Interface:Standard axioms of propositional logic

The standard axioms of propositional calculus are those used by metamath, Hamilton's 1988 Logic for Mathematicians, and Margaris's 1967 First Order Mathematical Logic. They were stated in their current form by Łukasiewicz around 1930 (building on earlier work by Frege), and consist of three axioms and one inference rule.

These axioms are equivalent to other axiomizations of propositional calculus such as Principia Mathematica's.

The only kind that we need to define is for formulas (called by some authors well-formed formulas):  kind (formula) var (formula p q r) 

Of the standard five logical connectives of classical propositional logic, only implication ($$\rightarrow$$-symbol) and negation ($$\neg$$-symbol) are needed to state the axioms. The other three can be defined in terms of those two.

 term (formula (¬ formula)) # Negation term (formula (→ formula formula)) # Implication 

The first axiom is Simp, for the principle of simplification (following the name given to it by Principia).

 stmt (Simp  (p → (q → p))) 

The second axiom is known as Frege or antecedent distribution.

 stmt (Frege  ((p → (q → r)) → ((p → q) → (p → r)))) 

The third axiom is the principle of transposition or Transp.

 stmt (Transp  (((¬ p) → (¬ q)) → (q → p))) 

The inference rule is modus ponens.  stmt (applyModusPonens (p (p → q)) q) 