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Logics of False Belief and Radical Ignorance

Jie Fan

TL;DR

This work tackles the problem of axiomatizing the transitive logic of false belief and shows that the standard doxastic operator $\\Box$ is almost definable by the false belief operator $W$ via an Almost Definability Schema, enabling core axioms for transitive and Euclidean logics and motivating a canonical relation for uniform completeness. It then constructs comprehensive axiom systems for false belief across multiple frame classes (K^W, KD^W, T^W, K4^W, K5^W) and provides uniform completeness proofs, with a novel canonical model based on $W(\\varphi \\land \\psi)$, reflecting the schema. The paper further extends these ideas to radical ignorance via the interdefinability $I_R \leftrightarrow (W \\varphi \\lor W \\neg\\varphi)$, develops minimal and serial/transitive axiom systems for $I_R$, and shows how translating between $W$ and $I_R$ yields robust, though not always complete, translation outcomes. Overall, the results advance frame definability and completeness techniques for false belief and radical ignorance, offering tools for extending the analysis to symmetric frames and related epistemic-doxastic logics.

Abstract

In the literature, the question about how to axiomatize the transitive logic of false belief is thought of as hard and left as an open problem. In this paper, among other contributions, we deal with this problem. In more details, although the standard doxastic operator is undefinable with the operator of false belief, the former is {\em almost definable} with the latter. On one hand, the involved almost definability schema guides us to find the desired core axioms for the transitive logic and the Euclidean logic of false belief. On the other hand, inspired by the schema and other considerations, we propose a suitable canonical relation, which can uniformly handle the completeness proof of various logics of false belief, including the transitive logic. We also extend the results to the logic of radical ignorance, due to the interdefinability of the operators of false belief and radical ignorance.

Logics of False Belief and Radical Ignorance

TL;DR

This work tackles the problem of axiomatizing the transitive logic of false belief and shows that the standard doxastic operator is almost definable by the false belief operator via an Almost Definability Schema, enabling core axioms for transitive and Euclidean logics and motivating a canonical relation for uniform completeness. It then constructs comprehensive axiom systems for false belief across multiple frame classes (K^W, KD^W, T^W, K4^W, K5^W) and provides uniform completeness proofs, with a novel canonical model based on , reflecting the schema. The paper further extends these ideas to radical ignorance via the interdefinability , develops minimal and serial/transitive axiom systems for , and shows how translating between and yields robust, though not always complete, translation outcomes. Overall, the results advance frame definability and completeness techniques for false belief and radical ignorance, offering tools for extending the analysis to symmetric frames and related epistemic-doxastic logics.

Abstract

In the literature, the question about how to axiomatize the transitive logic of false belief is thought of as hard and left as an open problem. In this paper, among other contributions, we deal with this problem. In more details, although the standard doxastic operator is undefinable with the operator of false belief, the former is {\em almost definable} with the latter. On one hand, the involved almost definability schema guides us to find the desired core axioms for the transitive logic and the Euclidean logic of false belief. On the other hand, inspired by the schema and other considerations, we propose a suitable canonical relation, which can uniformly handle the completeness proof of various logics of false belief, including the transitive logic. We also extend the results to the logic of radical ignorance, due to the interdefinability of the operators of false belief and radical ignorance.
Paper Structure (14 sections, 28 theorems, 34 equations)

This paper contains 14 sections, 28 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. False belief
  3. Syntax and Semantics
  4. Expressivity and Frame Definability
  5. Axiomatizations
  6. Minimal Logic
  7. Serial Logic
  8. Reflexive Logic
  9. Transitive Logic
  10. Euclidean Logic
  11. Radical Ignorance
  12. Minimal logic
  13. Serial and transitive logic
  14. Conclusion and Discussions

Key Result

Proposition 4

$\vDash W\psi\to (\Box\varphi\leftrightarrow W(\varphi\land\psi)).$

Theorems & Definitions (59)

  • Definition 1
  • Definition 2
  • Definition 3: Semantics
  • Proposition 4
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 49 more