Sentence (mathematical logic)
The following example in first-order logic
formula_1
 a sentence. This sentence is true in the positive real numbers ℝ+, false in the real numbers ℝ, and true in the complex numbers ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the square of a member of that particular structure.) On the other hand, the formula
formula_2
is a sentence, because of the presence of the free variable "y". In the structure of the real numbers, this formula is true if we substitute (arbitrarily) "y" = 2, but is false if "y" = –2.
It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even in the structure of the complex numbers, where the statement is always true, it is still not considered a sentence. Such a formula may be called a predicate instead.
Example
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In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that "must" be true or false. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value.