The Interdefinability of Expansions of Belnap-Dunn Logic

Core Concepts

Belnap-Dunn logic explores the interdefinability of expansions with classical connectives, shedding light on their relationships and definability. The author delves into the forgotten notion of defining connectives in logics, emphasizing the importance of understanding these relationships.

Abstract

The content discusses the interdefinability of expansions of Belnap-Dunn logic with classical connectives, focusing on defining connectives and exploring their relationships. It highlights the forgotten notion of defining connectives in logics and provides insights into the logical consequence relations and equivalence relations within these logics. The study showcases how various expansions are interconnected and how certain non-classical connectives can be defined within a framework of classical propositional logic.

Stats

For all A1, A2 ∈ FormBD: A1 ⇔BD A2 iff A1 |=BD A2, A2 |=BD A1, ¬A1 |=BD ¬A2, and ¬A2 |=BD ¬A1.
BD⊃,F ≃ BD∆; BD∆ ≃ BD◦∗; BD⊃,F ≃ BD◦∗.
∆ is definable in BD⊃,F; ◦∗ is definable in BD◦∗; ∗ is not definable in BD⊃,F.
F is definable in BD⊃,B,N,F; ◦ and ∗ are not definable in BD◦,∗.

Quotes

"The close relationship between their logical consequence relations is illustrated using a sound and complete sequent calculus proof system for BD⊃,F."
"Almost all publications that refer to the definability of connectives concern logics whose logical consequence relation is defined using a logical matrix."

Key Insights Distilled From

The interdefinability of expansions of Belnap-Dunn logic

by C. A. Middel... at arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04641.pdf

Deeper Inquiries

How does the concept of interdefinability impact traditional logic systems

The concept of interdefinability has a significant impact on traditional logic systems by highlighting the relationships between different logics. When two logics are interdefinable, it means that they can define each other's connectives within their own framework. This not only demonstrates the equivalence between the two logics but also shows how they can be translated or understood in terms of one another. In traditional logic systems, this concept helps bridge the gap between different logical frameworks and allows for a deeper understanding of their structures and operations.

What implications does the forgotten notion of defining connectives have on modern logic studies

The forgotten notion of defining connectives in modern logic studies sheds light on an essential aspect that may have been overlooked over time. Understanding how connectives are defined within a logic is crucial for analyzing its properties, relationships with other logics, and overall expressive power. By revisiting this notion, modern logic studies can benefit from a more comprehensive approach to defining and comparing different logical systems. It provides a foundational understanding of how connectives operate within a given logic and opens up avenues for exploring connections with other logics.

How can understanding the interdefinability between different logics enhance our approach to complex problem-solving

Understanding the interdefinability between different logics offers valuable insights into complex problem-solving approaches. By recognizing how various logics can define each other's connectives or concepts, we gain a broader perspective on problem-solving strategies across diverse domains. This knowledge enables us to leverage multiple logical frameworks to tackle complex issues effectively by drawing upon the strengths and nuances of each system. Additionally, exploring interdefinability enhances our ability to translate problems from one logical context to another, facilitating innovative solutions through interdisciplinary approaches that combine insights from various fields of study.