SEP presents the dilemma as follows:
4.3 Hempel’s dilemma
One might object that any formulation of physicalism which utilizes the theory-based conception will be either trivial or false. Carl Hempel (cf. Hempel 1969, see also Crane and Mellor 1990) provided a classic formulation of this problem: if physicalism is defined via reference to contemporary physics, then it is false — after all, who thinks that contemporary physics is complete? — but if physicalism is defined via reference to a future or ideal physics, then it is trivial — after all, who can predict what a future physics contains? Perhaps, for example, it contains even mental items. The conclusion of the dilemma is that one has no clear concept of a physical property, or at least no concept that is clear enough to do the job that philosophers of mind want the physical to play.
One response to this objection is to take its first horn, and insist that, at least in certain respects contemporary physics really is complete or else that it is rational to believe that it is (cf. Smart 1978, Lewis 1994 and Melnyk 1997, 2003). There is an element of truth in this. It may be rational to believe that contemporary science is true, even if not that it is complete. Nevertheless, it also seems mistaken to define physicalism with respect to the physics that happens to be true in this world. The reason is that whether a physical theory is true or not is a function of the contingent facts; but whether a property is physical or not is not a function of the contingent facts. For example, consider medieval impetus physics. Medieval impetus physics is false (though of course it might not have been) and thus it is irrational to suppose it true. Nevertheless, the property of having impetus — the central property that objects have according to impetus physics — is a physical property, and a counterfactual world completely described by impetus physics would be a world in which physicalism is true. But it is hard to see how any of this could be right if physicalism were defined by reference to the physics that we have now or by the physics that happens to be true in our world. (For development of this point, and for a dilemma that is similar to Hempel’s but which casts the issue in modal rather than temporal terms, see Stoljar 2010; for discussion, see Baltimore 2013, Fiorese 2016)
A different response to Hempel’s dilemma is that what it shows, if it shows anything, is that a particular proposal about how to define a physical property — namely, via reference to physics at a particular stage of its development — is mistaken. But from this one can hardly conclude that we have no clear understanding of the concept at all. As we have seen, we have many concepts that we don’t know how to analyze. So the mere fact — if indeed it is a fact — that a certain style of analysis of the notion of the physical fails does not mean that there is no notion of the physical at all, still less that we don’t understand the notion.
One might object that, while these remarks are perfectly true, they nevertheless don’t speak to something that is right about Hempel’s dilemma, namely, that for the theory-conception to be complete one needs to know what type of theory a physical theory is. Perhaps one might appeal here to the fact that we have a number of paradigms of what a physical theory is: common sense physical theory, medieval impetus physics, Cartesian contact mechanics, Newtonian physics, and modern quantum physics. While it seems unlikely that there is any one factor that unifies this class of theories, perhaps there is a cluster of factors — a common or overlapping set of theoretical constructs, for example, or a shared methodology. If so, one might maintain that the notion of a physical theory is a Wittgensteinian family resemblance concept. However, whether this is enough to answer the question of what kind of theory a physical theory is remains to be seen. (For further discussion of Hempel’s dilemma, see the papers in Elpidorou 2018a.)
In short, either we define physicalism in terms of current physics, which is clearly imperfect and incomplete (e.g., The Crisis in String Theory is Worse Than You Think | Leonard Susskind), or we define physicalism in terms of a vague, idealized, hypothetical notion of physics that we will hopefully figure out in the future, but in that case we have no way to know in advance what such an idealized notion of physics will actually look like (e.g., perhaps it will include consciousness as fundamental, perhaps it will include spirits, etc.).
Is there any way out of this dilemma?