 Indeed the real moral is more general still. We did not need to use, in our worked example, more than the fact that there was some determinant respected by every valuation in Vf with the opposite determinant respected by every valuation in Vg, where V respects a determinant ⟨x1,…,xn,xn+1⟩ for # (with each xi ∈ {T, F}) just in case for all formulas φ1,…,φn, v(φi) = xi for i = 1,…, n implies v(#(φ1,…,φn) = xn+1, and determinants are opposite if they differ in precisely their final positions. (Actually, inspection of the worked example shows that we also need the classes playing the role of Vf and Vg also to contain enough valuations to assign different truth-values to any pair of distinct propositional variables.) Take the 1-ary connective ☐ of any familiar modal logic (e.g., S5), for example. We expect it to satisfy the determinant ⟨F, F⟩, in the sense that the appropriate consequence relation should satisfy the condition induced by this determinant, since what is necessarily true is true—and thus if φ is false, ☐φ should be false—while we do not want any determinant of the form ⟨T, x⟩ to be satisfied. In this case we can describe ☐ as partially determined according to the consequence relation in question. On the other hand, if a consequence relation ⊢ (or a generalized consequence relation ⊩) satisfies, for each n-tuple of truth-values, the condition induced by exactly one determinant for the n-ary connective #, whose first n entries are as given by that n-tuple (the final, n+1st, entry being either a T or an F), then we say # is fully determined according to ⊢ (or ⊩). By the determinant-induced conditions on a consequence relation here is meant the conditions on a consequence relation derived as explained (or at least illustrated) above from the directly determinant-induced conditions on a generalized consequence relation. As one might expect, this property of connectives relative to consequence relations has an intimate relation with the similar property introduced at the start of this section: