Any difference between a consistent set of sentences and a consistent set of formulas?

Question

In several model theory textbooks consistency is only defined for a set of sentences and never for a set of formulas. However, when I researched on-line, there didn't seem to be any problems transferring the exact same definition and theory/theorems from sets of sentences to sets of formulas? Are there any traps for sets of formulas that I am not seeing?

This is within first order logic.

Answer 1

The potential trap is that not everybody agrees about how to treat free variables: if I am thinking about satisfiability I find it natural to treat them as existentially quantified and if I am thinking about consistency (or whether a formula has a model) I would find it natural to treat them as universally quantified. However, there is no general agreement about this. Sticking with sentences saves having to worry about this issue.

Answer 2

Well, the truth value of a sentence does not depend on the assignment of variables.

For example, $x\leq y$ and $y\leq x$ are both true in $\Bbb R$ with the usual order if the assignment gives both $x$ and $y$ the same value, but a not in general.