Definition:Axiom/Formal Systems

Definition

Let $\mathcal L$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its axioms.

An axiom of $\mathscr P$ is a well-formed formula of $\mathcal L$ that $\mathscr P$ approves of by definition.

Axiom Schema

An axiom schema is a well-formed formula $\phi$ of $\LL$, except for it containing one or more variables which are outside $\LL$ itself.

This formula can then be used to represent an infinite number of individual axioms in one statement.

Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs.

Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.

Also see

• Definition:Proof System
• Definition:Rule of Inference

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.2$: The Construction of an Axiom System
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 3.1$: Definition $3.1$