Interpretation of relational symbols 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?
 A: I don't see why you think this would be more consistent, and I think the analogy to the comprehension axiom (schema) doesn't fit.
First, I wouldn't say that the axiom schema of comprehension assigns sets to formulas, at least not in the same sense as sets are assigned to symbols in the definition of a structure; rather, it generates one closed formula as an axiom for each open formula with at least two free variables, and this formula happens to assert the existence of a set with a certain property -- if anything can be said to be assigned here, I'd say a closed formula is assigned to each open formula with at least two free variables.
Second, the axiom schema of comprehension does this for every open formula with at least two free variables; in your case, you want to assign sets only to very specific formulas, namely the atomic formulas corresponding to each of the relational symbols. In fact, to be precise you'd need to assign sets to equivalence classes of these atomic formulas so you don't distinguish between free variables with different names. This problem doesn't occur with respect to the axiom schema of comprehension since there's no harm done in asserting the existence of a set multiple times with different variables names, which shows again that asserting the existence of sets and assigning sets are two quite different things. If you do form the equivalence classes, there's no longer any real difference between assigning a set to each relational symbol and assigning a set to each equivalence class of atomic formulas with relational symbols, since they're in one-to-one correspondence, so you'd just be complicating the matter without changing the content. The same is not true for the axiom schema of comprehension, which is fundamentally about formulas and can't be equivalently formulated with respect to (equivalence classes of) symbols.
