TITLE: Two styles of semantics for a first-order language: what's to choose?
QUESTION [17 upvotes]: The usual classical semantics for FOL gets presented in two styles. Suppressing details irrelevant for the headline question, suppose the $L$-wff $\varphi(x)$ has only $x$ free, and let $I$ be a fixed interpretation of the non-logical vocabulary of $L$. Then one story comes to this:

$\exists x\varphi x$ is true on interpretation $I$ if, for some assignment $s$ of an object in the domain as value to $x$, $\varphi x$ is true on $I + s$.

The other story comes to this:

$\exists x\varphi x$ is true on interpretation $I$, if for some name $c$ in $L^+$ -- $L$ augmented with a name for every object in the domain -- $\varphi c$ is true on $I^+$ (the interpretation $I$ augmented with interpretations of all the new names).

We either keep $L$ fixed, and spin the objects assigned to variables; or we augment $L$ with a fixed name for every object.  On the face of it there is little (nothing?) to choose. Different standard texts go in different ways. But are there subtle techie or conceptual reasons to prefer one line to the other?
[There was some relevant discussion among some philosophers over 35 years ago about Tarskian vs Fregean styles of semantics: but I'm interested here in the views of  mathematicians (and computer scientists!).]

REPLY [5 votes]: The approach that proceeds via an extended language is sometimes more elegant. With the other "variable interpretation function" approach, it is no longer true that a formula is true or false in a structure; instead one has to speak of truth in a structure relative to a variable interpretation. One has to prove as a separate theorem that two variable interpretations that agree on the free variables of a formula will give it the same truth value, and thus the truth of a sentence in a structure is independent of the variable interpretation. However, to check the truth of a sentence, we must still begin with some arbitrary variable interpretation. That separate theorem, the need to choose an arbitrary assignment that will not affect anything, and the notational complications of variable interpretation functions are entirely eliminated with the augmented-language approach.
Also, the definition by variable interpretation functions fails for empty structures (i.e. empty domain of discourse), which do not have any variable interpretation functions. In light of the theorem mentioned before, we might define a sentence to be true in a structure when it is satisfied by every variable interpretation - but then every sentence is true in the empty model. If we say that a sentence is true when it is satisfied by some variable interpretation, then no sentence is true. Either way, the semantics via variable interpretation functions breaks down in empty structures. By contrast, the augmented-language version treats the quantifiers more disquotationally and has no problem with empty structures. 
The main downside of the augmented-language version is that it abuses the naive meaning of "language". When a structure is uncountable, its augmented language is also uncountable. That doesn't cause any mathematical problems if we treat the language abstractly, but if we think that a key aspect of a formal language is that we can write down each sentence with a finite number of symbols from a finite alphabet, then uncountable languages are no good.