[Paper] Vague Knowledge: Information without Transitivity and Partitions

Source: arXiv - 2512.05833v1

Overview

Kerry Xiao’s paper challenges two long‑standing assumptions in economic models of information—transitivity (if you can’t tell A from B and B from C, you can’t tell A from C) and the partition view (knowledge cleanly splits the world into distinct, non‑overlapping states). By relaxing these constraints, the author formalizes vague knowledge: a form of information where indistinguishability between states is non‑transitive, leading to “blurred” boundaries. The work bridges economics, logic, and finance, offering a fresh lens on why natural language and qualitative reasoning dominate real‑world communication.

Key Contributions

• Formal definition of vague knowledge – introduces a mathematical structure for non‑transitive indistinguishability over state spaces.
• Proof that vague knowledge is still informative – despite not forming a partition, it can separate some states while leaving others ambiguous.
• Characterization of “vague communication” – shows that only communication channels with fuzzy, overlapping messages can faithfully convey vague knowledge.
• Micro‑foundations for natural language – provides a theoretical basis for why everyday language, with its inherent vagueness, is an efficient medium for information exchange.
• Cross‑disciplinary synthesis – integrates concepts from economic theory, formal logic, and quantitative finance to model real‑world informational environments.

Methodology

1. State‑space setup – The paper starts with a finite set of possible worlds (states).
2. Indistinguishability relation – Instead of a classic equivalence relation (which is reflexive, symmetric, transitive), the author defines a non‑transitive relation ( \sim ) that captures when an agent cannot distinguish two states.
3. Vague knowledge operator – Building on modal logic, a knowledge operator ( K ) is defined such that ( K\phi ) holds in a state if the agent knows ( \phi ) under the vague indistinguishability relation.
4. Communication model – A signaling framework is introduced where messages are sets of states with overlapping boundaries, mimicking natural language categories (e.g., “high”, “moderate”).
5. Theoretical analysis – Using lattice theory and fixed‑point theorems, the author proves properties like monotonicity of ( K ) and the impossibility of representing vague knowledge with crisp partitions.

The approach stays high‑level enough for non‑experts: think of it as replacing a clean “yes/no” decision tree with a fuzzy map where some branches overlap.

Results & Findings

• Informative power retained – Even without transitivity, agents can correctly rule out certain states, enabling decision‑making that is better than pure ignorance.
• No partition representation – There exists no way to map vague knowledge onto a traditional partition of the state space without losing information.
• Necessity of vague messages – Only communication schemes that allow overlapping categories can preserve the agent’s knowledge; crisp, binary messages inevitably discard nuance.
• Alignment with linguistic practice – The mathematical constraints mirror how humans use adjectives like “large”, “small”, or “likely”, which lack sharp cut‑offs but still guide actions effectively.

Practical Implications

For developers, the takeaway is that embracing vagueness—through fuzzy data structures, probabilistic type systems, or overlapping message schemas—can lead to more robust, human‑aligned applications.

Limitations & Future Work

• Finite state assumption – The current theory is built on a finite set of worlds; extending to continuous or high‑dimensional spaces remains open.
• Computational tractability – Determining the exact vague knowledge operator can be costly; approximations or algorithmic shortcuts are needed for large‑scale systems.
• Empirical validation – The paper is primarily theoretical; future work could test the framework on real linguistic corpora or financial market data.
• Integration with existing probabilistic models – How vague knowledge interacts with Bayesian updating or reinforcement learning is an exciting direction for interdisciplinary research.

Authors

• Kerry Xiao

Paper Information

• arXiv ID: 2512.05833v1
• Categories: econ.TH, cs.CL, math.LO, q-fin.GN
• Published: December 5, 2025
• PDF: Download PDF