Axiomatizing Logics of False Belief and Radical Ignorance

Core Concepts

The paper deals with the problem of axiomatizing the transitive logic of false belief, which was previously thought to be difficult. It proposes an "almost definability" schema that guides the discovery of core axioms for the transitive and Euclidean logics of false belief. It also introduces a suitable canonical relation that enables uniform completeness proofs for various logics of false belief, including the transitive logic. Furthermore, the paper extends the results to the logic of radical ignorance, showing that the operators of false belief and radical ignorance are interdefinable.

Abstract

The paper focuses on the logical study of false belief and radical ignorance. It starts by introducing the language and semantics of the logic of false belief, and proposes an "almost definability" schema that relates the standard doxastic operator and the false belief operator. The paper then investigates the expressive power and frame definability of the logic of false belief, showing that it is less expressive than standard doxastic logic on various classes of models. It also identifies a class of narcissistic frames where the two logics are equally expressive. The main contribution of the paper is the axiomatization of various logics of false belief, including the minimal logic, the serial logic, the transitive logic, and the Euclidean logic. The axiomatization of the transitive logic solves an open problem raised in previous work. The key ideas are: (1) the "almost definability" schema guides the discovery of the core axioms, and (2) a suitable canonical relation is introduced to enable uniform completeness proofs. Finally, the paper extends the results to the logic of radical ignorance, showing that the operators of false belief and radical ignorance are interdefinable. It axiomatizes the minimal logic and the serial/transitive logic of radical ignorance.

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Key Insights Distilled From

Logics of False Belief and Radical Ignorance

by Jie Fan at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07442.pdf

Deeper Inquiries

What are the philosophical motivations and implications behind the notions of false belief and radical ignorance

The philosophical motivations behind the notions of false belief and radical ignorance stem from the fundamental distinctions in epistemology and cognitive science. False belief, as explored in the paper, delves into the concept that individuals can hold beliefs that do not align with reality. This notion challenges the traditional understanding of knowledge, which requires beliefs to be true. By introducing false belief into logical frameworks, researchers aim to capture the complexity of human cognition and reasoning, where individuals can hold false beliefs despite evidence to the contrary. On the other hand, radical ignorance, as proposed in the paper, seeks to address the phenomenon of the Dunning-Kruger effect, where individuals lack awareness of their own incompetence in a particular domain. Radical ignorance goes beyond mere lack of knowledge to encompass a deeper level of unawareness and misjudgment. By formalizing radical ignorance in logical systems, researchers aim to model the complexities of human cognition, particularly in situations where individuals overestimate their abilities or knowledge. The implications of these notions extend to various fields, including philosophy, psychology, and artificial intelligence. By formalizing false belief and radical ignorance, researchers can gain insights into human reasoning processes, decision-making, and the limitations of cognitive abilities. These formalizations can also inform the development of AI systems that mimic human-like reasoning and decision-making, taking into account the nuances of false beliefs and radical ignorance.

Can the results on the interdefinability of false belief and radical ignorance be extended to other related epistemic/doxastic notions, such as reliable belief

The results on the interdefinability of false belief and radical ignorance may provide insights into the relationship between these notions and other related epistemic or doxastic concepts, such as reliable belief. While the paper focuses on the interdefinability of false belief and radical ignorance, similar approaches could be applied to explore the connections between reliable belief and these notions. Reliable belief, which pertains to beliefs that are trustworthy or dependable, contrasts with false belief and radical ignorance. By investigating the interdefinability of reliable belief with false belief and radical ignorance, researchers could uncover how these concepts interact and influence each other within logical frameworks. This exploration could shed light on the nuances of belief formation, justification, and reliability in different epistemic contexts. Extending the results to include reliable belief would offer a more comprehensive understanding of the relationships between different belief states and cognitive phenomena. By examining how these notions can be defined in terms of each other, researchers can deepen their insights into the complexities of human belief systems and the factors that influence the reliability and accuracy of beliefs.

Are there any interesting classes of frames where the logics of false belief and standard doxastic logic are equally expressive, beyond the narcissistic frames considered in the paper

The paper discusses the expressive power and frame definability of the logics of false belief and standard doxastic logic on various classes of frames. While the results highlight the differences in expressivity between these logics on most frame classes, the notion of equally expressive logics on certain interesting classes of frames opens up avenues for further exploration. Beyond the narcissistic frames considered in the paper, it would be intriguing to investigate other classes of frames where the logics of false belief and standard doxastic logic exhibit equal expressive power. Identifying such classes could provide valuable insights into the structural properties of frames that influence the expressivity of these logics. By exploring these classes, researchers could uncover new perspectives on the relationships between false belief and standard doxastic logic, potentially leading to novel findings and implications in the field of epistemic and doxastic logic.