 If we reflect on these alternatives more, the role of a further basic dichotomy becomes apparent (within the category of non-eliminative structuralism): that between ante rem positions and either in re or post rem positions. Shapiro’s position is explicitly a form of  ante rem structuralism. In contrast, Russellian abstractionist structuralism can be seen as a form of post rem structuralism, because the equivalence classes used as the relevant structures are “built out of” their elements as classes, to which they  are thus posterior. In Dedekindian abstractionist structuralism, there is a form of posteriority involved as well. Here too, we start with more concrete relational systems, usually systems of sets or urelements, and we introduce abstract structures on that basis. But the prior versus posterior relation is different now. (It is not based on the element-class relation, but on a more basic argument-function-value relation.) We also end up with abstract structures that are neither standard sets nor classes.