Sentence (mathematical logic)

For the interpretation of formulas, consider these structures: the positive real numbers, the real numbers, and complex numbers. The following example in first-order logic

${\displaystyle \forall y\ \exists x\ (y=x^{2})}$

is a sentence. This sentence means that for every y, there is an x such that ${\textstyle y=x^{2}.}$ This sentence is true for positive real numbers, false for real numbers, and true for complex numbers.

However, the formula

${\displaystyle \exists x\ (y=x^{2})}$

is not a sentence because of the presence of the free variable y. For real numbers, this formula is true if we substitute (arbitrarily) ${\textstyle y=2,}$ but is false if ${\textstyle y=-2.}$

It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even for complex numbers, where the formula is always true, it is still not considered a sentence. Such a formula may be called a predicate instead.

• Ground expression
• Open formula
• Statement (logic)
• Proposition

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.
• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.