TOPICS
Search

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

 F=>F v G
(1)

in propositional calculus represents the axioms

A=>A v B,A=>A v A,¬A=>¬A v B
(2)
(A=>B)=>(A=>B) v (D ^ E),
(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.


See also

Axiom, First-Order Logic, Propositional Calculus

This entry contributed by Alex Sakharov (author's link)

Explore with Wolfram|Alpha

References

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

Referenced on Wolfram|Alpha

Axiom Schema

Cite this as:

Sakharov, Alex. "Axiom Schema." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AxiomSchema.html

Subject classifications