 Let us note in view of the recent reference to ⊢∧ ∩ ⊢∨, there has been an interest in how one might form something like the union rather than the intersection of the two consequence relations involved here. Not quite the union, since that would not be a consequence relation, but something answering to the intuitive idea of a consequence relation embodying the combined logic of conjunction and disjunction. (This is the theme of Béziau and Coniglio (2010), further taken up in Humberstone (2015).) The situation is not exactly dual to that of hybrids, to which topic we return in the folowing paragraph, because the envisaged hybrid logic contains one connective with the common logical properties of (in the present instance) conjunction and disjunction, whereas the combined logic is based on the language with the two connectives. (The precise dual would be a consequence relation on a language with a single connective, roughly speaking giving that connective all the logical powers possessed by either disjunction or conjunction, and by a result of Wolfgang Rautenberg cited below, according to this consequence relation any formula follows from any other. Indeed, this particular instance of Rautenberg’s observation is already well known in the literature on Tonk, recalled in §3 below: any consequence relation in a language including a binary connective which satisfies, relative to that consequence relation, both an ∨-Introduction condition and also an ∧-Elimination condition will be one of these trivial consequence relations.