How do formation rules determine whether an expression is a well-formed formula?

In metalogic, formation rules serve as a mechanical specification that determines whether an expression qualifies as a well-formed formula (wff). Because these rules are syntactic, they focus exclusively on the arrangement of symbols rather than their meanings, allowing a machine to verify if a candidate expression satisfies the system's requirements.1

This specification typically consists of three components:

• Primitive Symbols: A defined list of the basic units allowed within the language.1
• Atomic Formulas: Specific combinations of these symbols that are designated as the simplest meaningful expressions.12
• Inductive Clauses: Rules that stipulate how complex expressions can be constructed from simpler ones. For example, these clauses define how logical connectives (such as disjunction or negation) or quantifiers can be applied to existing wffs to generate new, valid wffs.13

By following these recursive or inductive steps, a formal system ensures that only expressions built according to the established syntactic structure are recognized as well-formed.12