 The second-order appearance mentioned in the title is as follows: when defining the conversion of structures into MSL we will add new universes containing all the categories of mathematical objects you can talk about in PML (or PDL); therefore, we add all sets and relations defined in these logics with their respective formulas and programs. And so, we are somehow shifting to SOL structures. We can add as well membership relation symbols to the language and membership relations to the structures, as we did in the conversion of SOL into MSL; we will end up in MSL but the inspiration is SOL.