Field Knowledge as a Dual to Distributed Knowledge: A Characterization by Weighted Modal Logic

TL;DR

This paper introduces field knowledge as the dual to distributed knowledge within a weighted modal logic framework that captures agents' epistemic abilities. It develops eight logics formed by combining common, distributed, and field knowledge, and provides eight corresponding axiom systems with soundness results and varying completeness (strong for the logics without common knowledge, weak for those with common knowledge). The semantics leverages similarity models that extend Kripke semantics with an edge-based representation of epistemic abilities and a nuanced interpretation of group modalities. Through translation, canonical-model, path-based canonical-model, and finitary-model techniques, the authors establish completeness results and show how field knowledge can be integrated with distributed knowledge without collapsing existing epistemic intuitions. The framework offers a flexible tool for characterizing knowledge after considering agents’ professional abilities and shared disciplines, with potential for richer similarity conditions and comparisons to related epistemic formalisms.

Abstract

The study of group knowledge concepts such as mutual, common, and distributed knowledge is well established within the discipline of epistemic logic. In this work, we incorporate epistemic abilities of agents to refine the formal definition of distributed knowledge and introduce a formal characterization of field knowledge. We propose that field knowledge serves as a dual to distributed knowledge. Our approach utilizes epistemic logics with various group knowledge constructs, interpreted through weighted models. We delve into the eight logics that stem from these considerations, explore their relative expressivity and develop sound and complete axiomatic systems.

Figures (3)

• Figure 1: Illustration of the model in Example \ref{['ex1']}.
• Figure 2: Illustration of a model for Example \ref{['ex2']}. We do not draw a line between two nodes when the edge between them is with no label (i.e., labeled by an empty set, e.g., between $s_1$ and $s_3$).
• Figure 3: The above two diagrams illustrate the relative expressive power of the languages. An arrow pointing from one language to another implies that the second language is at least as expressive as the first. The "at least as expressive as" relationship is presumed to be reflexive and transitive, meaning that a language is considered at least as expressive as another if a path of arrows exists leading from the second to the first (self-loops exist for all, but omitted). A lack of a path of arrows from one language to another indicates that the first language is not at least as expressive as the second. This implies that either the two languages are incomparable or that the first language is more expressive than the second.

Theorems & Definitions (38)

• definition 1
• definition 2
• definition 3
• proposition 1
• theorem 1: soundness
• definition 4
• lemma 1
• theorem 2
• definition 5
• definition 6