But the most influential objection to supervenience physicalism (and to modal formulations generally) is what might be called _the sufficiency problem_. This alleges that, while (1) articulates a necessary condition for physicalism it does not provide a sufficient condition. The underlying rationale is that, intuitively one thing can supervene on another and yet be of a completely different nature. To use Fine’s famous (1994) example, consider the difference between Socrates and his singleton set, the set that contains only Socrates as a member. The facts about the set supervene on the facts about Socrates; any world that is like ours in respect of the existence of Socrates is like ours in respect of the existence of his singleton set. And yet the set is quite different from Socrates. This in turn raises the possibility that something might be of a completely different nature from the physical and nevertheless supervene on it.One may bring out this objection further by considering positions in philosophy which entail supervenience and yet deny physicalism. A good example is _necessitation dualism_ , which is an approach that weaves together elements of both physicalism and its traditional rival, dualism. On the one hand, the necessitation dualist wants to say that mental facts and physical facts are metaphysically distinct—just as a standard dualist does. On the other hand, the necessitation dualist wants to agree with the physicalist that mental facts are necessitated by, and supervene on, the physical facts. If this sort of position is coherent, (1) does not articulate a sufficient condition for physicalism. For if necessitation dualism is true, any physical duplicate of the actual world is a duplicate _simpliciter_. And yet, if dualism of any sort is true, including necessitation dualism, physicalism is false.How to respond to the sufficiency problem? Some respond by denying the coherence of the position that causes the problem. Necessitation dualism as we have just described it violates (what is known as) Hume’s dictum that there are no necessary connections between distinct existences. According to necessitation dualism, mental and physical properties are metaphysically distinct, and yet are necessarily connected. However, Hume’s dictum is itself a matter of controversy (see Jackson 1993, Stalnaker 1996, Stoljar 2010, and Wilson 2005, 2010). Another approach appeals to a priori physicalism which will examine below (see Jackson 2006). But by far the most common response has been to concede that the sufficiency problem shows that supervenience formulation of physicalism is too weak (e.g. Kim 1998), and to look for an alternative. ### 2.2 Identity PhysicalismSuppose then that (1) provides a necessary condition for physicalism but not a sufficient condition; how might we strengthen it to make it more plausible? The most obvious thing to do is to appeal to identity. Indeed, in the history of attempts to answer the completeness question, the appeal to identity predates the appeal to supervenience. Nevertheless, this version of physicalism— _identity physicalism_ as we may call it—runs into serious problems._2.2.1. Token Physicalism_In fact there are two different versions of identity physicalism, _type physicalism_ and _token physicalism_. Token physicalism is the view that every particular thing in the world is a physical particular. So the token physicalist says that physicalism should be formulated in the following way: (3) Physicalism is true at a possible world _w_ iff for every particular (object, event or process) _x_ that exists at _w_ , there is some physical particular _y_ such that _x_ = _y_But (3) offers neither necessary nor sufficient conditions for physicalism. To see that it is not sufficient, consider the variety of dualism usually called _property dualism_. Property dualism says that (a) every particular is a physical particular but (b) some particulars (e.g. human beings) have psychological properties wholly distinct from any physical properties. The contrast here is with _substance dualism_. The substance dualist agrees with the property dualist that some particulars have psychological properties wholly distinct from any physical properties, but they will add that such particulars are themselves non-physical.