On the Meanings of the Logical Constants and the Justifications of the Logical Laws
📜 Abstract
This paper is about the semantics of logical constants and the justifications of the laws of logic. It is based on intuitionistic type theory, which allows us to treat concepts like constructivity, evidence, and the meaning of logical constants in precise mathematical terms. The approach is philosophical and foundational, aiming to give a constructive interpretation of the logical laws. Instead of treating logic as a collection of inexplicable laws, this work tries to demonstrate how logical constants can be understood meaningfully by associating them with operations on evidence. The goal is to elucidate how logical laws can be justified in terms of the meanings of logical constants, grounded in a type-theoretical framework.
✨ Summary
Per Martin-Löf’s 1983 paper, “On the Meanings of the Logical Constants and the Justifications of the Logical Laws,” offers an in-depth exploration of the semantics of logical constants within the framework of intuitionistic type theory. The paper proposes a constructive interpretation of logical laws, emphasizing their justification through the meanings of logical constants and the role of evidence. This work has been significant in philosophical logic and proof theory, shaping discussions around constructivism and the foundational aspects of mathematics. Intuitionistic type theory has influenced the development of proof assistants and automated reasoning systems. While there are no direct citations of this paper in modern literature found during my search, the concepts articulated by Martin-Löf continue to underpin much of the research in constructive mathematics and type theory. The influence of this paper is also evident in the ongoing discourse in the philosophy of logic, particularly concerning the roles of semantics and constructivism.