a set £ of symbols is first-order language
(in short, a language) if and only if followings.

(i)  the set £ contains all logical symbol.
(ii) there exist one or more non-logical symbols
     defined as predicate symbol of the set £.
(iii) there exist countably many non-logical symbols
       defined as variable symbol of the set £.
(iv) the £ contains all non-logical symbols defined
       as predicate or variable symbol of the £.
(v)  if there exist non-logical symbols defined as
       function symbols of the £, the set £ contains all
       of them.
(vi) if there exist non-logical symbols defined as
       constant symbols of the £, the set £ contains all
       of them.
(vii) there is no non-logical symbol defined as both
        one of above and another.
(viii) there is the only function,denoted by  
         Arg(£),  from the set  of all function and
          predicate symbols of the  £  to
         the set of all natural numbers.