 Thus (LE) requires that each v ∈ Val(⊩) “treats truths alike” when they are embedded under #, in the sense that when v(φ) = v(ψ) = T, then v(#φ) = v(#ψ), while (RE) requires that falsehoods be treated alike in that v(φ) = v(ψ) = F likewise implies that v(#φ) = v(#ψ). (It may help to remark that for ↔-classical ⊢ together these amount to requiring that φ ↔ ψ ⊢ #φ ↔ #ψ. But we are trying to keep the discussion ‘pure’ so that definitions make sense even when the connective—here #—with the property being defined is the sole connective in the language.) The conditions (LEi) already make sense as conditions on consequence relations (writing ⊢ for ⊩) since there is a single formula on the right, and we will speak of # as left-extensional according to ⊢ when they are all satisfied, while the conditions (REi) need to be converted into conditional conditions as above in the discussion of determinant-induced conditions. We will illustrate this only for the n = 1 case, with (RE) becoming (RE′), understood as demanding for all Γ, φ, ψ, χ: