 Francez motivates these principles by certain natural language discourses and a “dual Ramsey Test” that modifies the Ramsey test by assuming that in the course of arguing “If p will q?,” ~p is hypothetically added to a stock of knowledge. Francez’ natural deduction rules are straightforwardly obtained by modifying the natural deduction rules for the negation and implications fragment of David Nelsons’s four-valued constructive logic N4, cf. Kamide and Wansing 2012, in the manner that leads from N4 to the constructive connexive logic C from Wansing 2005, cf. section 4.5.1. In Francez 2019 the natural deduction system that gives AT, AT′, BT, and BT′ is relevantized as in the familiar natural deduction proof system for the implication fragment of the relevant logic R by introducing subscripts for book-keeping in order to avoid empty, irrelevant implication introductions. Omori (2016b) adds conjunction and disjunction to the language of Francez 2016, gives an axiomatization and a characterizing semantics for the natural deduction system that allows to prove B3 and B4, and observes that although AT and AT′ are valid, BT, and BT′ are invalid, which prompts him to call the provable equivalence ~(A → B) ↔ (~A → B) “half-connexive”.