Let be a language of the first-order logic. Assume that the language has the following sets of nonlogical symbols:
1. is the set of constant symbols of . (These are nullary function symbols.)
2. is the set of predicate symbols of , and for each , is the arity of . The symbols in are also called relation symbols of the language .
3. is the set of function symbols of , and for each , is the arity of . The symbols in are also called operation symbols of the language .
4. is the universal quantifier symbol of .
A structure for is a tuple , , where is a set (called the underlying set of ), and the following hold:
1. For each , ,
2. For each , ,
3. For each , .