Is it too pedantic to ask, why in the definition of a structure in model theory sets are assigned to the relational symbols $P, R, ...$ of a language and not to corresponding formulas $Px, Rxy, ...$ (modulo choice of variables)? It would seem to me more consistent with the interpretation of arbitrary open formulas inside model theory and compared to set theory where by the comprehension axiom sets are assigned to open formulas, not to symbols.
Is it just a notational abbreviation - to name a set by $P$ instead of $Px$ - or is there something deeper behind it? If it's an abbreviation: Why is this so seldom (if ever) made explicit?