Implicational Propositional Calculus - Axiom System

Axiom System

  • Axiom schema 1 is P → (QP).
  • Axiom schema 2 is (P → (QR)) → ((PQ) → (PR)).
  • Axiom schema 3 (Peirce's law) is ((PQ) → P) → P.
  • The one non-nullary rule of inference (modus ponens) is: from P and PQ infer Q.

Where in each case, P, Q, and R may be replaced by any formulas which contain only "→" as a connective. If Γ is a set of formulas and A a formula, then means that A is derivable using the axioms and rules above and formulas from Γ as additional hypotheses.

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