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The Axiomatization of Propositional Logic

Mariusz Giero
Mariusz Giero
   | Feb 23, 2017
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Published Online: Feb 23, 2017
Page range: 281 - 290
Received: Oct 18, 2016
DOI: https://doi.org/10.1515/forma-2016-0024
Keywords
© 2016 Mariusz Giero, published by De Gruyter Open
This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

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.

eISSN:
1898-9934
ISSN:
1426-2630
Language:
English
Publication timeframe:
4 times per year
Journal Subjects:
Computer Sciences, other, Mathematics, General Mathematics
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