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

In a formal system, primitive symbols and formation rules serve as the foundational components for constructing a formal language.

Primitive Symbols

These are the basic, undefined building blocks of the system, such as variables, constants, and logical operators (e.g., $~$, $\vee$, $\forall$). All other symbols within the system are defined in terms of these primitives.12

Formation Rules

Formation rules are specific instructions that dictate how these primitive symbols can be combined to create meaningful expressions. They function as follows:

• Defining Terms: Rules specify which combinations of symbols constitute valid terms, such as constants, variables, or functions.1
• Constructing Sentences: Rules establish how terms are combined to form atomic sentences (e.g., $a=b$) and how these can be further expanded into complex sentences using logical connectives.1
• Well-Formed Formulas: By adhering to these rules, the system ensures that only "well-formed" formulas are produced. These formulas form the basis for axioms and subsequent theorems derived through inference.2

Essentially, these rules allow the system to organize symbols into a rigorous structure for analyzing deduction without relying on external interpretations.2