A sentenceMathworldPlanetmath is a formulaMathworldPlanetmathPlanetmath with no free variablesMathworldPlanetmathPlanetmath.

Simple examples include:

  • xy[x<y]
  • z[z+7-43=0]
  • 1+2<2+3

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:


Remark. In first-order logic, the main differencePlanetmathPlanetmath 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 interpretationsMathworldPlanetmathPlanetmath 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 quantifiersMathworldPlanetmath in front of it. Given a formula φ, the universal closure of φ is the sentence


where {x1,,xn} is the set of all free variables occurring in φ.

The existential closure of a formula φ may be defined similarly.

For example, the universal closure of x+2=3 is


and its existential closure is


Note that the first sentence is false, while the second is true.

Title sentence
Canonical name Sentence
Date of creation 2013-03-22 13:00:24
Last modified on 2013-03-22 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