Remark on the hypothetical judgment

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. 1. Judgments and Propositions The distinction between the propositions and the judgments (assertions) is an old one, but prior to Martin-Löf, the significance of assertions was limited to the affirmation of the truth of propositions. Following Martin-Löf [8], forms of judgment other than P true are recognized, including P prop. What is the difference between a judgment (assertion) on the one hand, and a proposition on the other hand? A judgment is an act or an experience, whereas a proposition is a mathematical object which may be experienced in different ways. For instance, the assertion of the truth of a proposition (i.e. P true) consists in the fulfillment of the intention expressed by the proposition, while the recognition of an object as a proposition (i.e. P prop) is the act of understanding this intention. In addition to the categorical judgments above, higher-order forms of judgment are also explained, including the hypothetical judgment and the general judgment. Now, the primitive hypothetical judgment J2 (J1), 1 was explained by Martin-Löf in terms of hypothetical proof or demonstration, which he defined as follows: The notion of hypothetical proof [demonstration], in turn, which is a primitive notion, is explained by saying that it is a proof [demonstration] which, when supplemented by proofs Hypothetical judgment is to be distinguished from the sequent judgments Γ ⊢ · · · , which are not even higher-order judgments at all.