 In several respects, this second abstractionist alternative is close to Shapiro’s ante rem structuralism (who occasionally appeals to “abstraction” himself,  e.g., in Shapiro 1997); it is also close to Parsons’  form of non-eliminative structuralism. Yet it neither involves a separate  structure theory à  la Shapiro nor an appeal to Parsons’ meta-linguistic procedure. Instead, abstract structures are introduced  “by abstraction” from more concrete systems, e.g., from set-theoretic relational systems. The relevant abstraction can be explicated further in terms of an “abstraction  operator” and a corresponding “abstraction principle”.  Another comparison that suggests itself then is with the use of  abstraction principles in contemporary neo-logicism. In fact, that  connection has already been explored in Linnebo &amp;  Pettigrew (2014) and Reck (2018a). What we arrive at along such lines is an abstractionist form of non-eliminative structuralism. In contrast, the first  alternative mentioned above is an abstractionist form of eliminative structuralism. (Given their historical roots, these positions might be labeled “Russellian abstractionist  structuralism” and “Dedekindian abstractionist  structuralism”, respectively; cf. Reck 2018a.)