Imagine a very fi ne grid or graph on which dots are placed at various coordinates so that, as a consequence, this or that shape materializes there. Depending on the coordinates of the dots, different shapes will appear, and for every shape there will be a pattern in the coordinates that guarantees its appearance. Take, for example, the diagonal line that slopes rightward and upward at an angle of 45 degrees from the origin. This line is bound to make an appearance so long as the coordinates satisfy the condition or pattern that as they move away from the origin, (0,0), the coordinates are progressively larger pairs of equal numbers: (1,1), (3,3), and so on. In the world of such dots and shapes, it is going to be in principle possible, for any array of dots that realizes a relevant shape, to derive the presence of the shape from the numerical coordinates of the dots. More particularly, it is going to be possible to derive that shape without reliance on anything other than, fi rst, the empirical fact that the given array of coordinates instantiates this or that pattern; and second, the a priori knowable fact that the pattern guarantees the presence of the shape in question. The nature of the shapes on any grid—if indeed there are any relevant shapes present—is going to be a priori derivable from the positions of the dots; it is going to be possible in principle to derive the one from the other. The simplest and most appealing version of physicalism parallels this sort of doctrine about dots and shapes (Pettit 1994, 1995). It holds that just as the positions of the dots determine the nature of the shapes a priori, so the way the natural world is physically organized a priori determines the way it presents itself in psychological and other terms. The way things are physically confi gured entails the presence of psychological and other realities, and it does this without reliance on anything other than what a...