the relevant chapter earned the four truth values {T, F, B, N} (the relevant definition, I.D21), and the relevant chapter proved the explosion barrier (the relevant theorem, I.T13): the overdetermined value B does not collapse the logic to triviality. A natural question follows: when is the full four-valued logic necessary, and when does classical Boolean logic suffice? This chapter answers both questions. First, we construct a forgetful map f : Truth4 → Bool that collapses B ↦ T and N ↦ F, recovering classical two-valued logic as a quotient. The Boolean recovery theorem (Proposition [prop:boolean-recovery], I.P13) identifies the exact condition under which this collapse is lossless: when the spectral decomposition assigns only T or F to a predicate, Truth4 reduces to Bool with no information loss. Second, we define Ω_τ := Truth4 as the subobject classifier (the relevant definition, I.D41), previewing its role as the truth-value object of the earned topos in Part XIII.