Sentence (mathematical logic)

To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.

For first-order theories, interpretations are commonly called structures.

Given a structure or interpretation, a sentence will have a fixed truth value.

A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true.

For real numbers, this formula is true if we substitute (arbitrarily)

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.