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diff --git a/Master/texmf-dist/doc/plain/gfnotation/GFnotation-doc.tex b/Master/texmf-dist/doc/plain/gfnotation/GFnotation-doc.tex new file mode 100644 index 00000000000..d57cc991057 --- /dev/null +++ b/Master/texmf-dist/doc/plain/gfnotation/GFnotation-doc.tex @@ -0,0 +1,1186 @@ +% This file is part of the CTAN package named gfnotation. +% +% This is the users guide to the TeX macros ``GFnotation.tex''. +% GFnotation.tex: typeset G. Frege's notation of the Begriffsschrift or +% the Grundgesetze der Arithmetik in plain TeX +% +% Copyright (C) 2015 Udo Wermuth (author) +% +% This program is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% This program is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with this program. If not, see <http://www.gnu.org/licenses/>. +% %%% +% %%% GFnotation +% %%% +\input amssym.def +\catcode`\ß=11 +\let\nosubst=f +\let\showrotation=t \input rotate +\let\fgefontsknown=t +\input GFnotation +\gfbslognotationtrue +\gfbszeigestatstrue +% %%% +% %%% verbatim macros (from manmac.tex) +% %%% +\newskip\ttglue {\tt\global\ttglue=0.5em plus 0.25em minus 0.15em } +\def\ttverbatim{\begingroup \frenchspacing + \catcode`\\=12 \catcode`\{=12 \catcode`\}=12 \catcode`\$=12 \catcode`\&=12 + \catcode`\#=12 \catcode`\%=12 \catcode`\~=12 \catcode`\_=12 \catcode`\^=12 + \obeyspaces \obeylines \tt} +\def\verbatimspace{\ifvmode\indent\fi\space} +{\obeyspaces \gdef\makespaceverbspace{\def {\verbatimspace}}} +\outer\def\verbatim{$$\ifdim\parskip>0pt + \abovedisplayskip=\parskip \abovedisplayshortskip=\parskip + \belowdisplayskip=\parskip \belowdisplayshortskip=\parskip + \else + \abovedisplayskip=3pt \abovedisplayshortskip=3pt + \belowdisplayskip=3pt \belowdisplayshortskip=3pt + \fi + \let\par=\endgraf \ttverbatim \makespaceverbspace \parskip=0pt + \catcode`\§=0 \advance\leftskip by 10pt \ttfinish} +{\catcode`\§=0 §catcode`§\=12 % § is temporary escape character + §obeylines % end of line is active + §gdef§ttfinish#1^^M#2\endverbatim{§vbox{#2}§endgroup$$}} +\catcode`\€=\active +{\obeylines \gdef€{\ttverbatim \spaceskip\ttglue \let^^M=\ \let€=\endgroup}} +% %%% +% %%% Examples +% %%% +\def\Econt{\goodbreak\bgroup + \parskip=0pt \leftskip=3\parindent \rightskip=\leftskip \parindent=0pt } +\def\Epause{\par\egroup} +\def\Example#1{\bigskip\Econt\noindent{\bf Example #1.}\ } +\def\Eend#1{\ifnum#1>0 \vskip-\baselineskip\fi + \hfill\vrule height 7pt depth 1pt width 3pt \Epause\bigskip} +% %%% +% %%% hyphenation exceptions +% %%% +\hyphenation{Gottlob Frege Begriffs-schrift Grund-gesetze} +% %%% +% %%% other +% %%% +\def\\{\par\vskip0pt plus.3\vsize\penalty-253\vskip0pt plus-.3\vsize\relax} +\def\AmSTeX{$\cal A\kern-.1667em\lower.5ex\hbox{$\cal M$}\kern-.075em S$-\TeX} +\def\La{L\kern-3pt \raise.42ex\hbox{$\scriptstyle A$}} +\def\LaTeX{\La\kern-.15em\TeX} +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\parskip=0.25\baselineskip +\settabs 6 \columns +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\font\titlefont=cmssdc10 at 36pt +\font\subtitlefont=cmssdc10 at 17pt +% +\centerline{\titlefont Users Guide} +\bigskip +\centerline{\subtitlefont GFnotation.tex (version 2.9, 29.03.2015)} + +% %%%%%%%%%%% +\beginsection 1. Description + +This package implements macros to typeset formulas and inferences in +Gottlob Frege's notation in the style of his ``Begriffsschrift, eine der +arithmetischen nachgebildete Formelsprache des reinen Denkens'' (see +[1]) or in the style of his main work ``Grundgesetze der Arithmetik'' +(see [2]). Here is an example of a single +formula: \frege|{..\gA.{*.a.{f(\da)}}}. And this is a complete +inference: +\nlp4\*_[-\*-:-\ce{$\gA$} +\nlp4\*___\*_'-\ce{$\gB$} +\nlp4\*_[-\*---\ce{$\gB$} +\nlp4\*---\*---\*--- +\nlp4\*_[-\*---\ce{$\gA$} . + +The package has two input formats, the so-called ``symbolic +representation'' and the ``short form.'' The short form outputs the +formulas either in the style of Frege's {\it Begriffsschrift\/} or in +the style that is used in his {\it Grundgesetze}. The above formulas +are in the style of the {\it Begriffsschrift\/}; here is the first one +in the style of the {\it Grundgesetze\/}: +\toggleGGstyle\frege|{..\gA.{*.a.{f(\da)}}}.\toggleGGstyle + +{\it Note\/}: This users guide shows how to work with the package. +It does not explain the constructions and symbols in Frege's +notation. + + +% %%%%%%%%%%% +\beginsection 2. Elements of Frege's notation + +{\bf a) Letters.}\ Frege used lowercase roman letters, $a$, $b$, +$c$, \dots, uppercase Greek letters $\gA$, $\gB$, $\gG$, \dots, +lowercase Greek letters $\ka$, $\kb$, $\kg$, \dots, and Fraktur +letters $\da$, $\de$, $\dF$, \dots\thinspace. + +\noindent +{\bf b) Formulas (Begriffsschrift).}\ +The following notation is used by Frege to build formulas: +\smallskip +\+&{\it Content stroke\/}: \*---\ze{$\gA$}. + &&{\it Judgment\/}: \*[--\ze{$\gA$}.\cr +\smallskip +\+&{\it Negation\/}: \*-~-\ze{$\gA$}. + &&{\it Affirmation\/}: \*-+-\ze{$\gA$}.\cr + \smallskip +\+&{\it Condition\/}: \frege .{..\gB.\gA}. + &&{\it Generality\/}: \frege .{*.a.{\gF(\da)}}.\cr +\+&{\it Condition stroke\/}: \*_!_. + &&{\it Definition\/}: \*"3- $((\gA)\equiv\gB)$.\cr +\smallskip +\+&{\it Substitution\/}: $a$\*_|_$f(a)$.\cr + +\noindent +{\bf c) Inferences (Begriffsschrift).}\ +For inferences two types of lines are used: +\smallskip +\+&{\it Single line\/}: \*---\*---. + &&{\it Double lines\/}: \*===\*===.\cr +\smallskip +\+&{\it Separator between inference chains\/}:\cr +\decollator4 + +\noindent +{\bf d) Terminal symbols (Begriffsschrift).}\ +Four special constructions appear in the {\it Begriffsschrift\/}: +\smallskip +\+&$\gfvererbe\kd\ka{{x(\ka)}}{{f(\kd,\ka)}}$, \quad$\gffolgt\kg\kb{f(x_\kg,y_\kb)}$, +\quad$\gfgehoertan\kg\kb{f(x_\kg,z_\kb)}$, \quad$\gfeindeutig\kd\ke{f(\kd,\ke)}$.\cr + +\toggleGGstyle +\gfbskeinekompaktehoehlungtrue +\noindent +{\bf e) Formulas (Grundgesetze).}\ +The following notation is used by Frege to build formulas: +\smallskip +\+&{\it Content stroke\/}: \frege.{=.\gA}. + &&{\it Judgment\/}: \frege|{=.\gA}.\cr +\smallskip +\+&{\it Negation\/}: \frege.{=-\gA}. + &&{\it Affirmation\/}: \frege.{=+\gA}.\cr + \smallskip +\+&{\it Condition\/}: \frege .{..\gB.\gA}. + &&{\it Generality\/}: \frege .{*.a.{\gF(\da)}}.\cr +\+&{\it Condition stroke\/}: \*_!_. + &&{\it Definition\/}: \*"39 $\gA=\gB$.\cr + +\noindent +{\bf f) Inferences (Grundgesetze).}\ +For inferences eight types are used: +\smallskip +\+&{\it Single line\/}: \*?--\*?--. + &&{\it Double lines\/}: \*?==\*?==.\cr +\smallskip +\+&{\it Dashed line\/}: \*?-_\*?-_. + &&{\it Dashed double lines\/}: \*?=_\*?=_.\cr +\smallskip +\+&{\it Single/double line\/}: \*?=*\*?=*. + &&{\it Dot/dashed line\/}: \*?.-\*?.-.\cr +\smallskip +\+&{\it Contraposition\/}: \*?>x. + &&{\it Quantification\/}: \*?>u.\cr +\smallskip +\+&{\it Separator\/}: \gfggtrenner.\cr + +\noindent +{\bf g) Special symbols (Grundgesetze).}\ +Several special characters appear in the {\it Grundgesetze\/}: +\smallskip +\+&{\it symbols\/}: $\Gbf$ \G0 \G1 $\Guu$\cr +\smallskip +\+&{\it unary operators\/}: \G! $\kasl$ \GI\ $\G>$ \G- \G~ $\G*$ + $\G6$ $\Gf$ $\Gp$ $\GUorg$ $\Gnorg$ +&&&{\it last two from the fge fonts\/}: $\GU$ $\Gn$\cr +\smallskip +\+&{\it binary operators\/}: $\G^$ $\G_$ $\G;$ $\Gfm$ $\Gsm$ + $\G<$ $\Gmf$ $\Gl$ +&&&{\it rotated\/}: $\Gy$ $\Ge$ $\Gc$ $\GF$ $\Gss$\cr +\toggleGGstyle + + +% %%%%%%%%%%% +\beginsection 3. How to load the package + +The package requires characters that come with \AmSTeX\ (see [3]); +the following statement must be given before the package is loaded: +€\input amssym.def€. Then the package is loaded with the statement +€\input GFnotation€. + +The package has two styles: the one that is used in the {\it +Begriffsschrift\/} and the one that is used in the {\it Grundgesetze}. +If you want to work with the style of the {\it Grundgesetze\/} several +external flags are available. + +First, if only inferences without substitutions are needed the line +€\let\nosubst=t€ should be entered before €GFnotation€ is loaded. This +avoids typesetting errors and saves some \TeX\ registers. Second, the +flag €\let\showrotation=t€ can be set to output some terminal symbols +of the {\it Grundgesetze\/} rotated. Then €\input rotate€ (see [4]) +must be stated too. Otherwise the symbols are not shown as rotated +symbols (some DVI screen drivers can't do PS-rotation). The affected +symbols are shown in Section~2g. Third, the flag +€\let\fgefontsknown=t€ can be set to indicate that the fge-fonts by +J.~J.~Green (see~[5]) are available. Then some symbols of these fonts +are used in the output. The affected symbols are shown in Section~2g. + +After the package has been loaded two flags can be set (independent of the +style). +\item{1.}€\gfbslognotationtrue€ writes all formulas, which are given in the +short form, in their symbolic representation into the log file of the +\TeX\ run. +\item{2.}€\gfbszeigestatstrue€ writes a \TeX\ message for each formula in +short form if its number of lines is a new maximum. + +By default the package activates the style that is used in the {\it +Begriffsschrift}. To switch to the style of the {\it Grundgesetze\/} +the global acting command €\toggleGGstyle€ shall be given. + +Therefore a typical start is coded in the following way if the style +of the {\it Begriffschrift\/} is required: +\verbatim +\input amssym.def +\input GFnotation +\gfbslognotationtrue % log short form formulas in symbolic representation +\gfbszeigestatstrue % inform about maximum number of lines of short form formulas +\endverbatim + +Otherwise, the style of the {\it Grundgesetze\/} is started with +\verbatim +\input amssym.def +%%% external flags: +\let\nosubst=t % substitutions aren't needed +\let\showrotation=t \input rotate % activate this line if PS-rotation works +%\let\fgefontsknown=t % activate this line if the fge-fonts are available +\input GFnotation +%%% internal flags: +\gfbslognotationtrue % log short form formulas in symbolic representation +\gfbszeigestatstrue % inform about maximum number of lines of short form formulas +%%% define the style of the output: +\toggleGGstyle % use the style of the Grundgesetze +\endverbatim + + +% %%%%%%%%%%% +\beginsection 4. Shortcuts for letters + +The characters that Frege used (except the roman letters) are coded +with two letter shortcuts. The first letter indicates how the second +letter is output. Uppercase Greek letters are introduced by `g', +lowercase Greek letters by `k', and the Fraktur letters by `d': +€$\gA$€ outputs~$\gA$, €$\ka$€ outputs~$\ka$, and €$\da$€ +outputs~$\da$. + +Here is the list of the uppercase Greek letters; the coding follows~[6]. +\smallskip +\+&€$\gA$€: $\gA$& €$\gB$€: $\gB$& €$\gG$€: $\gG$& €$\gD$€: $\gD$& €$\gE$€: $\gE$\cr +\+&€$\gZ$€: $\gZ$& €$\gH$€: $\gH$& €$\gJ$€: $\gJ$& €$\gI$€: $\gI$& €$\gK$€: $\gK$\cr +\+&€$\gL$€: $\gL$& €$\gM$€: $\gM$& €$\gN$€: $\gN$& €$\gX$€: $\gX$& €$\gO$€: $\gO$\cr +\+&€$\gP$€: $\gP$& €$\gR$€: $\gR$& €$\gS$€: $\gS$& €$\gT$€: $\gT$& €$\gU$€: $\gU$\cr +\+&€$\gF$€: $\gF$& €$\gQ$€: $\gQ$& €$\gY$€: $\gY$& €$\gW$€: $\gW$\cr + +For the Fraktur letters only a subset of the alphabets is provided; +these are the letters that appear in the {\it Begriffsschrift\/} or in +part~1 of Vol.~1 of the {\it Grundgesetze\/}: +\smallskip +\+&€$\da$€: $\da$& €$\dc$€: $\dc$& €$\dd$€: $\dd$& €$\de$€: $\de$& €$\df$€: $\df$\cr +\+&€$\dg$€: $\dg$& €$\dm$€: $\dm$& €$\dn$€: $\dn$& €$\ddo$€: $\ddo$&€$\ddp$€: $\ddp$\cr +\+&€$\dq$€: $\dq$& €$\dr$€: $\dr$& €$\dt$€: $\dt$& €$\du$€: $\du$& €$\dF$€: $\dF$\cr + +{\it Note\/}: Commands like €\dp€ or €\do€ shouldn't change their +plain \TeX\ meaning. Therefore the commands for the letters `o' and +`p' are build with two `d's. + +\begingroup\def\ds{{\frak s}} % keep this definition local +You can add more letters using a simple +€\def€. For example, to add $\ds$ declare €\def\ds{{\frak s}}€. +\endgroup + + +% %%%%%%%%%%% +\beginsection 5. Input of formulas using the + ``symbolic representation'' (Begriffsschrift) + +This input format gives complete control over the output of the +formulas. But it requires some amount of typing. All formulas are +built from symbols and three of them are grouped into the argument of +a control symbol. The control symbol is the €\*€. (The original +meaning as discretionary multiplication is kept in the new control +word €\discretionarytimes€.) + +In total there are 35 symbols. In the following list the parentheses +contain the sequence €\*"#"€ if symbol `{\tt\#}' is explained. The +first collection lists the important symbols that are used for +hand-coding the formulas in Frege's notation. +\smallskip +% 11 symbols +\item{\tt\char95 }(\*"_") is the empty space; +\item{\tt-}(\*"-") is a horizontal line for the content stroke and the inference rule; +\item{\tt\~}(\*"~") is a horizontal line with centered negation; +\item{\tt+}(\*"+") is a horizontal line with two centered negation indicators (affirmation); +\item{\tt:}(\*":") is the then-part of the condition; +\item{\tt'}(\*"'") is the if-part of the condition; +\item{\tt!}(\*"!") is a vertical line for the condition stroke; +\item{\tt[}(\*"[") is a vertical line placed left and starts a horizontal line; +\item{\tt|}(\*"|") is the vertical line for the substitution; +\item{\tt3}(\*"3") are two vertical lines used for {\it definitions\/}; +\item{\tt=}(\*"=") are two horizontal lines. + +\smallskip\noindent +The following symbols extend the basic list and can be used to +fine-tune the output. +\smallskip +% 8 symbols (sum=19) +\item{\tt"}(\*""") is a skip, that is, no output at all; +\item{\tt(}(\*"(") negation but the indication of negation is moved to the left; +\item{\tt)}(\*")") negation with the indicator moved to the right; +\item{\tt<}(\*"<") negation: the indicator is placed at the left end; +\item{\tt>}(\*">") negation: the indicator is placed at the right end; +\item{\tt\^{}}(\*"^") affirmation with indicators moved to the left; +\item{\tt/}(\*"/") affirmation with indicators moved to the right; +\item{\tt]}(\*"]") is a vertical line placed right, that is, only the judgment stroke is printed. + +For the generality a letter must be given. Therefore the parentheses +contain the sequence €\*#a.€ (€\*@a"€ for {\tt @}). Instead of the +letter `a' any other letter can be used that is available in the +Fraktur font (see Section~4). A triple for the generality must be +started with one of the following symbols {\tt.,`;@}. If it is not +the {\tt @} then the symbol after the letter must be one of +{\tt.,`;}\thinspace. +\smallskip +% 5 symbols (sum=24) +\item{\tt.}(\*.a.) is a short vertical line; +\item{\tt,}(\*,a.) is a short vertical line with a centered negation indicator; +\item{\tt`}(\*`a.) represents negation but the indicator is moved to the left; +\item{\tt;}(\*;a.) is a short vertical line with affirmation; +\item{\tt @}(\*@a") signals (without any output) that a letter for the generality follows. + +{\it Note\/}: The letter `{\tt\ss}' is special. It uses the content of +the token register €\dsz€. For example, with the default €\dsz={\xi}€ +€\*.€{\tt\ss}€.€ outputs \*.ß.. Of course, {\tt\ss} must be a letter +so for plain \TeX\ the command €\catcode`\€{\tt\ss}€=11€ must be +entered before €\input GFnotation€ is done. + +\Example1 +The formula \ +\vtop{% + \hbox{\strut\*[-:\*--- $\gA$} + \hbox{\strut\*__'\*.a. $f(\da)$}} +of Section~1 can be coded in the following way: +\verbatim +\vtop{% + \hbox{\strut\*[-:\*--- $\gA$} + \hbox{\strut\*__'\*.a. $f(\da)$}} +\endverbatim\Eend1 + +The last list contains symbols that are used by the short form. The +symbols do not try to have a mnemonic representation for their +output. Most of these symbols can ``change'' and stand for one of two +symbols from the above lists. +\smallskip +% 11 symbols (sum=35) +\item{\tt0}negation either moved to the left or the right: {\tt0=()} +\item{\tt1}affirmation: {\tt1=\^{}/} +\item{\tt2}again affirmation: {\tt2=\^{}+} +\item{\tt4}not required content strokes: {\tt4=-"} +\item{\tt5}not required empty space: {\tt5=\char95"} +\item{\tt6}judgment strokes: {\tt6=[]} +\item{\tt7}negated generality: {\tt7=`,} +\item{\tt8}again negation: {\tt8=<\~{}} +\item{\tt9}represents a content stroke that has only one third of its usual length; +\item{\tt*}is a skip whose length is multiplied by $5/3$ to get back into alignment after a {\tt9}; +\item{\tt\$} is used only with a generality and represents either a short content stroke or an empty space. + + +% %%%%%%%%%%% +\beginsection 6. Input for inference chains in the + ``symbolic representation'' (Begriffsschrift) + +In order to typeset complete inferences several formulas must be +positioned in a chain and the rule for the inference has to be entered. +The construction of Example~1 with €\vtop€, €\hbox€, and €\strut€ is +shortened to a single command €\nlp€. The terminal symbols become +an argument of a second command €\ce€. +\smallskip +\item{1.}€\nlp<n>€ starts a new line and sets the following text {\tt<n>} units to the right; +\item{2.}€\ce{<s>}€ outputs the string {\tt<s>} with some spacing at the end of a line. + +\Example2 +This is the implementation of the inference in Section~1, that is +\nlp4\*_[-\*-:-\ce{$\gA$} +\nlp4\*___\*_'-\ce{$\gB$} +\nlp4\*_[-\*---\ce{$\gB$} +\nlp4\*---\*---\*--- +\nlp4\*_[-\*---\ce{$\gA$}\qquad: +\smallskip +\verbatim +\nlp4\*_[-\*-:-\ce{$\gA$} +\nlp4\*___\*_'-\ce{$\gB$} +\nlp4\*_[-\*---\ce{$\gB$} +\nlp4\*---\*---\*--- +\nlp4\*_[-\*---\ce{$\gA$} . +\endverbatim\Eend1 + +\noindent +For complete inferences with substitutions more commands are +required: +\smallskip +\item{3.}€\ci{<s>}€ is like €\ce€ but must be used in the line and spaces that follow are ignored; +\item{4.}€\nlpc<n>{<s>}€ acts like €\nlp<n>€ and sets the string {\tt<s>} to the left in the white space; +\item{5.}€\fono{<m>}€ outputs {\tt<m>} as the number of the formula at the right end of the previous line; +\item{6.}€\rep<n><f>€ repeats the €\*€-formula {\tt<f>} {\tt<n>} times; +\item{7.}€\bcc<n>/<p><m>:<c><f>\ecc€ typesets an inference rule. +{\tt<n>} is the position to the right, {\tt<m>} is a formula number used in the inference, +and {\tt<f>} is the rule itself. The parameter {\tt<p>} is either `l', 'm', or `r' to +position the formula number to the left, in the middle, or to the right. +And {\tt<c>} is either `.'\ or `:'\ to output one or two colons after the formula number. + +\Example3 +The following code +\Epause +\verbatim + \nlp4\ci{\hss\thinspace58\hss}\rep3\*___\*_[-\*-:-\*---\*-:-\ce{$f(b)$} +\nlpc4{$f(A)$}\*_|_\*-:-\ci{$f(A)$}\rep2\*___\*_!_\*___\*_'-\ce{$g(b)$} + \nlp4\*_|_\*_'-\ci{$g(A)$}\rep2\*___\*_'-\*.a.\*-:-\ce{$f(\da)$} + \nlpc4{$c$}\*_|_\ci{$b$} \rep3\*___\*___\*___\*_'-\ce{$g(\da)$} + \bcc4/m30:.\rep3\*___\null \rep6\*---\ecc + \nlpc4{$a$}\*_|_\ci{$f(b)$} \rep2\*___\*_[-\*-:-\*-:-\*,a.\*-:-\ce{$f(\da)$} + \nlpc4{$c$}\*_|_\ci{$g(b)$} \rep2\*___\*___\*_!_\*_!_\*___\*_'-\ce{$g(\da)$} + \nlpc4{$b$}\*_|_\*.a.\*-:-\ci{$f(\da)$} \*___\*_!_\*_'-\*---\*--(\ce{$f(b)$} + \nlp4\*_|_\*___\*_'-\ci{$g(\da)$} \*___\*_'-\*---\*---\*---\ce{$g(b)$} +\fono{59} +\endverbatim\Econt + +outputs formula~59 of the {\it Begriffsschrift}: +\smallskip + \nlp4\ci{\hss\thinspace58\hss}\rep3\*___\*_[-\*-:-\*---\*-:-\ce{$f(b)$} +\nlpc4{$f(A)$}\*_|_\*-:-\ci{$f(A)$}\rep2\*___\*_!_\*___\*_'-\ce{$g(b)$} + \nlp4\*_|_\*_'-\ci{$g(A)$}\rep2\*___\*_'-\*.a.\*-:-\ce{$f(\da)$} + \nlpc4{$c$}\*_|_\ci{$b$} \rep3\*___\*___\*___\*_'-\ce{$g(\da)$} + \bcc4/m30:.\rep3\*___\null \rep6\*---\ecc + \nlpc4{$a$}\*_|_\ci{$f(b)$} \rep2\*___\*_[-\*-:-\*-:-\*,a.\*-:-\ce{$f(\da)$} + \nlpc4{$c$}\*_|_\ci{$g(b)$} \rep2\*___\*___\*_!_\*_!_\*___\*_'-\ce{$g(\da)$} + \nlpc4{$b$}\*_|_\*.a.\*-:-\ci{$f(\da)$} \*___\*_!_\*_'-\*---\*--(\ce{$f(b)$} + \nlp4\*_|_\*___\*_'-\ci{$g(\da)$} \*___\*_'-\*---\*---\*---\ce{$g(b)$} +\fono{59}\Eend0 + +\noindent +One more command is required for collections of inferences: +\item{8.}€\decollator<l>€ outputs a thick line of width €0.<l>\hsize€. The line is +centered. It is preceded and followed by vertical space. + +\Example4 +Here are some inferences of the {\it Begriffsschrift\/}: +\outof p3,0"1"with\thatis +\formula p9|{..{..a.b}.a} +\followswith p3"8"a.p8s5 +\substituting p3 d:a \whichgives +\formula p9|{..{..a.a}.b} +\named "26" +\decollator4 +\outof p3,0"26"with b:{..{..a.b}.a} \thatis +\formula p8|{..{=.{..a.a}}.{..{..a.b}.a}} +\followswith p3"1"a:p8s5 +\formula p8|{=.{=.{..a.a}}} +\named "27" + +The command €\decollator4€ appears after formula~26. +\Eend0 + + +% %%%%%%%%%%% +\beginsection 7. Input of formulas using the ``short form'' + +This input format reduces the amount of typing compared to the +symbolic representation. It follows given rules and produces as an +intermediate step the code of the symbolic representation. + +The input is recursively entered using four constructions: +\smallskip +\item{1.}{\tt=<}sign{\tt>\char"7B<}formula{\tt>\char"7D} to add a +signed content stroke in front of a formula; +\item{2.}{\tt<}sign{\tt><}sign{\tt>\char"7B<}formula{\tt>\char"7D<}% + sign{\tt>\char"7B<}formula{\tt>\char"7D} +to build a signed condition from two signed formulas; +\item{3.}{\tt*<}sign{\tt><}character{\tt><}sign{\tt>\char"7B<}% + formula{\tt>\char"7D} to add a signed +concavity in front of a signed formula; +\item{4.}{\tt!\char92<}nameofdef{\tt><}arguments{\tt>} to insert a +previously defined subformula. + +\noindent +Here {\tt<}formula{\tt>} is either a valid formula constructed with +the four rules or a terminal string (both are typeset in math mode) +and {\tt<}sign{\tt>} is one of the following symbols: +\item{\tt.} represents no sign; +\item{\tt-} represents the negation; +\item{\tt+} represents the affirmation. + +In rule~4 {\tt<}nameofdef{\tt>} is together with the preceding +backslash a defined control word that has at most three arguments +given in {\tt<}arguments{\tt>}. The control word must define a part +of a formula. This rule implements {\sl macro expansion\/} to avoid +retyping of often used parts in formulas. + +Most formulas are created by rules~2 and~3; rule~1 is only needed to +type a single line. + +{\it Important note\/}: The braces around {\tt<}formula{\tt>} can be +omitted if the formula is a terminal string that consists of a single +token. Double curly braces are required around terminal strings that +have more than five tokens. But single braces must always be used for +formulas in rules 1--3, which are not terminal strings. (In Example~5 +below case~i shows that a single token needs no braces. The cases ii +and~iii show the use of single and double curly braces.) + +Formulas that are created by the above rules must be given to the +following commands: +\smallskip +\item{1.}€\frege<j>{<f>}€ where {\tt<f>} is a formula created by the above +stated rules and {\tt<j>} is either `{\tt.}'\ or `{\tt|}'. +\item{2.}€\formula p<n><j>{<f>}€ adds a position parameter {\tt<p>} to +move the formula {\tt<n>} units to the right. + +\noindent +The formula {\tt<f>} is output as a judgment if {\tt<j>} equals +{\tt|}. The command €\frege€ shall be used in running text, +€\formula€ in vertical mode. But €\frege€ doesn't start a paragraph; +use €\indent€ (or €\noindent€) if €\frege€ is the first item +in a paragraph. + +\Example5 +\def\0#1{*.a.{#1(\da)}} +Here are a few examples for all four rules: +\smallskip +\advance\leftskip by 20pt +\item{i)}€\frege|{=-\gA}€ outputs \frege|{=-\gA}. +\item{ii)}€\frege.{-.\gA-{{g(\ka,\kb)}}}€ outputs \frege.{-.\gA-{{g(\ka,\kb)}}}. +\item{iii)}€\frege.{*-a+{f(\da)}}€ outputs \frege.{*-a+{f(\da)}}. +\item{iv)}€\def\0#1{*.a.{#1(\da)}}€ together with €\frege|{-.\gA-{!\0f}}€ +outputs \frege|{-.\gA-{!\0f}}. +\smallskip +\advance\leftskip by -20pt +The output of iv) is nearly the formula of Section~1 (\frege|{..\gA.{!\0f}}); +it can be coded (without €\0€) as €\frege|{..\gA.{*.a.{f(\da)}}}€. +\Eend0 + +Of course, a formula in Frege's notation that has several lines is not +easy to read in a sequential form. The periods and braces do not help +to make the structure of that formula easy to recognize. +I suggest to copy a Frege formula from left to right and from bottom +to top and to use more than one line for the sequential coding. + +\Example6 +This example shows how to construct in a structured manner the short +form of a given formula, which is written in Frege's notation. +The formula is \frege|{.-{..\gA.\gB}.{*.a.{..\gG-\gD}}}. + +Step~1: The formula is a judgment and starts at the left with a +condition, whose upper line contains a negation indicator. +So the coding starts with +\verbatim +\frege|{.-{}.{}} +\endverbatim + +Or write it in two lines to make the structure visible: +\verbatim +\frege|{.-{} + .{}} +\endverbatim + +Step~2: The bottom part starts with a concavity, so the input becomes +\verbatim +\frege|{.-{} + .{*.a.{}}} +\endverbatim + +Step~3: A simple condition comes after the concavity +\verbatim +\frege|{.-{} + .{*.a.{..\gG-\gD}}} +\endverbatim + +And the final step is to add the condition in the upper part: +\verbatim +\frege|{.-{..\gA.\gB} + .{*.a.{..\gG-\gD}}} +\endverbatim +\medskip +{\it Note\/}: The input +\verbatim +\frege|{..{-.\gA.\gB} + .{*.a.{..\gG-\gD}}} +\endverbatim +outputs +\frege|{..{-.\gA.\gB} + .{*.a.{..\gG-\gD}}}, +which looks very similar to the given formula but, of course, +the negation indicator in the upper part has moved a little bit +to the right. +\Eend0 + + +% %%%%%%%%%%% +\beginsection 8. Inferences of the Begriffsschrift in the ``short form'' + +Five commands are available to typeset inferences with the short form: +\smallskip +\item{1.}€\outof p<n>,<o>"<m>"with<r>\thatis€ starts the inference chain. +The parameter {\tt<n>} is the number of units the output is moved to +the right, {\tt<m>} is a number of a formula, {\tt<o>} is an offset to +position the number {\tt<m>} {\tt<o>} lines lower, and {\tt<r>} is +either empty or a space separated list of pairs {\tt v:F} to indicate +the substitution of {\tt v} by the formula {\tt F}. +\item{2.}€\use p<n>"<m>"€ is a shortcut of €\outof€ if the parameters {\tt<o>} +and {\tt<r>} are not needed; {\tt<o>} is set 0 and {\tt<r>} is empty. +\item{3.}€\followswith p<n>"<m>"a<c>p<N>s<l>€ draws the inference rule and +can be expressed in the €\bcc€ macro of Section~6, no~7, as +€\bcc{<n>}/m{<m>}:<c>\rep(<N>-<n>)\*___\rep{<l>}\*---\ecc€. +\item{4.}€\substituting p<n> <r>\whichgives€ has the usual position parameter +{\tt<n>} and a list of substitutions {\tt<r>} in the above described form {\tt +v:F}. +\item{5.}€\named "<m>"€ outputs the number {\tt<m>} as the number of the +previous formula. + +An inference chain is typically typed with the sequence: +\verbatim +\outof ... with ... \thatis +\formula ... +\followswith ... +\substituting ... \whichgives +\formula ... +\named ... +\endverbatim + +\noindent +or if no substitution in the first formula is required, the €\outof€ line +is replaced by the simpler €\use€. + +\Example7 +The formula of Example~3 +\smallskip +\outof p4,0"58"with f(A):{..{f(A)}.{g(A)}} + c:b +\thatis +\formula p8|{..{..{f(b)}.{g(b)}} + .{*.a.{..{f(\da)}.{g(\da)}}}} +\followswith p4"30"a.p8s6 +\substituting p4 a:{f(b)} + c:{g(b)} + b:{*.a.{..{f(\da)}.{g(\da)}}} +\whichgives +\formula p8|{..{..{*-a.{..{f(\da)}.{g(\da)}}} + -{f(b)}} + .{g(b)}} +\named "59" + +\bigskip +has this coding: +\verbatim +\outof p4,0"58"with f(A):{..{f(A)}.{g(A)}} + c:b +\thatis +\formula p8|{..{..{f(b)}.{g(b)}} + .{*.a.{..{f(\da)}.{g(\da)}}}} +\followswith p4"30"a.p8s6 +\substituting p4 a:{f(b)} + c:{g(b)} + b:{*.a.{..{f(\da)}.{g(\da)}}} +\whichgives +\formula p8|{..{..{*-a.{..{f(\da)}.{g(\da)}}} + -{f(b)}} + .{g(b)}} +\named "59" +\endverbatim\Eend1 + + +% %%%%%%%%%%% +\beginsection 9. Terminal symbols of the Begriffsschrift + +In part~3 of the {\it Begriffsschrift\/} four symbols are +defined. They are available as \TeX\ commands. The commands get more +flexibility than the {\it Begriffsschrift\/} requires as all possible +content can be addressed by parameters. + +\item{1.}€\gfvererbe€ with four parameters: +€$\gfvererbe\kd\ka{x(\ka)}{f(\kd,\ka)}$€ outputs +$\gfvererbe\kd\ka{x(\ka)}{f(\kd,\ka)}$. +\item{2.}€\gffolgt€ with three parameters: +€$\gffolgt\kg\kb{f(x_\kg,y_\kb)}$€ outputs $\gffolgt\kg\kb{f(x_\kg,y_\kb)}$. +\item{3.}€\gfgehoertan€ with three parameters: +€$\gfgehoertan\kg\kb{f(x_\kg,z_\kb)}$€ outputs +$\gfgehoertan\kg\kb{f(x_\kg,z_\kb)}$. +\item{4.}€\gfeindeutig€ with three parameters: +€$\gfeindeutig\kd\ke{f(\kd,\ke)}$€ outputs $\gfeindeutig\kd\ke{f(\kd,\ke)}$. + +\noindent +I use shortcuts for these special terminal symbols and reduce the +number of parameters to the amount that is used in the text; +see Example~11 below. + + +% %%%%%%%%%%% +\beginsection 10. Output style of the Grundgesetze + +\toggleGGstyle +Frege made changes to the notation in his main work. A command and +several flags are defined to switch from the style of the {\it +Begriffsschrift\/} to the style of the {\it Grundgesetze}. +\smallskip +\item{1.}€\toggleGGstyle€ switches the styles; the default is the style of +the {\it Begriffsschrift}. Several of the changes are global so even +if the switch is made inside a group it shall be called a second time +to switch it back. This happens when it is called for the first time: +\itemitem{a)}€\gfbsuseGGstyletrue€ is set. Together with a call to +€\gfbsneuestriche€ the thickness of the lines is changed. +\itemitem{b)}€\gfbskompakttrue€ is set. This reduces the width of the formulas +that are input with the short form. +\itemitem{c)}€\gfggschlussweisetrue€ is set. This activates more symbols +to express the eight inferences of the {\it Grundgesetze\/} (see Section~12). +\itemitem{d)}€\gfbsfonoohnepunkttrue€ is set. No period after the number of +a formula is set in €\fono€ or €\named€. +\itemitem{e)}€\gfbsnegdirekttrue€ is set. The negation indicator touches +the content stroke. +\item{2.}The flags can be set individually, for example, to get a compact output +but keep the line thicknesses etc.\ of the {\it Begriffsschrift}. +\item{3.}To control the compact output of the generality when the flag +€\gfbskompakttrue€ is set more flags are available. Examples are given to +show how these flags influences the output. +\itemitem{$\alpha$)}The output is not compact if €\gfbskeinekompaktehoehlungtrue€ +is set. At the left the flag is false and at the right true; the source code +is €\frege|{..{*-a.{\gY(\da)}}.{*.e.{\gF(\de)}}}€: +\smallskip\indent\indent +{\gfbskompakttrue \gfbskeinekompaktehoehlungfalse +\frege|{..{*-a.{\gY(\da)}}.{*.e.{\gF(\de)}}} +}\quad +{\gfbskompakttrue \gfbskeinekompaktehoehlungtrue +\frege|{..{*-a.{\gY(\da)}}.{*.e.{\gF(\de)}}} +}. +\itemitem{$\beta$)}The output of the short form rule~1 (with the {\tt=}) can +have no effect except if the flag named €\gfbsaussagesichtbartrue€ is set. +At the left the flag is false and at the right true; the line reads +€\frege.{=.a} vs \frege.{=.{=.a}}€: +{\gfbskompakttrue \gfbsaussagesichtbarfalse +\frege.{=.a} vs \frege.{=.{=.a}} +}\quad +{\gfbskompakttrue \gfbsaussagesichtbartrue +\frege.{=.a} vs \frege.{=.{=.a}} +} + +{\it Note\/}: Both input forms, the symbolic representation and the +short form, can be used with this style. But only the short form +creates smaller formulas if €\gfbscompakttrue€ is set. The width of a +formula in symbolic representation must be changed manually if it is +not coded with the symbols of the last list in Section~5. + +\Example8 +Here is the example of Sections 1 and~5; it is coded in the +short form as €\frege|{..\gA.{*.a.{f(\da)}}}€: +\frege|{..\gA.{*.a.{f(\da)}}}. +\Eend0 + + +% %%%%%%%%%%% +\beginsection 11. Setting formulas in parentheses and brackets (Grundgesetze) + +In the {\it Grundgesetze\/} some formulas must be placed in +parentheses or brackets. A few commands are provided to make this an +easy task. +\smallskip +\item{1.}€\pfrege<x><j>{<f>}€ acts like €\frege<j>{<f>}€ and adds parentheses around +this formula. The result is stored in a box with the number {\tt<x>}. In total +there are four boxes for this task. +\item{2.}€\bfrege<y><j>{<f>}€ is like €\pfrege€ but the formula is placed between +brackets. The pool of boxes is independent from the €\pfrege€ boxes and there are +four boxes. +\item{3.}€\pparens<x><M>€ places the math {\tt<M>} with parentheses in the box +numbered {\tt<x>}. The boxes are taken from the pool that is available for +€\pfrege€. +\item{4.}€\bparens<y><M>€ acts like €\pparens€ but it uses brackets and the +pool of boxes of €\bfrege€. +\item{5.}€\pfbox<x>€ outputs box number {\tt<x>} from the €\pfrege€ pool. +\item{6.}€\bfbox<y>€ outputs the box {\tt<y>} from the pool used by €\bfrege€. + +\Example9 +The following formula appears on page~115 of the {\it Grundgesetze}, +vol.~1. +\Epause +\smallskip +\begingroup +\pfrege1.{.-{\ke=c}.{\ke\G^v}} +\pfrege2.{.-{\ke=b}.{\ke\G^u}} +\bfrege1.{-.{\ke=n}.{{\ke\G^\kesl\pfbox1}}} +\bfrege2.{-.{\ke=m}.{{\ke\G^\kesl\pfbox2}}} +\bfrege3.{--{{\ke\G^(\ka\G^q)}}-{..{\ke=b}-{\ka=c}}} +\pfrege1.{-.{\ke=n}.{{\ke\G^\kesl\pfbox1}}} +\pfrege2.{-.{\ke=m}.{{\ke\G^\kesl\pfbox2}}} +\pparens3{\kesl\bfbox1\G^\G>q} +\pparens4{\bfbox1\G^\G>\kasl\kesl\bfbox3} + +$$\frege|{..{..{..{{\kesl\pfbox2\G^\pfbox4}} + -{{b\G^\kesl\pfbox2}}} + -{{c\G^\kesl\pfbox1}}} + .{{\kesl\pfbox2\G^\pfbox3}}} +$$ +\endgroup +\Econt +And here is the source: +\verbatim +\pfrege1.{.-{\ke=c}.{\ke\G^v}} +\pfrege2.{.-{\ke=b}.{\ke\G^u}} +\bfrege1.{-.{\ke=n}.{{\ke\G^\kesl\pfbox1}}} +\bfrege2.{-.{\ke=m}.{{\ke\G^\kesl\pfbox2}}} +\bfrege3.{--{{\ke\G^(\ka\G^q)}}-{..{\ke=b}-{\ka=c}}} +\pfrege1.{-.{\ke=n}.{{\ke\G^\kesl\pfbox1}}} +\pfrege2.{-.{\ke=m}.{{\ke\G^\kesl\pfbox2}}} +\pparens3{\kesl\bfbox1\G^\G>q} +\pparens4{\bfbox1\G^\G>\kasl\kesl\bfbox3} +\endverbatim\smallskip\verbatim +$$\frege|{..{..{..{{\kesl\pfbox2\G^\pfbox4}} + -{{b\G^\kesl\pfbox2}}} + -{{c\G^\kesl\pfbox1}}} + .{{\kesl\pfbox2\G^\pfbox3}}} +$$\endverbatim\Eend1 + + +% %%%%%%%%%%% +\beginsection 12. Inferences of the Grundgesetze + +The list entry~1c of Section~10 needs some explanation as it extends +the list of available symbols for inferences. Two of the symbols have +no output ({\tt?}\ and {\tt>}). In the following list the text in +parentheses contains the sequence €\*?#"€ or €\*?>#€ for the symbol +`{\tt\#}'. +\smallskip +\item{\tt ?} (must be the first in the triple) signals that the next +two symbols build an inference line in the style of the {\it +Grundgesetze\/}. Such symbols use 50\% more space than ordinary symbols. +\item{\tt -}(\*?-") is a single horizontal stroke; +\item{\tt .}(\*?.") is a single (centered) period; +\item{\tt =}(\*?=") is a double stroke; +\item{\tt *}(\*?*") is a single stroke in the height of the upper line of the +double stroke; +\item{\tt\char95 }(\*?_") is an empty space; +\item{\tt "}(\*?"") is a skip; +\item{\tt >}has no output; the next symbol must be a {\sl transition-sign\/}: +\itemitem{\tt x}(\*?>x) it must follow a {\tt >}; +\itemitem{\tt u}(\*?>u) it must follow a {\tt >}. + +\noindent +Inferences of the {\it Grundgesetze\/} are separated by a new type of line: +\item{1.}€\separator€ outputs a thick line with a centered dot. + +\Example{10} +Here is an inference chain from the {\it Grundgesetze\/}: +\smallskip +\advance\rightskip by 2\rightskip +\use p3"III" +\formula p5|{.-{*.f.{..{\df(a)}.{\df(a)}}}-{a=a}} +\nlp{6.5}\*?>x +\formula p5|{..{a=a}.{*.f.{..{\df(a)}.{\df(a)}}}} +\named "$\ka$" +\separator +\use p3"I" +\formula p5|{..{f(a)}.{f(a)}} +\nlp5\*?_"\*?>u +\formula p5|{*.f.{..{\df(a)}.{\df(a)}}} +\named"$\kb$" +\bcc3/m$\ka$:.\*__"\rep3\*?--\ecc +\formula p5|{=.{a=a}} +\named"IIIe" + +\rightskip=\leftskip +\medskip +And this is the code: +\verbatim +\use p3"III" +\formula p5|{.-{*.f.{..{\df(a)}.{\df(a)}}}-{a=a}} +\nlp{6.5}\*?>x +\formula p5|{..{a=a}.{*.f.{..{\df(a)}.{\df(a)}}}} +\named "$\ka$" +\separator +\use p3"I" +\formula p5|{..{f(a)}.{f(a)}} +\nlp5\*?_"\*?>u +\formula p5|{*.f.{..{\df(a)}.{\df(a)}}} +\named"$\kb$" +\bcc3/m$\ka$:.\*__"\rep3\*?--\ecc +\formula p5|{=.{a=a}} +\named"IIIe" +\endverbatim\Eend1 + + +% %%%%%%%%%%% +\beginsection 13. Terminal symbols of the Grundgesetze + +The {\it Grundgesetze\/} has not such complex constructions as the +{\it Begriffsschrift\/} (see Section~9). But it contains a lot of +special symbols. +\smallskip +\settabs 2 \columns +\def\hsnor#1.{\hbox to 3em{\hss#1.}\ } +\+\hsnor1.€\G!€: $\G!\kx$.& +\hsnor16.€\Gfm€: $u\Gfm q$.\cr +\+\hsnor2.€\kasl€: $\kasl\gF(\ka)$.& +\hsnor17.€\Gsm€: $p\Gsm q$.\cr +\+\hsnor3.€\G^€: $a\G^b$.& +\hsnor18.€\G<€: $p\G<q$.\cr +\+\hsnor4.€\GI€: $\GI p$.& +\hsnor19.€\Gy€: $p\Gy q$.\cr +\+\hsnor5.€\G>€: $\G>p$.& +\hsnor20.€\Ge€: $p\Ge q$.\cr +\+\hsnor6.€\GU€: $\GUorg p$ (or $\GU p$).& +\hsnor21.€\Gc€: $p\Gc q$.\cr +\+\hsnor7.€\Gn€: $\Gnorg u$ (or $\Gn u$).& +\hsnor22.€\Gmf€: $p\Gmf q$.\cr +\+\hsnor8.€\G0€: $\G0$.& +\hsnor23.€\G*€: $\G*t$.\cr +\+\hsnor9.€\G1€: $\G1$.& +\hsnor24.€\G6€: $\G6s$.\cr +\+\hsnor10.€\Gbf€ or €\Gff€: $\Gbf$.& +\hsnor25.€\Gf€: $\Gf s$.\cr +\+\hsnor11.€\G-€: $\G-q$.& +\hsnor26.€\GF€: $s\GF u$.\cr +\+\hsnor12.€\G~€: $\G~q$.& +\hsnor27.€\Gl€: $s\Gl u$.\cr +\+\hsnor13.€\G_€: $p\G_q$.& +\hsnor28.€\Gp€: $\Gp s$.\cr +\+\hsnor14.€\Guu€: $\Guu$.& +\hsnor29.€\Gss€ or €\Gsz€: $s\Gss p$.\cr +\+\hsnor15.€\G;€: $o\G;a$.\cr + +\noindent +The symbols are constructed with the Computer Modern +and \AmSTeX\ fonts. But with the external flag €\fgefontsknown€ the +second symbol in numbers 6 and 7 is selected (see Section~3). + +Several symbols are rotated and this requires the package {\tt +rotate.tex}. The output of unrotated symbols is activated by default; +the external flag €\let\showrotation=t€ must be given to activate +rotation (see Section~3). +\toggleGGstyle + + +% %%%%%%%%%%% +\beginsection 14. Typesetting definitions + +The following command is provided to typeset definitions: +\smallskip +\item{1.}€\gfdefinition p<n><J> <j>{<f>}=<e>€ outputs the formula +€\frege<j>{<f>}€ in parentheses and sets it equal to the new symbol +{\tt<e>} if the parameter {\tt<J>} is either {\tt|} to type a judgment +or {\tt.}\ for a definition. In the style of the {\it Grundgesetze\/} +the parameter {\tt<J>} can also be {\tt:}. In this case write €<j><f>€ +and this is interpreted as a math formula and not as a formula created +by €\frege€. The parameter {\tt<n>} is the position parameter to move +the output the given amount of units to the right. + +\noindent +The output of a definition depends on the style. + +\Example{11} +The definition in formula~69 of the {\it Begriffsschrift\/} is: +\smallskip +\def\1#1{{\gfvererbe\kd\ka{#1(\ka)}{f(\kd,\ka)}}} +\gfdefinition p4. .{*.d.{..{*.a.{..{F(\da)}.{{f(\dd,\da)}}}}.{F(\dd)}}}={\1F} +\smallskip +In the style of the {\it Grundgesetze\/} the output is: +\smallskip +\toggleGGstyle +\def\1#1{{\gfvererbe\kd\ka{#1(\ka)}{f(\kd,\ka)}}} +\gfdefinition p4. .{*.d.{..{*.a.{..{F(\da)} + .{{f(\dd,\da)}}}} + .{F(\dd)}}}={\1F} +\toggleGGstyle +\smallskip +And here is the source (note that €\1€ is a normal macro and needs no +macro expansion): +\verbatim +\def\1#1{{\gfvererbe\kd\ka{#1(\ka)}{f(\kd,\ka)}}} +\gfdefinition p4. .{*.d.{..{*.a.{..{F(\da)} + .{{f(\dd,\da)}}}} + .{F(\dd)}}}={\1F} +\endverbatim\Eend1 + +\Example{12} +Here is an example of the {\it Grundgesetze}, in which the {\tt<J>} is +a colon. +\smallskip +\advance\rightskip by 2\rightskip +\toggleGGstyle +\gfdefinition p4: \kesl(o\G^(a\G^\ke))={o\G;a} +\named "$\gX$" +\toggleGGstyle +\smallskip +\rightskip=\leftskip +And this is the source: +\verbatim +\gfdefinition p4: \kesl(o\G^(a\G^\ke))={o\G;a} +\named "$\gX$" +\endverbatim\Eend1 + + +% %%%%%%%%%%% +\beginsection 15. Commands to improve page breaks + +The formulas are never broken. A page or column break must appear +at a decollator, separator, or after an inference rule. To gain some +space the formulas can be typeset in a smaller height and depth. +(€\offinterlineskip€ should be set.) +\smallskip +\item{1.}€\gfbssetzeabstand<h><d>€ sets the height and depth between the lines to +{\tt<h>}\thinspace pt and {\tt<d>}\thinspace pt, resp.; +\item{2.}€\gfbsreduziereabstandum<nn>€ reduces the height and depth each by {\tt<nn>}\% +({\tt<nn>} is a two-digit number); +\item{3.}€\gfbsabstandzuruecksetzen€ sets height and depth to the values of €\strutbox€. + +\Example{13} +The right formula uses €\gfbsreduziereabstandum20€: +\smallskip +\begingroup +\offinterlineskip +\setbox0=\hbox to .3\hsize{\vtop to 30pt{% + \nlp4\*_[-\*-:-\*---\ce{$a$} + \nlp4\*___\*_'-\*-:-\ce{$c$} + \nlp4\*___\*___\*_'-\ce{$e$}\vss}\hss} +\setbox1=\hbox to .3\hsize{\vtop to 30pt{% + \nlp1\*_[-\*-:-\*---\ce{$a$} + \gfbsreduziereabstandum20 + \nlp1\*___\*_'-\*-:-\ce{$c$} + \nlp1\*___\*___\*_'-\ce{$e$}\vss}\hss} +\centerline{\box0 \box1 } +\gfbsabstandzuruecksetzen +\endgroup + +And this is the code: +\verbatim +\offinterlineskip +\setbox0=\hbox to .3\hsize{\vtop to 30pt{% + \nlp4\*_[-\*-:-\*---\ce{$a$} + \nlp4\*___\*_'-\*-:-\ce{$c$} + \nlp4\*___\*___\*_'-\ce{$e$}\vss}\hss} +\setbox1=\hbox to .3\hsize{\vtop to 30pt{% + \nlp1\*_[-\*-:-\*---\ce{$a$} + \gfbsreduziereabstandum20 + \nlp1\*___\*_'-\*-:-\ce{$c$} + \nlp1\*___\*___\*_'-\ce{$e$}\vss}\hss} +\centerline{\box0 \box1 } +\gfbsabstandzuruecksetzen +\endverbatim\Eend1 + + +% %%%%%%%%%%% +\beginsection 16. List of parameters + +Nearly all aspects of the lines that build the structure of Frege's +notation is controlled by parameters. They can be changed to fit +better with a certain type of font. But note that a couple of tests +have been made with these values to guarantee, for example, that the +affirmation can be placed in front of a quantification.\\ + +\noindent +Here is the list of registers that make up the style of the +{\it Begriffsschrift}. +\item{1.}$€\gfbsvolleeinheit€=\rm0.57\,em$; width of one €\*€ triple +\item{2.}$€\gfbsraise€=\rm0.5\,ex$; height of the horizontal line +\item{3.}$€\gfbsneg€=\rm0.25\,ex$; height of the negation indicator +\item{4.}$€\gfbsstrichdicke€=\rm0.5\,pt$; thickness of the horizontal line +\item{5.}$€\gfbsurteildicke€=\rm1\,pt$; thickness of the judgment stroke\\ +\item{6.}$€\gfbsuht€=\rm1.5\,ex$; height of the judgment stroke\\ +\item{7.}$€\gfbsudp€=\rm0.5\,ex$; depth of the judgment stroke\\ +\item{8.}$€\gfbsersetzungdicke€=\rm0.8\,pt$; thickness of the substitution line\\ +\item{9.}$€\gfbsdefdicke€=\rm0.75\,pt$; thickness of the definition strokes +\item{10.}$€\gfbsschlussabstand€=\rm2.5\,pt$; distance of double lines in an inference +\item{11.}$€\gfbsschlussdicke€=\rm0.8\,pt$; thickness of the second line in an inference +\item{12.}$€\gfbsdpabstand€=\rm0.2\,em$; distance between colons in front of an inference +\smallskip +\item{13.}$€\gfbsmaxanzahlzeilen€=25$; the maximal amount of lines +in a single formula of the short form (the package declares for each +line one or two token and skip registers; two registers of each kind +are needed if substitutions are allowed); +\item{14.}$€\gfbsnegpct€=70$; when €\ifgfbsnegdirect€ is set to false the negation +indicator has only a height of this percentage so that a gap occurs. +\smallskip +\noindent +And this is the list of additional dimen parameters for the style of +the {\it Grundgesetze}. +\item{15.}$€\gfggstrichdicke€=\rm0.58\,pt$; common line thickness +\item{16.}$€\gfggraise€=\rm0.14\,ex$; height of horizontal line +\item{17.}$€\gfggneg€=\rm0.47\,ex$; height of negation indicator +\item{18.}$€\gfgguht€=\rm1.4\,ex$; height of the judgment stroke +\item{19.}$€\gfggudp€=\rm0.9\,ex$; depth of the judgment stroke +\item{20.}$€\gfggdpabstand€=\rm0.06\,em$; distance between colons in front of an inference + + +% %%%%%%%%%%% +\beginsection 17. Related work + +Other people have worked on the problem to typeset Frege's notation +in \TeX\ too. J.~Parsons has developed a package {\it begriff.sty\/} +for \LaTeX\ (see [7]). Later the package was improved by +Q.~Pamp~([8]). A team that translated the {\it Grundgesetze\/} created +its own package from {\it begriff.sty\/} and called it {\it +grundgesetze.sty\/} (see [9]). It uses the fonts [5] that J.~J.~Green +has developed for the special terminal symbols in the {\it +Grundgesetze}. + + +% %%%%%%%%%%% +\beginsection 18. References + +\raggedright\frenchspacing +\item{[1]}Gottlob Frege, + {\sl Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache + des reinen\hfil\break Denkens\/} + (Halle an der Saale: Louis Nebert, 1879) +\item{}``Begriffsschrift, a formula language, modeled upon that of arthmetic, for + pure thought,'' in: Jean van Heijenoort (ed.), + {\sl From Frege to G\"odel: A Sourcebook in Mathematical Logic, + 1879--1931\/},\hfil\break(Cambridge, MA: Harvard University Press, 1967), 1--82; + translation by S. Bauer-Mengelberg + +\item{[2]}Gottlob Frege, + {\sl Grundgesetze der Arithmetik --- begriffsschriftlich abgeleitet\/} + (Jena: Hermann Pohle, Volume 1 1893, Volume 2 1903) +\item{}{\sl Basic Laws of Arithmetic\/} (Oxford: Oxford Univ. Press, 2013); + translation by Philip A. Ebert, Marcus Rossberg, Crispin Wright + +\item{[3]}American Mathematical Society, {\tt amstex}, + €http://www.ams.org/publications/authors/tex/tex€ or + €http://www.ctan.org/tex-archive/macros/amstex€ + (accessed: 2014-11-29) + +\item{[4]}{\tt rotate}, + €http://www.ctan.org/tex-archive/macros/plain/contrib/misc€ + (accessed: 2014-11-29) + +\item{[5]}J. J. Green, {\tt fge}, + €http://www.ctan.org/tex-archive/fonts/fge€ + (accessed: 2014-11-29) + +\item{[6]}Silvio Levy, ``Using Greek fonts with \TeX,'' + TUGboat {\bf 9,1} (1988), 20--24 + +\item{[7]}Josh Parsons, {\tt begriff.sty}, + €http://www.ctan.org/tex-archive/macros/latex/contrib/begriff€ + (accessed: 2014-11-29) + +\item{[8]}Quirin Pamp, {\tt frege.sty}, + €http://www.ctan.org/tex-archive/macros/latex/contrib/frege€ + (accessed: 2014-11-29) + +\item{[9]}Marcus Rossberg, {\tt grundgesetze.sty}, + €http://www.ctan.org/tex-archive/macros/latex/contrib/€\hfil\break€grundgesetze€ + (accessed: 2014-11-29) + +\bye +% Local Variables: +% coding: iso-latin-9-unix +% End: |