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Full Idea
A formula of a first-order language is a 'sentence' just in case it has no free variables.
Gist of Idea
A first-order 'sentence' is a formula with no free variables
Source
José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2)
Book Ref
Zalabardo,José L.: 'Introduction to the Theory of Logic' [Westview 2000], p.89