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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 24, Issue 4 (Dec 2016)

Issues

The Axiomatization of Propositional Logic

Mariusz Giero
  • Faculty of Economics and Informatics, University of Białystok, Kalvariju 135, LT-08221 Vilnius, Lithuania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-02-23 | DOI: https://doi.org/10.1515/forma-2016-0024

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.

Keywords: completeness; formal system; Lindenbaum’s lemma

MSC 2010: 03B05; 03B35

References

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  • [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

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  • [9] Mariusz Giero. Propositional linear temporal logic with initial validity semantics. Formalized Mathematics, 23(4):379–386, 2015. doi:10.1515/forma-2015-0030.CrossrefGoogle Scholar

  • [10] Witold Pogorzelski. Dictionary of Formal Logic. Wydawnictwo UwB - Bialystok, 1992. Google Scholar

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  • [14] Anita Wasilewska. An Introduction to Classical and Non-Classical Logics. SUNY Stony Brook, 2005. Google Scholar

  • [15] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Google Scholar

About the article

Received: 2016-10-18

Published Online: 2017-02-23

Published in Print: 2016-12-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0024.

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© 2016 Mariusz Giero, published by De Gruyter Open. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License. BY-SA 3.0

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