I am sitting on the couch thinking about one of my trivial and (almost!) meaningless mathematical problems. Let me explain: terms are recursively defined in First Order Logic starting from symbols for individual variables and symbols for individual constants.
As I was thinking about considering constants as nullary functions I ran into the problem of accepting the empty string to be a term. Is this assumption correct?
Look at the recursive definition of "term." The base clauses are:
Variables are terms, and
Constant symbols (if present in the language) are terms.
The recursion clause meanwhile only lets you build longer terms: it says that whenever $f$ is an $n$-ary function symbol in the language and $t_1,...,t_n$ are terms, then $f(t_1,...,t_n)$ is a term.
(Incidentally, if we view constant symbols as nullary function symbols, then the second base clause is subsumed by the recursion clause.)
At no point does the empty string enter: we start with nonempty strings and only go "less empty" from there.