Definition:Word (Formal Systems)

This page is about a finite string of symbols from a given alphabet. For other uses, see Definition:Word.

Definition

Let $\mathcal F$ be a formal language.

Let $\mathcal A$ be the alphabet of $\mathcal F$.

Then a word in $\mathcal A$ is defined as a finite string of symbols of $\mathcal A$.

Other Terms for the Same Concept

Different treatments of formal languages use different terms for this concept, for example:

• Formula;
• Sentence.

Some sources use the term string in this limited context.

Note

Not all words formed of symbols of $\mathcal A$ necessarily belong to $\mathcal F$.

Whether it does or not depends on the grammar of $\mathcal F$.

A word which does belong to $\mathcal F$ is sometimes called a well-formed word.

Sources

• E.J. Lemmon: Beginning Logic (1965): $\S 2.1$ (in the context of the Propositional Calculus)
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.2$