Sentence (mathematical logic)

A sentence can be viewed as expressing a proposition, something that must be true or false.

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

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.