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diff --git a/Master/texmf-dist/doc/latex/principia/principia.tex b/Master/texmf-dist/doc/latex/principia/principia.tex index 76618c0b0b9..86d5ce3a223 100644 --- a/Master/texmf-dist/doc/latex/principia/principia.tex +++ b/Master/texmf-dist/doc/latex/principia/principia.tex @@ -1,11 +1,13 @@ \documentclass[12pt]{article} \NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{principia}[2020/10/25 principia package version 1.1] %This package supports typesetting the Peanese notation in Volume I of Whitehead and Russell’s 1910 ``Principia Mathematica". +\ProvidesPackage{principia}[2021/04/19 principia package version 1.3] %This is the principia package is for representing notations in Whitehead and Russell's ``Principia Mathematica" close to their appearance in the original. %Version 1.0 (superseded by Version 1.1): Covers typesetting of notation through Volume I. 2020/10/24 -%Version 1.1 (updates): fixed the spacing of scope dots around parentheses; fixed spacing of theorem sign; fixed spacing around primitive proposition and definition signs. 2020/10/25 -%Version 1.2 (updates): boldfaced (`thickened') the truth-functional connectives, existential quantifier, set and relation symbols; added numerous commands for typesetting brackets and substitutions into theorems. 2021/02/23 +%Version 1.1 (superseded by Version 1.2) minor updates: fixed the spacing of scope dots around parentheses; fixed spacing of theorem sign; fixed spacing around primitive proposition and definition signs. 2020/10/25 %Licensed under LaTeX Project Public License 1.3c. -%Copyright Landon D. C. Elkind, 2020. (https://landondcelkind.com/contact/). +%Version 1.2 (superseded by Version 1.3) (minor updates): boldfaced (`thickened') the truth-functional connectives, existential quantifier, set and relation symbols; added numerous commands for typesetting brackets and substitutions into theorems. 2021/02/25 +%Version 1.3 (minor update): removes package dependency on marvosym; updates Section A and Section B notations. 2021/04/19 +%Licensed under LaTeX Project Public License 1.3c. +%Copyright Landon D. C. Elkind, 2021. (https://landondcelkind.com/contact/). \usepackage{fullpage} \usepackage[T1]{fontenc} @@ -13,14 +15,89 @@ \usepackage{setspace} %Principia package requirements -\usepackage{principia} %The package itself, and its dependencies: \usepackage{amssymb} %This loads the relation domain and converse domain limitation symbols. \usepackage{amsmath} %This loads the circumflex, substitution into theorems, \text{}, \mathbf{}, \boldsymbol{}, \overleftarrow{}, \overrightarrow{}, etc. \usepackage{pifont} %This loads the eight-pointed asterisk. -\usepackage{marvosym} %This loads the male and female symbol. \usepackage{graphicx} %This loads commands that flip iota for definite descriptions, Lambda for the universal class, and so on. The (superseded) graphics package should also work here, but is not recommended. -\title{\texttt{principia.sty }\\ A \LaTeXe \space Package for Typesetting Whitehead and Russell's \textit{Principia Mathematica} (Version 1.2)} +%Volume I +%Mathematical logic +%The theory of deduction +%Meta-logical symbols +\newcommand{\pmfd}{\begin{center} \rule{5cm}{.5pt} \end{center}} %Dividing line between introductory remarks in a starred number and the formal deductions. +\newcommand{\pmdem}{\textit{Dem}.} %This notation begins a proof. +\newcommand{\pmdemi}{\indent \pmdem} %This idents the notation that begins a proof. +\newcommand{\pmhp}{\text{Hp}} %This typesets Hp (short for antecedent), which occurs at the beginning of a proof. +\newcommand{\pmprop}{\text{Prop}} %This occurs at the end of a proof. +\newcommand{\pmithm}{\pmimp\;\pmthm} %This occurs when a meta-theoretic implication is asserted. +\newcommand{\pmbr}[1]{\bigg \lbrack \normalsize #1 \bigg \rbrack} %These are larger brackets for substitution. +\newcommand{\pmsub}[2]{\bigg \lbrack \small \begin{array}{c} #1 \\ \hline #2 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmsubb}[4]{\bigg \lbrack \small \begin{array}{c c} #1, & #3 \\ \hline #2, & #4 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmsubbb}[6]{\bigg \lbrack \small \begin{array}{c c c} #1, & #3, & #5 \\ \hline #2, & #4, & #6 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmsubbbb}[8]{\bigg \lbrack \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmSub}[3]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c} #2 \\ \hline #3 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmSubb}[5]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c} #2, & #4 \\ \hline #3, & #5 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmSubbb}[7]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c c} #2, & #4, & #6 \\ \hline #3, & #5, & #7 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmSubbbb}[9]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array} \bigg \rbrack} %This is the substitution command. +\newcommand{\pmsUb}[2]{\small \begin{array}{c} #1 \\ \hline #2 \end{array}} %This is the substitution command. +\newcommand{\pmsUbb}[4]{\small \begin{array}{c c} #1, & #3 \\ \hline #2, & #4 \end{array}} %This is the substitution command. +\newcommand{\pmsUbbb}[6]{\small \begin{array}{c c c} #1, & #3, & #5 \\ \hline #2, & #4, & #6 \end{array}} %This is the substitution command. +\newcommand{\pmsUbbbb}[8]{\small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array}} %This is the substitution command. +\newcommand{\pmSUb}[3]{\normalsize #1 \text{ } \small \begin{array}{c} #2 \\ \hline #3 \end{array}} %This is the substitution command. +\newcommand{\pmSUbb}[5]{\normalsize #1 \text{ } \small \begin{array}{c c} #2, & #4 \\ \hline #3, & #5 \end{array}} %This is the substitution command. +\newcommand{\pmSUbbb}[7]{\normalsize #1 \text{ } \small \begin{array}{c c c} #2, & #4, & #6 \\ \hline #3, & #5, & #7 \end{array}} %This is the substitution command. +\newcommand{\pmSUbbbb}[9]{\normalsize #1 \text{ } \small \begin{array}{c c c c} #1, & #3, & #5, & #7 \\ \hline #2, & #4, & #6, & #8 \end{array}} %This is the substitution command. +\newcommand{\pmthm}{\mathpunct{\text{\scalebox{.5}[1]{$\boldsymbol\vdash$}}}} %This is the theorem sign. +\newcommand{\pmast}{\text{\resizebox{!}{.75\height}{\ding{107}}}} %This is the sign introducing a theorem number. +\newcommand{\pmcdot}{\text{\raisebox{.05cm}{$\boldsymbol\cdot$}}} %This is a sign introducing a theorem sub-number. +\newcommand{\pmiddf}{\mathbin{=}} +\newcommand{\pmdf}{\quad \text{Df}} +\newcommand{\pmpp}{\quad \text{Pp}} + +%Square dots for scope, defined for up to six dots +\newcommand{\pmdot}{\mathrel{\hbox{\rule{.3ex}{.3ex}}}} +\newcommand{\pmdott}{\mathrel{\overset{\pmdot}{\pmdot}}} +\newcommand{\pmdottt}{\pmdott\hspace{.1em}\pmdot} +\newcommand{\pmdotttt}{\pmdott\hspace{.1em}\pmdott} +\newcommand{\pmdottttt}{\pmdott\hspace{.1em}\pmdott\hspace{.1em}\pmdot} +\newcommand{\pmdotttttt}{\pmdott\hspace{.1em}\pmdott\hspace{.1em}\pmdott} + +%Logical connectives +\newcommand{\pmnot}{\mathord{\ooalign{$\boldsymbol{\sim}\mkern.5mu$\hidewidth\cr$\boldsymbol{\sim}$\cr\hidewidth$\mkern.5mu\boldsymbol{\sim}$}}} +\newcommand{\pmor}{\mathbin{\ooalign{$\boldsymbol{\vee}\mkern.5mu$\hidewidth\cr$\boldsymbol{\vee}$\cr\hidewidth$\mkern.5mu\boldsymbol{\vee}$}}} +\newcommand{\pmimp}{\mathbin{\ooalign{$\boldsymbol{\supset}\mkern.5mu$\hidewidth\cr$\boldsymbol{\supset}$\cr\hidewidth$\mkern.5mu\boldsymbol{\supset}$}}} %1.01 +\newcommand{\pmand}{\mathrel{\hbox{\rule{.3ex}{.3ex}}}} %3.01 +\newcommand{\pmandd}{\overset{\pmand}{\pmand}} +\newcommand{\pmanddd}{\pmandd\hspace{.1em}\pmand} +\newcommand{\pmandddd}{\pmandd\hspace{.1em}\pmandd} +\newcommand{\pmanddddd}{\pmandd\hspace{.1em}\pmandd\hspace{.1em}\pmand} +\newcommand{\pmandddddd}{\pmandd\hspace{.1em}\pmandd\hspace{.1em}\pmandd} +\newcommand{\pmprod}{\mathbin{\ooalign{$\boldsymbol{\wedge}\mkern.5mu$\hidewidth\cr$\boldsymbol{\wedge}$\cr\hidewidth$\mkern.5mu\boldsymbol{\wedge}$}}} %Not in Principia, but added here as a dual of its symbol for disjunction. +\newcommand{\pmiff}{\mathbin{\ooalign{$\boldsymbol{\equiv}\mkern.5mu$\hidewidth\cr$\boldsymbol{\equiv}$\cr\hidewidth$\mkern.5mu\boldsymbol{\equiv}$}}} %4.01 + +%The theory of apparent variables +\newcommand{\pmall}[1]{(#1)} +\newcommand{\pmsome}[1]{(\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}#1)} %10.01 +\newcommand{\pmSome}{\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}} + +%Additional defined logic signs +\newcommand{\pmhat}[1]{\mathbf{\hat{\text{$#1$}}}} +\newcommand{\pmbreve}[1]{\mathbf{\breve{\text{$#1$}}}} +\newcommand{\pmcirc}[1]{\mathbf{\dot{\text{$#1$}}}} +\newcommand{\pmpf}[2]{#1#2} %for propositional functions of one variable +\newcommand{\pmpff}[3]{#1(#2, #3)} %for propositional functions of two variables +\newcommand{\pmpfff}[4]{#1(#2, #3, #4)} %for propositional functions of three variables +\newcommand{\pmshr}{\textbf{!}} %*12.1 and *12.11, used for predicative propositional functions +\newcommand{\pmpred}[2]{#1\pmshr#2} %for predicates (``predicative functions'') of one variable +\newcommand{\pmpredd}[3]{#1\pmshr(#2, #3)} %for predicates (``predicative functions'') of two variables +\newcommand{\pmpreddd}[4]{#1\pmshr(#2, #3, #4)} %for predicates (``predicative functions'') of three variables +\newcommand{\pmnid}{\mathrel{\ooalign{$=$\cr\hidewidth\footnotesize\rotatebox[origin=c]{210}{\textbf{/}}\hidewidth\cr}}} %*13.02 +\newcommand{\pmiota}{\ooalign{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}\cr\hidewidth\raisebox{.0125em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.025em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.0375em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}\cr\hidewidth\raisebox{.05em}{\rotatebox[origin=c]{180}{$\mathbf{\iota}$}}}} %the rotated Greek iota used in definite descriptions +\newcommand{\pmdsc}[1]{(\pmiota#1)} %*14.01 +\newcommand{\pmDsc}{\pmiota} +\newcommand{\pmexists}{\textbf{E}\hspace{.1em}\pmshr} %*14.02 + +\title{\texttt{principia.sty}\\ A \LaTeXe \space Package for Typesetting Whitehead and Russell's \textit{Principia Mathematica} (Version 1.3)} \author{Landon D. C. Elkind \texttt{elkind@ualberta.ca}} \date{\today} @@ -33,9 +110,11 @@ The \texttt{principia} package is designed for typesetting the Peanese notation In \textit{Principia Mathematica} some symbols occur with an argument and sometimes that same symbol occurs without an argument. For example, `$\pmsome{x}$' occurs in some formulas, but sometimes `$\pmSome$' occurs in the text when they talk about the symbol itself. \texttt{principia} is designed to accommodate these different occurrences of symbols. When a symbol is to occur without an argument, capitalize the first letter following the `\texttt{$\backslash$pm}' part of the command. E.g. \verb|\pmsome{x}| produces $\pmsome{x}$ and \verb|\pmSome| produces `$\pmSome$'. Note the former command requires an argument and the latter command does not. Not all commands in the \texttt{principia} package admit of such dual use because some symbols in \textit{Principia Mathematica} never occur without an argument or do not take an argument in the usual sense. For example, the propositional connectives do not take an `argument' in the way singular or plural descriptions do. -Version 1.2 (and 1.0-1.1) of \texttt{principia} is adequate to typeset all notations throughout Volume I of \textit{Principia} and includes some fixes to spacing, etc. See the package documentation for details. Updates to include all notations throughout Volumes II and III are planned. +Version 1.3 of \texttt{principia} is adequate to typeset all notations throughout Sections A and B of \textit{Principia}'s Volume I and includes some minor fixes. See the package documentation for details. + +\texttt{principia}'s dependencies are \texttt{amsmath}, \texttt{amssymb}, \texttt{pifont}, and \texttt{graphicx}. Make sure to load these package by typing \texttt{$\backslash$usepackage\{graphicx\}}, etc., into the document preamble. -\texttt{principia}'s dependencies are \texttt{amsmath}, \texttt{amssymb}, \texttt{pifont}, \texttt{marvosym}, and \texttt{graphicx}. Make sure to load these package by typing \texttt{$\backslash$usepackage\{graphicx\}}, etc., into the document preamble. To load \texttt{principia}, type \texttt{$\backslash$usepackage\{principia\}} in the document's preamble. +To load \texttt{principia}, type \texttt{$\backslash$usepackage\{principia\}} in the document's preamble. \noindent \begin{tabular}{@{}p{3cm} | p{5cm} | p{8.25cm}} \textbf{Symbol} & \textbf{\LaTeX command} & \textbf{Notes} \\ \hline @@ -70,105 +149,7 @@ Version 1.2 (and 1.0-1.1) of \texttt{principia} is adequate to typeset all notat \noindent \begin{tabular}{@{}p{3cm} | p{5cm} | p{8.25cm}} $=$, $\pmnid$ & \verb|=|, \verb|\pmnid| & Identity and its negation. \\ $\pmdsc{x}$ & \verb|\pmdsc{x}| & Definite description. \\ - $\pmexists$ & \verb|\pmexists| & Existence. \\ - $\pmcls{z}{\psi z}$ & \verb|\pmcls{z}{\psi z}| & The class of $z$s satisfying $\psi$. \\ - $\pmcin$ & \verb|\pmcin| & The class membership symbol. \\ - $\pmClsn{n}$, $\pmCls$ & \verb|\pmClsn{n}|, \verb|\pmCls| & The class of classes of individuals. \\ - $\pmscl{\alpha}$, $\pmsCl$ & \verb|\pmscl{\alpha}|, \verb|\pmsCl| & The subclasses of a class $\alpha$. \\ - $\pmsrl{R}$, $\pmsRl$ & \verb|\pmsrl{R}|, \verb|\pmsRl| & The sub-relations of a relation $R$. \\ - $\pmcuni$ & \verb|\pmcuni| & The universal class. \\ - $\pmcnull$ & \verb|\pmcnull| & The null class. \\ - $\pmcexists$ & \verb|\pmcexists| & The existence of a class. \\ - $\pmccmp{\alpha}$ & \verb|\pmccmp{\alpha}| & This command requires one argument. \\ - $\pmcmin{\alpha}{\beta}$ & \verb|\pmcmin{\alpha}{\beta}| & This command requires two arguments. \\ - $\pmccup$ & \verb|\pmccup| & Class union. \\ - $\pmccap$ & \verb|\pmccap| & Class intersection. \\ - $\pmcinc$ & \verb|\pmcinc| & Class inclusion. \\ - $\pmrel{x}{y}{\phi(x,y)}$ & \verb|\pmrel{x}{y}{\phi(x,y)}| & The relation in extension given by $\phi$. \\ - $\pmrele{a}{x}{y}{R}{b}$ & \verb|\pmrele{a}{x}{y}{R}{b}| & This command requires five arguments. \\ - $\pmrelep{a}{R}{b}$ & \verb|\pmrelep{a}{R}{b}| & This command requires three arguments. \\ - $\pmrin$ & \verb|\pmrin| & The relation membership symbol. \\ - $\pmReln{n}$, $\pmRel$ & \verb|\pmReln{n}|, \verb|\pmRel| & The class of relations ($n$-many `of relations'). \\ - $\pmruni$ & \verb|\pmruni| & The universal relation. \\ - $\pmrnull$ & \verb|\pmrnull| & The null relation. \\ - $\pmrexists$ & \verb|\pmrexists| & This symbol prefixes relations. \\ - $\pmrcmp{R}$ & \verb|\pmrcmp{\alpha}| & This command requires one argument. \\ - $\pmrmin{R}{S}$ & \verb|\pmcmin{R}{S}| & This command requires two arguments. \\ - $\pmrcup$ & \verb|\pmrcup| & Relation union. \\ - $\pmrcap$ & \verb|\pmrcap| & Relation intersection. \\ - $\pmrinc$ & \verb|\pmrinc| & Relation inclusion. \\ - $\pmcrel{R}$ & \verb|\pmcrel{R}| & The converse of a relation. \\ - $\pmCnv$ & \verb|\pmCnv| & The command for `Cnv'. \\ - $\pmdscf{R}{x}$ & \verb|\pmdscf{R}{x}| & A singular descriptive function. \\ - $\pmdscff{R}{\beta}$ & \verb|\pmdscff{R}{\beta}| & A plural descriptive function. \\ - $\pmdscfff{R}{\kappa}$ & \verb|\pmdscfff{R}{\kappa}| & A plural descriptive function. \\ - $\pmdscfe{R}{\beta}$ & \verb|\pmdscfe{R}{\beta}| & The existence of a plural descriptive function. -\end{tabular} - -\noindent \begin{tabular}{@{}p{3cm} | p{5cm} | p{8.25cm}} - $\pmdscfr{R}{x}$, `$\pmdscfR{R}$'& \verb|\pmdscfr{R}{x}|, \verb|\pmdscfR{R}| & The relation of $\pmdscfr{R}{\beta}$ to $\beta$. \\ - $\pmdm{R}$, $\pmDm$ & \verb|\pmdm{R}|, \verb|\pmDm| & The domain of a relation $R$. \\ - $\pmcdm{R}$, $\pmCdm$ & \verb|\pmcdm{R}|, \verb|\pmCdm| & The converse domain of a relation $R$. \\ - $\pmcmp{R}$, $\pmCmp$ & \verb|\pmcmp{R}|, \verb|\pmCmp| & The campus of a relation $R$. \\ - $\pmfld{R}$, $\pmFld$ & \verb|\pmfld{R}|, \verb|\pmFld| & The field of a relation $R$. \\ - $\pmrrf{R}{x}$, $\pmRrf{R}$ & \verb|\pmrrf{R}{x}|, \verb|\pmRrf{R}| & The referents of a given relation. \\ - $\pmrrl{R}{x}$, $\pmRrl{R}$ & \verb|\pmrrl{R}{x}|, \verb|\pmRrl{R}| & The relata of a given relation. \\ - $\pmsg{R}$, $\pmSg$ & \verb|\pmsg{R}|, \verb|\pmSg| & \\ - $\pmgs{R}$, $\pmGs$ & \verb|\pmgs{R}|, \verb|\pmGs| & \\ - $\pmrprd{R}{S}$, $\pmRprd$ & \verb|\pmrprd{R}{S}|, \verb|\pmrprd| & The relative product of $R$ and $S$. \\ - $\pmrprdn{R}{n}$ & \verb|\pmrprdn{R}{n}| & The $n$th relative product of $R$. \\ - $\pmrprdd{R}{S}$, $\pmRprdd$ & \verb|\pmrprdd{R}{S}|, \verb|\pmrprdd| & The double relative product of $R$ and $S$. \\ - $\pmrlcd{\alpha}{R}$ & \verb|\pmrld{\alpha}{R}| & The limitation of $R$'s domain to $\alpha$. \\ - $\pmrlcd{R}{\beta}$ & \verb|\pmrld{R}{\beta}| & The limitation of $R$'s converse domain to $\beta$. \\ - $\pmrlf{\alpha}{R}{\beta}$ & \verb|\pmrlf{\alpha}{R}{\beta}| & The limitation of $R$'s field to $\alpha$ and $\beta$, resp. \\ - $\pmrlF{P}{\alpha}$ & \verb|\pmrlF{\alpha}{R}{\beta}| & The limitation of $P$'s field to $\alpha$. \\ - $\pmrl{\alpha}{\beta}$ & \verb|\pmrl{\alpha}{\beta}| & The relation made of all $x$s in $\alpha$ and $y$s in $\beta$. \\ - $\pmop$ & \verb|\pmop| & The operation symbol. \\ - $\pmopc{\alpha}{y}$ & \verb|\pmopc{\alpha}{y}| & The relation of $x$s in $\alpha$ taken to $y$ by $\pmop$. \\ - $\pmccsum{\alpha}$ & \verb|\pmccsum{\alpha}| & The sum of a class of classes. \\ - $\pmccprd{\alpha}$ & \verb|\pmccprd{\alpha}| & The product of a class of classes. \\ - $\pmcrsum{\alpha}$ & \verb|\pmcrsum{\alpha}| & The sum of a class of relations. \\ - $\pmcrprd{\alpha}$ & \verb|\pmcrprd{\alpha}| & The product of a class of relations. \\ - $\pmrid$, $\pmrdiv$ & \verb|\pmrid|, \verb|\pmrdiv| & The relations of identity and diversity. \\ - $\pmcunit{x}$, $\pmcUnit$ & \verb|\pmcunit{x}|, \verb|\pmcUnit| & The unit class. \\ - $\pmcunits{\alpha}$ & \verb|\pmcunits{\alpha}| & The sum of unit classes of $\alpha$'s elements. \\ - $\pmrn{n}$ & \verb|\pmrn{n}| & The ordinal number $n$. \\ - $\pmdn{n}$ & \verb|\pmdn{n}| & The class of relations equal to an $n$-tuple. \\ - $\pmoc{x}{y}$ & \verb|\pmoc{x}{y}| & The ordinal number restricted to $R=(x,y)$. \\ - $\pmrt{x}$, $\pmrti{n}{x}$ & \verb|\pmrt{x}|, \verb|\pmrti{n}{x}| & The relative type of $x$ ($n$-many `type of's). \\ - $\pmrtc{n}{\alpha}$ & \verb|\pmrtc{n}{\alpha}| & The relative type of $\alpha$ ($n$-many `type of's). \\ - $\pmrtri{n}{R}$, $\pmrtrc{n}{R}$ & \verb|\pmrtri{n}{R}|, \verb|\pmrtrc{n}{R}| & The relative type of (with $n$-many `type of's) $R$ from individuals to individuals, or from classes to classes. `$nm$' can replace `$n$'. -\end{tabular} - -\noindent \begin{tabular}{@{}p{3cm} | p{5cm} | p{8.25cm}} - $\pmrtric{n}{m}{R}$, $\pmrtrci{n}{m}{R}$ & \verb|\pmrtric{n}{R}|, \verb|\pmrtrci{n}{R}| & The relative type of $R$ from individuals to classes, or from classes to individuals. \\ - $\pmrtdi{\alpha}{x}$, $\pmrtdri{R}{(x,y)}$ & \verb|\pmrtdi{\alpha}{x}|, \verb|\pmrtdri{R}{(x,y)}| & The result of determining that the members of $\alpha$ ($R$) belong to the relative type of $x$ (in the domain, and of $y$ in the converse domain). \\ - $\pmrtdc{\alpha}{x}$, $\pmrtdrc{R}{x,y}$ & \verb|\pmrtdc{\alpha}{x}|, \verb|\pmrtdrc{R}{x,y}| & The result of determining that the members of $\alpha$ ($R$) belong to the relative type of $\pmrt{x}$ (in the domain, and of $\pmrt{y}$ in the converse domain). \\ - $\pmrdc{\alpha}{\beta}$ & \verb|\pmrdc{\alpha}{\beta}| & The class of relations $R$ with domain contained in $\alpha$ and converse domain in $\beta$. \\ - $\pmoneone$, $\pmonemany$, $\pmmanyone$ & \verb|\pmoneone|, \verb|\pmonemany|, \verb|\pmmanyone| & The class of one-one, or one-many, or many-one, relations. Note \verb|\pmrdc| can be used here. \\ - $\pmsm$, $\pmsmbar$ & \verb|\pmsm|, \verb|\pmsmbar| & The similarity relation. \\ - $\pmselp{\kappa}$, $\pmSelp$ & \verb|\pmselp{\kappa}|, \verb|\pmSelp| & The $P$-selections from $\kappa$ \\ - $\pmsele{\kappa}$, $\pmSele$ & \verb|\pmsele{\kappa}|, \verb|\pmSele| & The $\pmcin$-selections from $\kappa$ \\ - $\pmself{\kappa}$, $\pmSelf$ & \verb|\pmself{\kappa}|, \verb|\pmSelf| & The $F$-selections from $\kappa$ \\ - $\pmexc$ & \verb|\pmexc| & The class of pairwise-disjoint classes. \\ - $\pmexcn$ & \verb|\pmexcn| & The class of pairwise-disjoint non-null classes. \\ - $\pmexcc{\gamma}$ & \verb|\pmexcc{\gamma}| & A class of mutually exclusive classes in $\gamma$. \\ - $\pmselc{P}{y}$ & \verb|\pmselc{P}{y}| & The class of couples $(y, \pmdscf{P}{y})$. \\ - $\pmmultc$ & \verb|\pmmultc| & The class of multipliable classes. \\ - $\pmmultr$ & \verb|\pmmultr| & The class of multipliable relations. \\ - $\pmmultax$ & \verb|\pmmultax| & The multiplicative axiom. \\ - $\pmanc{R}$, $\pmancc{R}$ & \verb|\pmanc{R}|, \verb|\pmancc{R}| & The ancestral and its converse. \\ - $\pmrst{R}$, $\pmrts{R}$ & \verb|\pmrst{R}|, \verb|\pmrts{R}| & The powers of the ancestral and its converse. \\ - $\pmmin{P}$, $\pmmax{P}$ & \verb|\pmmin{P}|, \verb|\pmmax{P}| & The minimum and maximum under $P$. \\ - $\pmpot{R}$, $\pmpotid{R}$ & \verb|\pmpot{R}|, \verb|\pmpotid{R}| & The products (strict and not) of an ancestral. \\ - $\pmpo{R}$ & \verb|\pmpo{R}| & The product of a class of ancestrals $R$. \\ - $\pmB$ & \verb|\pmB| & The relation of beginning under $P$. \\ - $\pmgen{P}$ & \verb|\pmgen{P}| & The generation of $P$. \\ - $\pmefr{P}{Q}$ & \verb|\pmefr{P}{Q}| & The equi-factor relation. \\ - $\pmipr{R}{x}$ & \verb|\pmipr{R}{x}| & The non-distinct posterity of $x$ under $R$. \\ - $\pmjpr{R}{x}$ & \verb|\pmjpr{R}{x}| & The distinct posterity of $x$ under $R$. \\ - $\pmfr{R}{x}$ & \verb|\pmfr{R}{x}| & The ancestry and posterity of $x$ under $R$. \\ - $\pmnc{\kappa}$, $\pmNc$ & \verb|\pmnc{\kappa}|, \verb|\pmNc| & The cardinal number of $\kappa$. + $\pmexists$ & \verb|\pmexists| & Existence. \end{tabular} \end{document}
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