Editing Axiom (section)

=====Propositional logic=====
In [[propositional logic]], it is common to take as logical axioms all formulae of the following forms, where <math>\phi</math>, <math>\chi</math>, and <math>\psi</math> can be any formulae of the language and where the included [[Logical connective|primitive connectives]] are only "<math>\neg</math>" for [[negation]] of the immediately following proposition and "<math>\to</math>" for [[Entailment|implication]] from antecedent to consequent propositions:

# <math>\phi \to (\psi \to \phi)</math>
# <math>(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))</math>
# <math>(\lnot \phi \to \lnot \psi) \to (\psi \to \phi).</math>

Each of these patterns is an ''[[axiom schema]]'', a rule for generating an infinite number of axioms.  For example, if <math>A</math>, <math>B</math>, and <math>C</math> are [[propositional variable]]s, then <math>A \to (B \to A)</math> and <math>(A \to \lnot B) \to (C \to (A \to \lnot B))</math> are both instances of axiom schema 1, and hence are axioms.  It can be shown that with only these three axiom schemata and ''[[modus ponens]]'', one can prove all tautologies of the propositional calculus.  It can also be shown that no pair of these schemata is sufficient for proving all tautologies with ''modus ponens''.

Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.<ref>Mendelson, "6. Other Axiomatizations" of Ch. 1</ref>

These axiom schemata are also used in the [[predicate calculus]], but additional logical axioms are needed to include a quantifier in the calculus.<ref>Mendelson, "3. First-Order Theories" of Ch. 2</ref>