This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. 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 ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ 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.
@article{bwmeta1.element.doi-10_1515_forma-2016-0024,
author = {Mariusz Giero},
title = {The Axiomatization of Propositional Logic},
journal = {Formalized Mathematics},
volume = {24},
year = {2016},
pages = {281-290},
zbl = {1357.03036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0024}
}
Mariusz Giero. The Axiomatization of Propositional Logic. Formalized Mathematics, Tome 24 (2016) pp. 281-290. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0024/