# The Axiomatization of Propositional Logic

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## Summary

This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φφ. 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 schemes

• α ⇒ (βα),

• (α ⇒ (βγ)) ⇒ ((αβ) ⇒ (αγ)),

• β ⇒ ¬α) ⇒ ((¬βα) ⇒ β).

Modus 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.

## References

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# Formalized Mathematics

## (a computer assisted approach)

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