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

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researchers in the fields of formal methods and computer-checked mathematics

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