 It is easy to construct complex predicates. But whether there really are corresponding properties of this kind is a much more difficult and controversial issue, tightly bound to the sparse/abundant distinction. In the sparse conception the tendency is to dispense with them. This is understandable since in this camp one typically postulates properties in rebus for empirical explanatory reasons. But then, if we explain some phenomena by attributing properties \(F\) and \(G\) to an object, while denying that it does not exemplify \(H\), it seems of no value to add that it also has, say, F and G, F or H, and non-H. Nevertheless, Armstrong, the leading supporter of sparseness, has defended over the years a mixed stance without disjunctive and negative properties, but with conjunctive ones. Armstrong’s line has of course its opponents (see, e.g., Paolini Paoletti 2014; 2017a; 2020a; Zangwill 2011 argues that there are negative properties, but they are less real than positive ones). On the other hand, in the abundant conception, all sorts of logically compound properties are acknowledged. Since the focus now is on meaning and mental content, it appears natural to postulate such properties to account for our understanding of complex predicates and of how they differ from simple ones. Even here, however, there are disagreements, when it comes to problematic predicates such the “does not exemplify itself” of Russell’s paradox. Some suggest that in this case one should deny there is a corresponding property, in order to avoid the paradox (Van Inwagen 2006). However, we understand this predicate and thus it seems ad hoc that the general strategy of postulating a property fails. It seems better to confront the paradox with other means (see  §6).