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Comparing Knowledge: An Analysis of the Relative Epistemic Powers of Groups

Baltag Alexandru, Smets Sonja

TL;DR

The paper extends multi-agent epistemic logic with a group-comparative operator $A\preceq B$ to analyze when one group knows at least as much as another, across KT, $S4$, and $S5$ models. It provides a formal language (LDC≼) and an extensive axiomatisation, showing that Known Superiority holds in $S5$ but fails in $S4$ (and KT in certain cases), and it explores how epistemic power transfers among overlapping groups and how free-riders can arise. Through semantic analyses and illustrative examples, the authors map out the landscape of what can be known about relative epistemic positions under different introspection assumptions. The work sets the stage for future work on dynamics, topology-based semantics, topic-relative comparisons, and broader notion of common distributed knowledge.

Abstract

We use a novel type of epistemic logic, employing comparative knowledge assertions, to analyze the relative epistemic powers of individuals or groups of agents. Such comparative assertions can express that a group has the potential to (collectively) know everything that another group can know. Moreover, we look at comparisons involving various types of knowledge (fully introspective, positively introspective, etc.), satisfying the corresponding modal-epistemic conditions (e.g., $S5$, $S4$, $KT$). For each epistemic attitude, we are particularly interested in what agents or groups can know about their own epistemic position relative to that of others.

Comparing Knowledge: An Analysis of the Relative Epistemic Powers of Groups

TL;DR

The paper extends multi-agent epistemic logic with a group-comparative operator to analyze when one group knows at least as much as another, across KT, , and models. It provides a formal language (LDC≼) and an extensive axiomatisation, showing that Known Superiority holds in but fails in (and KT in certain cases), and it explores how epistemic power transfers among overlapping groups and how free-riders can arise. Through semantic analyses and illustrative examples, the authors map out the landscape of what can be known about relative epistemic positions under different introspection assumptions. The work sets the stage for future work on dynamics, topology-based semantics, topic-relative comparisons, and broader notion of common distributed knowledge.

Abstract

We use a novel type of epistemic logic, employing comparative knowledge assertions, to analyze the relative epistemic powers of individuals or groups of agents. Such comparative assertions can express that a group has the potential to (collectively) know everything that another group can know. Moreover, we look at comparisons involving various types of knowledge (fully introspective, positively introspective, etc.), satisfying the corresponding modal-epistemic conditions (e.g., , , ). For each epistemic attitude, we are particularly interested in what agents or groups can know about their own epistemic position relative to that of others.

Paper Structure

This paper contains 9 sections, 17 theorems, 44 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Group Epistemic Logic: Distributed, Common, and Comparative Knowledge
  3. Syntax and Semantics
  4. Axiomatisation of $LDC\preceq$
  5. Reasoning About Epistemic Comparison
  6. Knowledge, Distributed Knowledge, and Common Knowledge About Epistemic Comparison in $S5$ Models
  7. Reasoning About Overlapping Groups and Epistemic Free-Riders
  8. "Known Superiority" Revisited
  9. Conclusions

Key Result

Proposition 1

All axioms in groups (I)--(IV) are sound on $KT$ models (i.e., $KT$-valid). Positive Introspection is sound on $S4$ models (thus, also on $S5$ models), while Negative Introspection and Known Superiority are valid only on $S5$-models.

Figures (3)

  • Figure 1: The drawing represents the epistemic $S5$ model for the penny--quarter box with three agents and a possible world for each of the four possible states of the box. All accessibility relations of agents are drawn except for the reflexive loops. At each world, we specified (a conjunction of) the atomic facts that are true and we gave an example of a true epistemic proposition of the form $K_{\{a,b,c\}} \varphi$.
  • Figure 2: The drawing represents the epistemic $S4$ model with four possible worlds, $s,t,u,v$, and two agents, $a,b$, for the penny--quarter box. Except for the reflexive loops, all other relations are drawn.
  • Figure 3: The drawing represents the epistemic $S5$ model $M$ with three possible worlds, $s,t,u$, for the penny--quarter box in which agents are told that $TT$ is not possible. Except for the reflexive loops, all other accessibility relations are drawn.

Theorems & Definitions (39)

  • proof
  • Example 1
  • Example 2
  • Proposition 1
  • Example 3
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 29 more