 As a substantive matter, how should we think of these “reductions”? For example, should we view them as giving us an analysis of what it is for something to be obligatory? Taking Kanger's approach first, it would seem that \(d\) must be read as a distinctive deontic ingredient, if we are to get the derivative deontic reading for the “reduced” deontic operators. Also, as our reading suggests, it is not clear that \(d\) does not, at least by intention, express a complex quantificational notion involving the very concept of obligation (demand) as a proper part, namely that all obligations have been fulfilled, so that the “reduction”, presented as an analysis, would appear to be circular. If we read \(d\) instead as “ideal circumstances obtain”, the claim of a substantive reduction or analysis appears more promising, until we ask: Are the circumstances ideal only with respect to meeting normative demands or obligations, or are they ideal in other (for example supererogatory) ways that go beyond merely satisfying normative demands? Anderson’s “liability to sanction” approach may appear more promising, since the idea that something is obligatory if (and only if) and because non-compliance necessitates (in some sense) liability to (or perhaps desert of) punishment does not appear to be circular (unless the notion of “liability” itself ultimately involves the idea of permissibility of punishment), but is it plausible? Alternatively, perhaps a norm that is merely an ideal cannot be violated, in which case perhaps norms that have been violated can be distinguished (as a subset) from norms that have not been complied with, and then the notion of an obligation as something that must obtain unless some norm is violated will not be obviously circular. There is a substantive philosophical question lingering here that the language of a “reduction” brings naturally to the surface, but is sloughed over. The formal utility of the reduction does not hinge on this, but its philosophical significance does.