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%*      Version:  1.5    Date: 22/05/2026 File: gn-logic.tex                 */
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%* 30/12/90 First version of documentation.                                  */
%* 21/04/92 new properties                                                   */
%* 22/05/26 updated                                                          */
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%* Title:                                                                    */
%* Author: Gerd Neugebauer                                                   */
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%* Usage:  latex gn-logic.tex                                                */
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\documentclass[11pt,a4paper]{article}
\usepackage{gn-logic}

\setlength{\unitlength}{1pt}


\begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newlength{\Width} \Width=\textwidth \advance\Width by -1.5em \divide\Width by 2

\section{The {\tt gn-logic} style option}
Description of Version 1.5 (5/2026) by Gerd Neugebauer \bigskip

The {\tt gn-logic} style option provides a facility to typeset logical
formulas of a certain kind. This style option provides an environment like
\verb|eqnarray|, an extended {\tt newtheorem} environment and several macros.



\subsection{Mathematical Symbols}

The following macros provide better usage of the junctors and quantifiers.
Especially the spacing  is improved.


\noindent\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}*{4}{l}}
\multicolumn{1}{c}{\small Symbol}
	   & \multicolumn{1}{c}{\small Macro}
			     & 	\multicolumn{2}{c}{\small Example}	  \\
	   &		     &			      & 		  \\
$\AND$	   & \verb|\AND|     & \verb$A\AND B$	      & $A\AND B$	  \\
$\OR$	   & \verb|\OR|	     & \verb$A\OR B$	      & $A\OR B$	  \\
$\XOR$	   & \verb|\XOR|     & \verb$A\XOR B$	      & $A\XOR B$	  \\
$\IMPLIES$ & \verb|\IMPLIES| & \verb$A\IMPLIES B$     & $A\IMPLIES B$	  \\
$\IMPL$	   & \verb|\IMPL|    & \verb$A\IMPL B$	      & $A\IMPL B$	  \\
$\IF$	   & \verb|\IF|	     & \verb$A\IF B$	      & $A\IF B$	  \\
$\IFF$	   & \verb|\IFF|     & \verb$A\IFF B$	      & $A\IFF B$	  \\
$\IFFdef$  & \verb|\IFFdef|  & \verb$A\IFFdef B$      & $A\IFFdef B$	  \\
$\ANDdots$ & \verb|\ANDdots| & \verb$A_1\ANDdots A_n$ & $A_1\ANDdots A_n$ \\
$\ORdots$  & \verb|\ORdots|  & \verb$A_1\ORdots A_n$  & $A_1\ORdots A_n$  \\
$\is$	   & \verb|\is|	     & \verb$x\is y$	      & $x\is y$	  \\
$\Nat$	   & \verb|\Nat|     & \verb$n\in\Nat$	      & $n\in\Nat$	  \\
$\Forall$  & \verb|\Forall|  & \verb$\Forall x P(x)$  & $\Forall x P(x)$  \\
$\Exists$  & \verb|\Exists|  & \verb$\Exists y P(x)$  & $\Exists y P(x)$  \\
\end{tabular*}


\newcommand{\bs}{{\tt\char"5C}}
\newcommand{\mac}[1]{The {\tt\char92 #1} Macro}
\newcommand{\macs}[2]{The {\tt\char92 #1} and the {\tt\char92 #2} Macros}
\newenvironment{compare}%
{\noindent\begin{center}%
  \begin{tabular}{@{}l@{\hspace*{1.5em}produces\hspace*{1.5em}}l@{}}}%
{\end{tabular}\end{center}}

\subsubsection*{\mac{AND}}
This macro can be used for the logical conjunction. In addition to the
\verb|\wedge| macro it adds more space and the formulas tend to be better
readable. Compare

\begin{compare}
\verb$x=1\AND y=x$	& $x=1\AND y=x$		\\
\verb$x=1\wedge y=x$	& $x=1\wedge y=x$	\\
\verb$x=1\land y=x$	& $x=1\land y=x$
\end{compare}

\subsubsection*{\mac{OR}}
This macro can be used for the logical disjunction. In addition to the
\verb|\vee| macro it adds more space. Compare

\begin{compare}
\verb$x=1\OR y=x$	& $x=1\OR y=x$	\\
\verb$x=1\vee y=x$	& $x=1\vee y=x$	\\
\verb$x=1\lor y=x$	& $x=1\lor y=x$
\end{compare}


\subsubsection*{\mac{XOR}}
This macro can be used for the exclusive disjunction. It has no common
counterpart. The spacing is like in all junctor macros.

\begin{compare}
\verb$x=1\XOR y=x$	& $x=1\XOR y=x$
\end{compare}


\subsubsection*{\macs{IMPL}{IMPLIES}}
These macros can be used for the logical implication. In addition to the
\verb|\rightarrow| macro it adds more space. Compare

\begin{compare}
\verb$x=1\IMPL y=x$		& $x=1\IMPL y=x$	\\
\verb$x=1\IMPLIES y=x$		& $x=1\IMPLIES y=x$	\\
\verb$x=1\rightarrow y=x$	& $x=1\rightarrow y=x$
\end{compare}


\subsubsection*{\mac{IF}}
 This macro can be used for the logical implication written in reverse order.
In addition to the \verb|\leftarrow| macro it adds more space. Compare

\begin{compare}
\verb$x=1\IF y=x$		& $x=1\IF y=x$	\\
\verb$x=1\leftarrow y=x$	& $x=1\leftarrow y=x$
\end{compare}

\subsubsection*{\mac{IFF}}
 This macro can be used for the logical equivalence.
In addition to the \verb|\leftrightarrow| macro it adds more space. Compare

\begin{compare}
\verb$x=1\IFF y=x$		& $x=1\IFF y=x$	\\
\verb$x=1\leftrightarrow y=x$	& $x=1\leftrightarrow y=x$
\end{compare}

\subsubsection*{\mac{IFFdef}}
 Like above but with a small ``def'' above the arrow.

\begin{compare}
\verb$x=1\IFFdef y=x$		& $x=1\IFFdef y=x$
\end{compare}

\subsubsection*{\mac{is}}
 This macro is for typesetting unifiers. In this case the predefined
\verb|\setminus| produces too much space.

\begin{compare}
\verb$\{y\setminus x, z\setminus 4\}$	& $\{y\setminus x, z\setminus 4\}$ \\
\verb$\{y\is x, z\is 4\}$		& $\{y\is x, z\is 4\}$		   \\
\verb$\{y\backslash x, z\backslash 4\}$	& $\{y\backslash x, z\backslash 4\}$
\end{compare}

\ifx\AmSTeX\undefined
\def\AmSTeX{$\cal A$\kern-.1667em\lower.5ex\hbox
 {$\cal M$}\kern-.125em$\cal S$-\kern-.1em\TeX}
\fi

\subsubsection*{The Number Macros}
These are macros for those who have no access to the \AmSTeX{}
fonts. It makes the symbols for the natural numbers, integers,
rationals, reals and complex numbers. The usual magnification commands
apply to it as well.


\def\BB#1{\csname bb#1\endcsname}
\def\Line#1{\LINE{#1}\(\BB{#1}_{\BB{#1}}\)}
\def\LINE#1{{\tt \char92bb#1}&%
  {\tiny\BB{#1}}&%
  {\scriptsize\BB{#1}}&%
  {\footnotesize\BB{#1}}&%
  {\small\BB{#1}}&%
  {\normalsize\BB{#1}}&%
  {\large\BB{#1}}&%
  {\Large\BB{#1}}&%
  {\LARGE\BB{#1}}&%
  {\huge\BB{#1}}&%
  {\Huge\BB{#1}}&%
  }
\begin{center}
  \begin{tabular}{c|cccccccccc|c}
    &\multicolumn{10}{|c|}{{\tt \char92tiny \hfill...\hfill\char92normalsize \hfill...\hfill\char92Huge}}&\verb|X_X|
    \\\hline
    \Line B\\
    \verb|\Complex|\Line C\\
    \Line D\\
    \Line E\\
    \Line F\\
    \Line G\\
    \Line H\\
    \Line I\\
    \Line J\\
    \Line K\\
    \Line L\\
    \Line M\\
    \verb|\Nat|  \Line N\\
    \Line O\\
    \Line P\\
    \verb|\Rat|  \Line Q\\
    \verb|\Real| \Line R\\
    \verb|\Int|  \Line Z\\
    \Line{One}
  \end{tabular}
\end{center}

Unfortunately the macros \verb|\bbC|, \verb|\bbG|, \verb|\bbO|, and
\verb|\bbQ| do not scale properly when used in subscripts or superscripts of
formulae. The following example shows how the sizing can be achieved manually

\noindent\begin{compare}
\verb$\bbQ_{\mbox{\scriptsize \bbQ}}$  & $\bbQ_{\mbox{\scriptsize \bbQ}}$
\end{compare}



\subsubsection*{\macs{Forall}{Exists}}

The general problem with quantifiers is that after the quantified variable the
following formula is not automatically separated with a small space. This can
be overcome by the following macros.

The \verb|\Forall| and the \verb|\Exists| macros take one argument. They
typeset the respective quantifier followed by the argument (i.e.\ the variable)
and finally a small space. As usual the argument has to be enclosed in braces
if it consists of more than one character. Otherwise the braces can be omitted.
This allows an elegant notation of short quantified formulas.

\noindent\begin{compare}
\verb$\Forall x P(x)$  & $\Forall x P(x)$\\
\verb$\Forall{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$%
&$\Forall{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$\\
\verb$\Exists x P(x)$  & $\Exists x P(x)$\\
\verb$\Exists{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$%
& $\Exists{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$
\end{compare}



\subsection{The {\tt Formula} Environment}

This environment allows to typeset logical formulas. The main problem with the
\verb|eqnarray| environment was the numbering. In multiline formulas my
intention was to have the number in the middle of the formula. Inside this
environment several macros are valid.

\begin{description}
 \item[{\tt\bs begin\{Formula\}[{\em label}] \bs end\{Formula\}}] \ \\
	Start the list of formulas. Optionally a label can be given. This label
	is used to reference the first formula.
 \item[{\tt\bs =}] \ \\
	Start a new line.
 \item[{\tt\bs >}{\em level}] \ \\
	Start a new line and indent to the given {\em level}. This indentation
	is done in quantities of \verb|\FormulaIndent| which can be set with
	the \verb|\setlength| command. The default value is {\tt 3em}.
 \item[{\tt\bs Form[{\em label}]}] \ \\
	Start a new formula. Optionally a {\em label} can be given. This {\em
	label} can be used to reference to the formula (see \verb|\ref|).
\end{description}

Now let's have a look at some examples. First, we see a single two-line formula.
Note that the number at the right side is centered between the two lines.
\medskip

\noindent
\begin{minipage}{\Width}
\small\begin{verbatim}
\begin{Formula}
    P(X) \IMPL
\=  Q(X) \IFF R_1(X) \OR R_2(X)
\end{Formula}
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{\Width}
\begin{Formula}
	P(X) \IMPL
\=	Q(X) \IFF R_1(X) \OR R_2(X)
\end{Formula}
\end{minipage}\medskip

Next we will see an example of several formulas. The first formula is split to
three lines and the third line is indented to level 1. Remark: \verb|\=| is in
reality an abbreviation for \verb|\>0|.
\medskip

\noindent
\begin{minipage}{\Width}
\small\begin{verbatim}
\begin{Formula}[form:1]
    P(X) \IMPL
\=  Q(X) \IFF R_1(X)
\>1 \OR R_2(X)
\Form[form:2]
    S(X) \IMPL
\=  \neg Q(X) \IFF R_1(X) \OR R_2(X)
\end{Formula}
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{\Width}
\begin{Formula}[form:1]
    P(X) \IMPL
\=  Q(X) \IFF R_1(X)
\>1 \OR R_2(X)
\Form[form:2]
    S(X) \IMPL
\=  \neg Q(X) \IFF R_1(X) \OR R_2(X)
\end{Formula}
\end{minipage}\medskip


\subsection{The {\tt NewTheorem} Environment}

My experience with the {\tt newtheorem} environment was that I had a
certain scheme to use it. First, every theorem got a label. Thus,
every {\em theorem} was followed by a {\tt label} command. Optionally
a {\em theorem} may have a name. This name is typeset right after the
number. The body of the {\em theorem} always started in the next
line. This led to the definition of an extended {\tt NewTheorem}
environment. The arguments are the same as those of the {\tt
  newtheorem} environment. But the environment defined by this
extended command takes two optional arguments. The first optional
argument is a label to be assigned to the {\em theorem}. This argument
has to be enclosed in parentheses. The second type of optional
argument has to be enclosed in brackets. It is typeset in
\verb|\small| after the title text. The third argument is optional and
enclosed in \verb|<>|. It is typeset in \verb|\small\bf| and
surrounded by parentheses.  \medskip

\noindent\begin{minipage}{\Width}
\small\begin{verbatim}

\NewTheorem{guess}{Conjecture}

\begin{guess}[Fermat](thm:fermat)
 There do not exist integers $n>2$,
 $x$, $y$, and $z$ such that
 $x^n+y^n=z^n$.
\end{guess}
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{\Width}
\NewTheorem{guess}{Conjecture}

\begin{guess}[Fermat](thm:fermat)
 There do not exist integers $n>2$,
 $x$, $y$, and $z$ such that
 $x^n+y^n=z^n$.
\end{guess}
\end{minipage}
\medskip

The commands used to typeset some of the optional argument can be customized
in the following way. The macros \verb|\TheoremTitle| and \verb|\TheoremName|
are used to typeset their argument in \verb|\small| and \verb|\small\bf| and
enclosed in parentheses respectively. These macros can be redefined using
\verb|\renewcommand| as shown in the following example:
\medskip

\noindent\begin{minipage}{\Width}
\footnotesize\begin{verbatim}
\NewTheorem{theorem}{Theorem}
\renewcommand{\TheoremTitle}[1]{{\sf [#1]}}
\renewcommand{\TheoremName}[1]{{\small(#1)}}
\begin{theorem}[Fermat]<conjecture>(thm:f2)
 There do not exist integers ...
\end{theorem}
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{\Width}
\NewTheorem{theorem}{Theorem}
\renewcommand{\TheoremTitle}[1]{{\sf #1}}
\renewcommand{\TheoremName}[1]{{\small(#1)}}
\begin{theorem}[Fermat]<conjecture>(thm:f2)
 There do not exist integers $n>2$,
 $x$, $y$, and $z$ such that
 $x^n+y^n=z^n$.
\end{theorem}
\end{minipage}

\end{document}

