 Then the truth-functions f for which every valuation consistent with ⊢ (as above) associates #f with f are precisely the projection-conjunction truth-functions. (Gabbay calls #f, as it behaves according to ⊢, strongly classical in these cases, though his own definition is not formulated in these terms.) By way of example: when n = 1, these are the identity and the constant-true functions, while for n = 2, they are the first projection, second projection, constant true and (Boolean) conjunction truth-functions. (The constant true cases arise by taking J as ∅.) The remaining 12 binary truth-functions thus exhibit the behaviour we saw with the disjunction truth-function; the corresponding connectives are what Gabbay calls weakly classical: determined by a class of valuations over which the connective is associated with a certain truth-function despite the existence of further consistent valuation on which it is not so associated. It is easy to see that the latter phenomenon does not arise in the case of generalized consequence relations ⊩, since we can easily write down unconditional conditions on such relations which enforce the association with a particular truth-function on valuations in Val(⊩). This is best seen as proceeding determinant by determinant. The unenforceable determinant in the case of disjunction is ⟨F, F, F⟩, dictating that when the disjuncts have the values listed in the first two positions, the disjunction has the value listed in the final position. This determinant induces in a familiar way (e.g., Segerberg 1982, §3.3) the following condition on ⊩: that for all φ, ψ, we have φ ∨ ψ ⊩ φ, ψ. Alternatively put, in Set-Set, as opposed to Set-Fmla, there is a set of formulas holding on precisely the ∨-Boolean valuations, namely that containing, for all formulas φ, ψ, the sequents φ ≻ φ ∨ ψ, and ψ ≻ φ ∨ ψ, and—the new case not already available in Set-Fmla—φ ∨ ψ ≻ φ, ψ. In view of this, we can say that in Set-Set though not in Set-Fmla, the class of ∨-Boolean valuations is sequent-definable, and the same goes for connectives associated with all the other truth-functions not meeting the projection-conjunction test (such as negation and material implication). As the ∨ example makes clear, by contrast with the case of consequence relations, Val(⊩), for a generalized consequence relation ⊩, need not be closed under conjunctive  combinations.[7]  Again, one can see this in the light of the considerations of Section  1,  thinking of the relational connection between Set-Set sequents and arbitrary valuations, with the relation being ‘holds on’. This connection supplies disjunctive combinations on the left, with σ + σ′ being the least common weakening of σ and  σ′.[8]  Thus it cannot also supply conjunctive combinations on the right, since it does not satisfy the crossover condition.