Axiom Schema

Propositional calculus, first-order logic, and other theories in mathematical logic are defined by their axioms (or axiom schemata, plural: axiom schemata) and inference rules. An axiom schema is a sentential formula representing infinitely many axioms. These axioms are obtained by replacing variables in the schema by any formula. For example, the axiom schema

(1)

in propositional calculus represents the axioms

	(2)
	(3)

and so on.

It is typical to define a theory by axiom schemata rather than axioms. If axioms but not their schemata are utilized, then substitution for variables should be incorporated into inference rules.

REFERENCES

Kleene, S. C. Mathematical Logic. New York: Dover, 2002.