@MISC{PL_AXIOM.MIZ,
AUTHOR = {Giero, Mariusz},
TITLE = {The Axiomatization of Propositional Logic},
SECTION1 = {Preliminaries},
SECTION2 = {The Syntax},
SECTION3 = {The Semantics},
SECTION4 = {The Axioms. Derivability.},
SECTION5 = {Soundness Theorem. Deduction Theorem.},
SECTION6 = {Strong Completeness Theorem},
DAY = {18},
MONTH = {October},
YEAR = {2016},
ADDRESS1 = {Faculty of Economics and Informatics in Vilnius\\University of Bia{\l}ystok\\
  Kalvariju 135, LT-08221 Vilnius\\Lithuania},
SUMMARY = {This article introduces propositional logic as a formal system
  (\cite{AniWas}).
  The formulae of the language are as follows
  $\phi::=\,\,\perp|\, p\,|\,\phi\rightarrow\phi$.
  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
  $\alpha\Rightarrow(\beta\Rightarrow\alpha),\,
  (\alpha\Rightarrow(\beta\Rightarrow\gamma))\Rightarrow((\alpha\Rightarrow\beta)\Rightarrow
  (\alpha\Rightarrow\gamma)),\,(\neg\beta\Rightarrow\neg\alpha)\Rightarrow
  ((\neg\beta\Rightarrow\alpha)\Rightarrow\beta)$.
  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.}}

@BOOK{AniWas,
AUTHOR={Wasilewska, Anita},
TITLE={An Introduction to Classical and Non-Classical Logics},
PUBLISHER={SUNY Stony Brook},
YEAR={2005}}
