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me property as that of being a member of the series 2, 4, 6, 8, The property of being even (being an even integer) is equivalent to the property of being divisible by 2 without remainder; but is that property identical with, or even equivalent to, the property of being a member of the set 2, 4, The members of the set in question are thus divisible; but is being a member of it the same property as being divisible by 2 without remainder? The former is a set-theoretical property, the latter a numerical (arithmetical) one. These seem different in kind. This example and others, such as those involving purported reductions of normative concepts or properties, bring out an important point. The relevant sense of reduction is no clearer than our understanding of when two predicates express the same property (or, given any properties F and G, under what conditions F = G). This remains a vexed question. Do two predicates express the same property provided they are (a) synonymous, (b) analytically equivalent, (c) conceptually equivalent (if that is not equivalent to one of the other categories), (d) logically equivalent, (e) metaphysically equivalent, (f) synthetically a priori equivalent, (g) explanationally equivalent, (h) nomically equivalent, (i) causally equivalent, or (j) something else again? Even if we have an answer to this question, the ten notions in question remain controversial, and they all need analysis. Whatever our criteria for property identity, it is far from clear that there are disjunctive properties at all, as opposed to disjunctions of predicates that each express a property. If there are disjunctive properties, we need an explanation of why specifying that a shape is either circular or elliptical is not an answer to the question What are its shape properties? or, especially, What is its shape? One could argue that the needed explanation is simply pragmatic, say that in asking for a shape we presuppose in normal contexts that a non-disjunctive specification will be given; but that is not confirmed by our explanatory or descriptive practices even when answering a particular question in a communicative situation is not a requirement for appropriateness.[18] To be sure, the disjunctively specified grounds of obligation cited above are significant; they yield multiple paths from certain descriptive notions to the crucial notion of obligatory action. Similarly for the grounds of justification, which might be argued to be disjunctively specifiable in terms of, say, the classically recognized basic sources of justification: perception, memory, introspective consciousness, and intuitive reason. But this point about the grounding of the normative shows-assuming the grounds in question are genuinely constitutive-only that the normative properties (and the concepts of them) are anchored in the natural world, not that those properties are genuinely natural or even equivalent to disjunctive properties, if there are such properties. This is a good place to note that what has been said of properties may well not apply to concepts. Concepts are essentially connected with subsuming things under the abstract equivalent of a description, an equivalent that may be disjunctive or otherwise compound. This does not require indicating something about what the relevant things are "like", especially where this is related to explanation, prediction, connections to other things, or all of these (and perhaps other elements). Viewed in this way, at least, concepts may be seen as disjunctive. In this, and in the very fine-grained way in which we individuate them, concepts are more like propositions than like properties. One might now wonder whether we might naturalize normative concepts like that of obligation disjunctively, even if not the property. First, it is doubtful that a naturalistic concept can express a non-naturalistic property. If this is so, then conceptual naturalization of the normative will apparently fail if naturalization of normative properties fails. But even apart from this question, the concept of obligation is surely not equivalent to the disjunctive concept of being either a promise-keeping or an avoidance of harm for any purportedly complete list of the constitutive sources of obligation. These elements may perhaps indicate all of the various grounds of obligation but not what, conceptually, obligation is. It is a kind of commitment, and the disjunction indicates ways of incurring it, but not what it is. 4. Non-Reductive Naturalism The view that non-natural properties are strongly supervenient on natural ones is sometimes called non-reductive naturalism, and the name is doubly appropriate if the supervenience is paired with a grounding relation. It normally is paired with one, though in the mental-physical case the grounding relation is empirical and presumably nomic, whereas in the normative case it is (on my view and that of many others) a priori and metaphysically necessary rather than nom