Remark on the hypothetical judgment

TL;DR

This paper examines the proper interpretation of intuitionistic hypothetical judgment and propositional implication, emphasizing the need for explanations that support the development of transcendental arguments in mathematics.

Contribution

It critically analyzes Brouwer and Heyting's conflicting explanations and proposes criteria based on the expression of the bar theorem for evaluating their adequacy.

Findings

Highlights ambiguities in historical interpretations of hypothetical judgment

Proposes criteria for acceptable explanations based on mathematical expression

Emphasizes importance for development of transcendental frameworks

Abstract

What is the proper explanation of intuitionistic hypothetical judgment, and thence propositional implication? The answer is unclear from the writings of Brouwer and Heyting, who in their lifetimes propounded multiple (sometimes conflicting) explanations of the hypothetical judgment. To my mind, the determination of an acceptable explanation must take into account its adequacy for the expression of the bar theorem and, more generally, the development of an open-ended framework for transcendental arguments in mathematics.