A sentence is a logic formula in which every variable is quantified. The concept of a sentence is important because formulas with variables that are not quantified are ambiguous.
The concept of the sentence can be illustrated as follows (Enderton 1977). The formula 
, in which
each variable is quantified, can be translated into English as the complete sentence
"There exists a set which has every set as an element." However, the formula
, in which 
is not quantified, can only be translated as the sentence
fragment "Every set is an element of ___," where "___" is unspecified
because 
is not quantified.
Because a "quantified variable" (or "quantifier") is just a more descriptive name for a bound variable, a sentence can also be defined as a logic formula with no free variables (Enderton 1977). A sentence can also be defined as a closed sentential formula (Carnap 1958, pp. 24 and 85), although in some language systems, open sentential formulas are also admitted as sentences (Carnap 1958, p. 25).
