The Axiomatization of Propositional Logic

This article introduces propositional logic as a formal system ([], [], []). The formulae of the language are as follows::= ⊥ ||→. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemesModus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented. 14 10 11 φ p φ φ

⇒ (⇒), α β α

(⇒ (⇒)) ⇒ ((⇒) ⇒ (⇒)), α β γ α β α γ

(¬⇒ ¬) ⇒ ((¬⇒) ⇒). β α β α β