alternatives to standard distributed belief () by leveraging maximally consistent subgroups. Cautious belief uses a universal quantification over maximally consistent subgroups, yielding a normal, relational modality that preserves some but not all doxastic properties; bold belief uses an existential quantification and adopts neighbourhood semantics, capturing credulous aggregation without a relational basis. The paper establishes definability and expressivity results, shows how these notions relate to ordinary belief, and characterizes when properties like seriality, reflexivity, and transitivity are preserved. The findings illuminate how distributed belief can be made resilient to conflicting information and highlight expressive gaps between the variants, guiding future axiomatizations and applications in distributed epistemic settings.

Abstract

Motivated by the search for forms of distributed belief that do not collapse in the face of conflicting information, this paper introduces the notions of cautious and bold distributed belief. Both notions rely on maximally consistent subgroups of agents, with cautious quantifying universally and bold quantifying existentially. As a result, while the cautious distributed belief of a group is inconsistent only when all group members are individually inconsistent, the bold distributed belief of a group is never inconsistent. The paper discusses these two notions, presenting their respective modalities and semantic interpretations, discussing some of their basic properties, studying whether they preserve doxastic properties from the members of the group, and comparing them not only with standard distributed belief but also with one another, both at the level of modalities and at the level of language expressivity.
Paper Structure (15 sections, 25 theorems, 2 equations, 2 figures, 3 tables)

This paper contains 15 sections, 25 theorems, 2 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Cautious distributed belief
  4. Semantic interpretation
  5. Some basic results about DA
  6. Preserving doxastic properties
  7. Relationship between DA and D and between DA and D
  8. Bold distributed belief
  9. Semantic interpretation
  10. Some basic results about DE
  11. Preserving doxastic properties
  12. Relationship between DE and D and between DB and D
  13. Relationship between DE and DA and between DE and DA
  14. Summary and further work
  15. Appendix

Key Result

Proposition 1

Take any groups $G \subseteq H \subseteq A$ and any $a \in A$.

Figures (2)

  • Figure 1: Counterexamples used in the proof of \ref{['pro:DA:rel-properties']}
  • Figure 2: Counterexamples used in the proof of \ref{['pro:DE:rel-properties']}

Theorems & Definitions (39)

  • Definition 1: Belief model
  • Definition 2: Distributed belief
  • Definition 3: Relative expressivity
  • Definition 4: Consistency, maximal consistency
  • Definition 5: Cautious distributed belief
  • Example 1
  • Proposition 1
  • Definition 6: Preservation
  • Proposition 2
  • Proposition 3
  • ...and 29 more