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\begin{document}

\title{Axioms of Propositional Calculus}
\author{Michael Meyling}
\email{principa@meyling.com}
\date{2002-07-27T20:50:51}

\begin{abstract}
This module includes the axioms of propositional calculus. These axioms (together with some rules) allow the deduction of all theorems of propositional calculus. To learn about possible conclusions click through the following `.. used by..' list. \par This file is part of the project `Principia Mathematica II' see \href{http://www.meyling.com/principia/principia.html}{project homepage}.
\end{abstract}

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\section*{module specification}

Author of this module: 
\begin{longtable}[h!]{l@{\extracolsep{\fill}}l}
Michael Meyling & michael@meyling.com \\
\end{longtable}


\medskip
This module has the following specification:

\mbox{}
\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & propaxiom \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{http://www.meyling.com/principia/0_00_51/propaxiom_1.00.00_1.00.00.qedeq} \\
\end{longtable}


\medskip
Is used by the following modules:

\mbox{}
\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & proptheo1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{proptheo1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{proptheo1_1.00.00_1.00.00.pdf}  \\
Name: & proptheo2 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{proptheo2_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{proptheo2_1.00.00_1.00.00.pdf}  \\
Name: & predtheo1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{predtheo1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{predtheo1_1.00.00_1.00.00.pdf}  \\
Name: & prophilbert1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{prophilbert1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{prophilbert1_1.00.00_1.00.00.pdf}  \\
Name: & subst \\
Version: & 1.00.00 \\
Rule version: & 1.01.00 \\
Orgin: & \url{subst_1.00.00_1.01.00.qedeq}  \\
pdf: & \url{subst_1.00.00_1.01.00.pdf}  \\
Name: & prophilbert1 \\
Version: & 1.00.00 \\
Rule version: & 1.02.00 \\
Orgin: & \url{prophilbert1_1.00.00_1.02.00.qedeq}  \\
pdf: & \url{prophilbert1_1.00.00_1.02.00.pdf}  \\
\end{longtable}



We only need the logical disjunction (`or') and the negation (`not'). The other operands will be defined by them. (It would be possible to use only one operator: `sheffer's or' or `sheffer's and'.) At first we note an abbreviation that defines the implication:




\begin{abr}
\hypertarget{impl}{}
\begin{displaymath}
(@F_{0}\ \Rightarrow \  @F_{1}) \equiv  (\neg @F_{0}\ \vee \  @F_{1})\end{displaymath}
\end{abr}


This means that we could transform a part formula that matches the pattern on the left side into the right side and we could transform a part formula that matches the pattern on the right side into the formula on the left side.



\bigskip




The logical conjunction (`and') could be defined with de Morgan:




\begin{abr}
\hypertarget{and}{}
\begin{displaymath}
(@F_{0}\ \wedge \  @F_{1}) \equiv  \neg (\neg @F_{0}\ \vee \  \neg @F_{1})\end{displaymath}
\end{abr}


This is still an abbreviation.



\bigskip




The logical equivalence is defined as usual:




\begin{abr}
\hypertarget{equi}{}
\begin{displaymath}
(@F_{1}\ \Leftrightarrow \  @F_{2}) \equiv  ((@F_{1}\ \Rightarrow \  @F_{2})\ \wedge \  (@F_{2}\ \Rightarrow \  @F_{1}))\end{displaymath}
\end{abr}



Now we come to the first axiom of propositional calculus. This axiom enables us to get rid of an unnecessary disjunction:




\begin{axm}
\hypertarget{axiom1}{}
\begin{displaymath}
((P\ \vee \  P)\ \Rightarrow \  P)\end{displaymath}
\end{axm}


This expresses the `idempotence' of the disjunction.



\bigskip




Axiom of weakening:




\begin{axm}
\hypertarget{axiom2}{}
\begin{displaymath}
(P\ \Rightarrow \  (P\ \vee \  Q))\end{displaymath}
\end{axm}


If a proposition is true, any alternative may be added without making it false.



\bigskip




The commutativity of the disjunction is expressed with the third axiom:




\begin{axm}
\hypertarget{axiom3}{}
\begin{displaymath}
((P\ \vee \  Q)\ \Rightarrow \  (Q\ \vee \  P))\end{displaymath}
\end{axm}



The last axiom is: 




\begin{axm}
\hypertarget{axiom4}{}
\begin{displaymath}
((P\ \Rightarrow \  Q)\ \Rightarrow \  ((A\ \vee \  P)\ \Rightarrow \  (A\ \vee \  Q)))\end{displaymath}
\end{axm}



\begin{dec}
modus ponens
\hypertarget{rule1}{}
\label{rule1}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#modusponens}{Modus Ponens}



\bigskip




\begin{dec}
add axiom
\hypertarget{rule2}{}
\label{rule2}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#addaxiom}{Add Axiom}



\bigskip




\begin{dec}
add sentence
\hypertarget{rule3}{}
\label{rule3}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#addsentence}{Add Sentence}



\bigskip




\begin{dec}
replace proposition variable
\hypertarget{rule4}{}
\label{rule4}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#replaceprop}{Replace Proposition Variable}



\bigskip




\begin{dec}
use abbreviation
\hypertarget{rule5}{}
\label{rule5}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#useabbreviation}{Use Abbreviation}



\bigskip




\begin{dec}
reverse abbreviation
\hypertarget{rule6}{}
\label{rule6}
\end{dec}



For an explanation see \href{http://www.meyling.com/principia/0_00_51/rules_1.00.00.html#reverseabbreviation}{Reverse Abbreviation}



\bigskip


\section*{}
This module used by the following modules:

\mbox{}
\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & proptheo1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{proptheo1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{proptheo1_1.00.00_1.00.00.pdf}  \\
Name: & proptheo2 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{proptheo2_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{proptheo2_1.00.00_1.00.00.pdf}  \\
Name: & predtheo1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{predtheo1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{predtheo1_1.00.00_1.00.00.pdf}  \\
Name: & prophilbert1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{prophilbert1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{prophilbert1_1.00.00_1.00.00.pdf}  \\
Name: & subst \\
Version: & 1.00.00 \\
Rule version: & 1.01.00 \\
Orgin: & \url{subst_1.00.00_1.01.00.qedeq}  \\
pdf: & \url{subst_1.00.00_1.01.00.pdf}  \\
Name: & prophilbert1 \\
Version: & 1.00.00 \\
Rule version: & 1.02.00 \\
Orgin: & \url{prophilbert1_1.00.00_1.02.00.qedeq}  \\
pdf: & \url{prophilbert1_1.00.00_1.02.00.pdf}  \\
\end{longtable}

\end{document}