Single Idea 13610

[catalogued under 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL]

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]