How do rules of inference produce theorems from axioms?

In a formal system, rules of inference function as the mechanism for deriving new theorems from a set of axioms. This process is central to the construction of a formal system, where the derivation depends entirely on the syntactic structure of the formulas rather than their meaning.123

The production of theorems occurs through the following steps:

• Axioms as Starting Points: Certain well-formed formulas are designated as axioms, which serve as the foundational theorems of the system.12
• Application of Rules: A rule of inference acts as an inductive clause. It specifies that if certain formulas are already established as theorems, then another formula related to them in a defined way is also a theorem.1
• Finite Sequences: A theorem is formally defined as any formula that can be reached through a finite sequence of steps. In this sequence, every formula must either be an axiom or be derived from previous formulas in the sequence by applying an accepted rule of inference.42

By following these syntactic procedures, a system can generate complex logical consequences from its initial premises.13