Core Concepts

The authors present a formalization of higher-order logic in the Isabelle proof assistant, building directly on the foundational Isabelle/Pure framework. The goal is to provide a gentle introduction to higher-order logic and proof assistants, without the complexity of Isabelle/HOL.

Abstract

The paper introduces a formalization of higher-order logic in the Isabelle proof assistant, called HOL_Pure, which is built directly on the Isabelle/Pure framework. This is presented as an alternative to the more complex Isabelle/HOL system, with the aim of providing a gentle introduction to higher-order logic and proof assistants.
The authors first discuss their approach, which involves a graceful extension from first-order to higher-order logic, and present their main contributions. They then discuss related work on using proof assistants for teaching formal reasoning.
The paper includes sample natural deduction proofs in propositional logic, first-order logic, and higher-order logic (Cantor's theorem) to showcase the Isabelle code and proof style. It also covers the formalization of first-order logic systems, including both implicational axiomatics and natural deduction, with formal proofs of soundness and completeness in Isabelle/HOL.
The authors then build up the intuitionistic and classical higher-order logic theories in Isabelle/Pure, starting from the fundamental concepts of implication and universal quantification. They discuss the differences between the various proof systems and logics presented, and how they are used in the authors' automated reasoning course.
Finally, the paper mentions further developments, including the addition of features like the axiom of choice and set comprehension, and concludes with remarks on the usefulness of the approach for teaching higher-order logic and proof assistants.

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Deeper Inquiries

The authors' approach to teaching higher-order logic using Isabelle can be extended to cover other advanced topics in logic and formal reasoning by following a similar structured and systematic methodology. For teaching type theory, the approach can involve introducing the foundational concepts of types, type systems, and type inference, building on the basic principles of higher-order logic. This extension would require defining type constructors, type variables, and type rules within the Isabelle framework, similar to how logical connectives and quantifiers were defined in the higher-order logic formalization.
When it comes to modal logics, the approach can focus on introducing modal operators, such as necessity and possibility, and defining the semantics and proof rules associated with modal logic within Isabelle. The formalization would involve incorporating modal axioms and modal inference rules, allowing students to reason about modal properties and modalities using the Isabelle proof assistant.
By structuring the teaching of these advanced topics in a similar manner to the higher-order logic formalization, students can gradually build their understanding from foundational concepts to more complex logical systems. Providing clear explanations, sample proofs, and interactive exercises within Isabelle can enhance the learning experience and help students grasp the intricacies of type theory and modal logics effectively.

The HOL_Pure formalization, while serving as a simplified and transparent introduction to higher-order logic in Isabelle, has certain limitations compared to the full Isabelle/HOL system. One significant limitation is the lack of automation and advanced features present in Isabelle/HOL, such as built-in theories for various mathematical concepts, extensive libraries, and sophisticated proof tactics. This limitation can impact the efficiency and convenience of using HOL_Pure for more complex formalizations and advanced theorem proving tasks.
Additionally, HOL_Pure may lack certain meta-properties and advanced functionalities that are available in Isabelle/HOL, such as the ability to reason about semantics, completeness, and soundness of the logic system. This limitation could restrict the depth of understanding that students can achieve when studying higher-order logic using HOL_Pure, especially when compared to the comprehensive capabilities of Isabelle/HOL.
Despite these limitations, HOL_Pure can still be a valuable teaching tool for introducing students to the fundamental concepts of higher-order logic and proof assistants. By focusing on simplicity, clarity, and minimalism, HOL_Pure can provide a solid foundation for beginners to grasp the core principles of logic and formal reasoning before transitioning to more advanced systems like Isabelle/HOL.

Several insights from the authors' experience teaching higher-order logic using Isabelle can be applied to teaching formal reasoning in other proof assistants or logical frameworks.
Gradual Progression: Following a structured approach that starts with foundational concepts and gradually builds up to more complex topics can be beneficial in teaching formal reasoning in other proof assistants. By introducing concepts in a logical sequence, students can develop a solid understanding of the material.
Interactive Learning: Incorporating interactive exercises, sample proofs, and hands-on activities within the proof assistant can enhance student engagement and comprehension. Providing opportunities for students to practice formal reasoning in a guided environment can improve their skills and confidence.
Clear Explanations: Offering clear explanations of logical rules, axioms, and inference techniques is essential for effective teaching. Breaking down complex concepts into digestible parts and providing detailed examples can aid in student learning and retention.
Utilizing Automation: Leveraging the automation capabilities of proof assistants to assist students in theorem proving tasks can streamline the learning process. Integrating automated proof tactics and tools can help students validate their reasoning and gain insights into efficient proof strategies.
By applying these insights to teaching formal reasoning in other proof assistants or logical frameworks, educators can create engaging and effective learning experiences for students interested in logic and formal methods.

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