Definition:Rule of Formation

Definition

Let $\FF$ be a formal language whose alphabet is $\AA$.

The rules of formation of $\FF$ are the rules which define how to construct collations in $\AA$ which are well-formed.

That is, the rules of formation tell you how to build collations featuring symbols from the alphabet $\AA$ which are part of the formal language $\FF$.

The rules of formation of a formal language together constitute its formal grammar.

There are no strict guidelines on what a rule of formation should look like, since they are employed to produce such strict guidelines.

Thus, these rules of formation are often phrased in natural language, and their exact form is to some extent arbitrary.

Examples

Not A

In the language of propositional logic, logical negation is implemented using the rule of formation:

$\mathbf W: \neg:$ If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF.

Also known as

Rules of formation are also referred to in some sources as rules of syntax.

Some sources prefer the construct formation rules.

Also see

• Definition:BNF Specification of Propositional Logic, in which one can see rules of formation employed.
• Definition:Bottom-Up Specification of Propositional Logic, a different approach to rules of formation.

• Definition:Formal Grammar

• Results about rules of formation can be found here.

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.2$: The Construction of an Axiom System
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $1$ Formation Rules
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): formal language
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): formation rules
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): formal language
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): formation rules