How does a formal language use primitive symbols and formation rules?

In a formal language, formation rules provide a mechanical specification for identifying which expressions count as well-formed formulas. This process is purely syntactic, focusing on the arrangement of symbols rather than their meaning.1

The specification typically involves three components:

• Primitive Symbols: A defined list of basic units, such as variables, constants, logical connectives (e.g., $\vee$, $~$), quantifiers (e.g., $\forall$, $\exists$), and grouping symbols like parentheses.12
• Atomic Sentences: Specific, simple combinations of these symbols that are identified mechanically as the foundational units of the language.12
• Inductive Clauses: Rules that allow for the construction of more complex sentences from simpler ones. For example, if $A$ and $B$ are sentences, these rules stipulate that combinations like $~A$ or $A\vee B$ are also valid, well-formed sentences.12

By applying these rules, a system can generate an infinite number of formulas through finite combinations of the primitive symbols, allowing for the analysis of deductive structures independent of intuitive meaning.3