 In a proof system in Set-Fmla such conditional requirements emerge as sequent-to-sequent rules, closure under which (along with containing all the sequents corresponding to the unconditional requirements) will suffice to secure completeness—in this case that any unprovable sequent fails to hold on some ¬-Boolean valuation—without the strong classicality property of having the provable sequents hold only on such valuations. (The process just described transforms the Set-Set condition, φ ∨ ψ ≻ φ, ψ (all φ, ψ), induced by the Set-Fmla unenforceable ∨ determinant ⟨F, F, F⟩, into the conditional condition of closure under the rule (∨E).) If a generalized consequence relation ⊩ with a Boolean connective # in its language satisfies all the conditions induced by the determinants of the associated truth-function we call ⊩ #-classical. Similarly for a consequence relation which satisfies the conditions on consequence relations—conditional conditions included—derived, as just described, from these determinant-induced conditions on generalized consequence relations. (These versions of #-classicality can often be simplified on a case-by-case basis. For example, the ∧-classicality of a consequence relation or generalized consequence relation—we use the former notation here—is equivalent to its being the case that for all φ, ψ, we have (1) φ, ψ ⊢ φ ∧ ψ, (2) φ ∧ ψ ⊢ φ, and (3) φ ∧ ψ ⊢ ψ. Here we see a three-part condition, instead of the four-part condition comprising the conditions induced by each of the four determinants.)