sentence
A sentence^{} is a formula^{} with no free variables^{}.
Simple examples include:

•
$$ 
•
$$\exists z[z+743=0]$$ 
•
$$
Note that the last sentence contains no variables.
A sentence is also called a closed formula. A formula that is not a sentence is called an open formula.
The following formula is open:
$$x+2=3$$ 
Remark. In firstorder logic, the main difference^{} between a sentence and an open formula, semantically, is that a sentence has a definite truth value, whereas the truth value of an open formula may vary, depending on the interpretations^{} of the free variables occurring in the formula. In the open formula above, if $x$ were $1$, then the formula is true. Otherwise, it is false.
Every open formula may be converted into a sentence by placing quantifiers^{} in front of it. Given a formula $\phi $, the universal closure of $\phi $ is the sentence
$$\forall {x}_{1}\forall {x}_{2}\mathrm{\cdots}\forall {x}_{n}\phi $$ 
where $\{{x}_{1},\mathrm{\dots},{x}_{n}\}$ is the set of all free variables occurring in $\phi $.
The existential closure of a formula $\phi $ may be defined similarly.
For example, the universal closure of $x+2=3$ is
$$\forall x[x+2=3],$$ 
and its existential closure is
$$\exists x[x+2=3].$$ 
Note that the first sentence is false, while the second is true.
Title  sentence 

Canonical name  Sentence 
Date of creation  20130322 13:00:24 
Last modified on  20130322 13:00:24 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  7 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 03B99 
Synonym  closed formula 
Defines  open formula 
Defines  universal closure 
Defines  existential closure 