Full Idea
For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ
Gist of Idea
A logic with ¬ and → needs three axiom-schemas and one rule as foundation
Source
David Bostock (Intermediate Logic [1997], 5.2)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.194
A Reaction
A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic.
Related Idea
Idea 13619 Quantification adds two axiom-schemas and a new rule [Bostock]