 Aristotle’s and Boethius’ theses express, as it seems, some pre-theoretical intuitions about meaning relations between negation and implication. But it is not clear that a language must contain only one negation operation and only one implication. The language of bi-intuitionistic logic contains two negations, the language of the bi-intuitionistic connexive logics in Wansing 2016b and Kamide &amp; Wansing 2016 contain three negations, and the language of systems of consequential implication comprises two implication connectives together with one negation, see Pizzi 1977, 1991, 1993, 1996, 1999, 2004, 2005, 2008, 2018, Pizzi and Williamson 1997, 2005. Pizzi (2008, p. 127) considers a notion of consequential relevance, namely that “[t]he antecedent and the consequent of a true conditional cannot have incompatible modal status,” and suggests to capture consequential relevance by requiring that in any true conditional A → B, (i) A strictly implies B and (ii) A and B have the same modal status in the sense that □A ⊃ □B, □B ⊃ □A, ◊A ⊃ ◊B, and ◊B ⊃ ◊A are ture, where ⊃ is material implication. Moreover, it is required that □A ⊃ ◊A is always true.