The Axiomatization of Propositional Logic

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.

Język:: Angielski

Częstotliwość wydawania:: 1 razy w roku
Dziedziny czasopisma:: Informatyka, Informatyka, inne, Matematyka, Matematyka ogólna

Kanał RSS czasopisma

The Axiomatization of Propositional Logic

Mariusz Giero

Data publikacji: 23 lut 2017

Zakres stron: 281 - 290

Otrzymano: 18 paź 2016

DOI: https://doi.org/10.1515/forma-2016-0024

Słowa kluczowecompleteness, formal system, Lindenbaum’s lemma

© 2016 Mariusz Giero, published by De Gruyter Open

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

Słowa kluczowe
completeness, formal system, Lindenbaum’s lemma