I recently came across a description of Frege's propositional calculus. It consists of six axioms and one rule of inference, described here. Hilbert's deductive system is similar. Frege's six axioms are:

A → (B → A)
(A → (B → C)) → ((A → B) → (A → C))
(A → (B → C)) → (B → (A → C))
(A → B) → (¬B → ¬A)
¬¬A → A
A → ¬¬A

Several articles, that wikipedia page included, say that Frege's axioms and the theorems derived from them are equivalent to the axioms of standard propositional logic. What are those axioms (I assume they're things like De Morgan's Law, material implication, etc.), and how do they relate to Frege's axioms? What do axioms like A → (B → A) or (A → (B → C)) → (B → (A → C)) mean intuitively?


closed as too broad by symplectomorphic, José Carlos Santos, Aweygan, Yanior Weg, Lee David Chung Lin Jun 12 at 1:40

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