How does a logical system use axioms and rules of inference to produce theorems?

In a formal system, axioms and rules of inference work together to generate theorems through a mechanical, syntactic process that does not rely on the meaning of the symbols involved.123

• Axioms: These are specific, well-formed formulas within the system that are designated as starting points or "basic theorems".12
• Rules of Inference: These are inductive clauses that dictate how new formulas can be derived from existing ones. A rule of inference specifies that if certain formulas (premises) are already established as theorems, then a new, related formula (the conclusion) is also a theorem.2
• Theorems: A theorem is defined as any formula that can be proven through a finite sequence of well-formed formulas. Each step in this sequence must either be an axiom or a formula derived from previous steps by applying the system's rules of inference.12

By following these procedures, a logical system can systematically build complex structures of knowledge from a foundational set of assumptions, allowing for the analysis of deductive structures independent of specific interpretations.13