 These rules are Cut-inductive, as the interested reader may confirm, but not Id-inductive, since they do not uniquely characterize Knot. To see this, note that both are derivable rules for ∧ and also for ∨, and of course we cannot prove φ ∨ ψ ≻ φ ∧ ψ in general. By contrast, neither ∧ nor ∨ satisfies both of the left and right rules for Tonk. The upshot of the Knot rules is best visualised in terms of the zero-premiss rules (sequent-schemata) with which they are interderivable given the structural rules: φ, ψ ≻ φ Knot ψ (the right rule) and φ Knot ψ ≻ φ, ψ (the left rule). An incidental point: since these are the sequent formulations of conditions induced by the determinants ⟨T, T, T⟩ and ⟨F, F, F⟩ one might think of them together as expressing idempotence and therefore as being interchangeable with the simpler pair φ ≻ φ Knot φ and φ Knot φ ≻ φ. But while the latter two can be derived from the earlier pair (just take ψ as φ), the converse derivation is not possible without further principles—conspicuously, a sequent version of the extensionality conditions on generalized consequence relations mentioned in Section  2.  We could call binary # for which φ, ψ ⊩φ # ψ and conversely, strongly idempotent according to the generalized consequence relation ⊩. Then ∧ and ∨ are not the only Boolean candidates for a connective behaving as the Knot rules dictate—strongly idempotent according to the generalized consequence relation ⊩CL determined by the class of Boolean valuations: we also have the binary projection to the first coordinate (φ # ψ equivalent to φ), to the second coordinate (φ # ψ equivalent to ψ), and venturing further afield, the Set-Fmla hybrids of any two of these connectives. Of these perhaps the most interesting is the hybrid of the two projection connectives, since this—let us write is as “•”—can be used to define all other hybrids explicitly from the connectives to be hybridized; for example if we want the common properties (as explained in Section  2)  of → and ∨, we have them in the following compound of φ and ψ: (φ → ψ) • (φ ∨  ψ).[32]