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Diffstat (limited to 'macros/latex/contrib/principia/principia.tex')
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1 files changed, 685 insertions, 15 deletions
diff --git a/macros/latex/contrib/principia/principia.tex b/macros/latex/contrib/principia/principia.tex index 86d5ce3a22..8c5eb4e7bf 100644 --- a/macros/latex/contrib/principia/principia.tex +++ b/macros/latex/contrib/principia/principia.tex @@ -1,13 +1,13 @@ \documentclass[12pt]{article} \NeedsTeXFormat{LaTeX2e} -\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. +\ProvidesPackage{principia}[2023/03/13 principia package version 2.0] %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 %Version 1.0 (superseded by Version 1.1): Covers typesetting of notation through Volume I. 2020/10/24 %Version 1.1 (superseded by Version 1.2) minor updates: fixed the spacing of scope dots around parentheses; fixed spacing of theorem sign; fixed spacing around primitive proposition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. -%Version 1.2 (superseded by Version 1.3) (minor updates): boldfaced (`thickened') the truth-functional connectives, existential quantifier, set and relation symbols; added numerous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update): removes package dependency on marvosym; updates Section A and Section B notations. 2021/04/19 +%Version 1.2 (superseded by Version 2.0) (minor updates): boldfaced (`thickened') the truth-functional connectives, existential quantifier, set and relation symbols; added numerous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 2.0 (major update): extends the package to cover typesetting of all notations in Volume II; removes package dependency on marvosym. 2023/03/13 %Licensed under LaTeX Project Public License 1.3c. -%Copyright Landon D. C. Elkind, 2021. (https://landondcelkind.com/contact/). +%Copyright Landon D. C. Elkind, 2022. (https://landonelkind.com/contact/). \usepackage{fullpage} \usepackage[T1]{fontenc} @@ -24,6 +24,26 @@ %Mathematical logic %The theory of deduction %Meta-logical symbols +\newcommand{\ie}{\textit{i}.\textit{e}.\ } +\newcommand{\Ie}{\textit{I}.\textit{e}.\ } +\newcommand{\eg}{\textit{e}.\textit{g}.\ } +\newcommand{\Eg}{\textit{E}.\textit{g}.\ } +\newcommand{\pmsch}[1]{\pmast#1} %Starred chapter +\newcommand{\pmschs}[2]{\pmast#1\text{---}\pmast#2} %Starred chapter +\newcommand{\pmsns}[3]{\pmast#1\pmcdot#2\text{---}\pmcdot#3}%Starred number +\newcommand{\pmpsn}[2]{(\pmast#1\pmcdot#2)} +\newcommand{\pmpsnn}[3]{(\pmast#1\pmcdot#2\pmcdot#3)} +\newcommand{\pmsn}[2]{\pmast#1\pmcdot#2} +\newcommand{\pmnsn}[1]{\text{#1}} +\newcommand{\pmsnn}[3]{\pmast#1\pmcdot#2\pmcdot#3} +\newcommand{\pmsnnn}[4]{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4} +\newcommand{\pmsnnnn}[5]{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4\pmcdot#5} +\newcommand{\pmsnnnnn}[6]{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4\pmcdot#5\pmcdot#6} +\newcommand{\pmsnb}[2]{\boldsymbol{\pmast#1\pmcdot#2}} %Starred number boldface +\newcommand{\pmsnnb}[3]{\boldsymbol{\pmast#1\pmcdot#2\pmcdot#3}} +\newcommand{\pmsnnnb}[4]{\boldsymbol{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4}} +\newcommand{\pmsnnnnb}[5]{\boldsymbol{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4\pmcdot#5}} +\newcommand{\pmsnnnnnb}[6]{\boldsymbol{\pmast#1\pmcdot#2\pmcdot#3\pmcdot#4\pmcdot#5\pmcdot#6}} \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. @@ -38,7 +58,7 @@ \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{\pmSubbbb}[9]{\bigg \lbrack \normalsize #1 \text{ } \small \begin{array}{c c c c} #2, & #4, & #6, & #8 \\ \hline #3, & #5, & #7, & #9 \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. @@ -46,12 +66,13 @@ \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{\pmSUbbbb}[9]{\normalsize #1 \text{ } \small \begin{array}{c c c c} #2, & #4, & #6, & #8 \\ \hline #3, & #5, & #7, & #9 \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{\pmDf}{\text{Df}} \newcommand{\pmpp}{\quad \text{Pp}} %Square dots for scope, defined for up to six dots @@ -74,6 +95,7 @@ \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 +\newcommand{\pminc}{\mathbin{|}} %8.01 %The theory of apparent variables \newcommand{\pmall}[1]{(#1)} @@ -81,24 +103,574 @@ \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{\pmhat}[1]{\boldsymbol{\hat{\text{$#1$}}}} +\newcommand{\pmbreve}[1]{\boldsymbol{\breve{\text{$#1$}}}} +\newcommand{\pmcirc}[1]{\boldsymbol{\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{\pmpffff}[5]{#1(#2, #3, #4, #5)} %for propositional functions of four variables (including ellipses) +\newcommand{\pmppf}[2]{#1\pmshr#2} %for propositional predicative functions of one variable +\newcommand{\pmppff}[3]{#1\pmshr(#2, #3)} %for propositional predicative functions of two 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{\pmpredddd}[5]{#1\pmshr(#2, #3, #4, #5)} %for predicates (``predicative functions'') of four variables +\newcommand{\pmpreddddd}[6]{#1\pmshr(#2, #3, #4, #5, #6)} %for predicates (``predicative functions'') of five variables +\newcommand{\pmpredddddd}[7]{#1\pmshr(#2, #3, #4, #5, #6, #7)} %for predicates (``predicative functions'') of six variables +\newcommand{\pmid}{\mathbin{=}} \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{\pmiota}{\ooalign{\rotatebox[origin=c]{180}{$\boldsymbol{\iota}$}\cr\hidewidth\raisebox{.0125em}{\rotatebox[origin=c]{180}{$\boldsymbol{\iota}$}}\cr\hidewidth\raisebox{.025em}{\rotatebox[origin=c]{180}{$\boldsymbol{\iota}$}}\cr\hidewidth\raisebox{.0375em}{\rotatebox[origin=c]{180}{$\boldsymbol{\iota}$}}\cr\hidewidth\raisebox{.05em}{\rotatebox[origin=c]{180}{$\boldsymbol{\iota}$}}}} %the rotated Greek iota used in definite descriptions \newcommand{\pmdsc}[1]{(\pmiota#1)} %*14.01 +\newcommand{\pmthe}[2]{(\pmiota#1)(#2 #1)} %*14.01 +\newcommand{\pmtheb}[2]{[(\pmiota#1)(#2 #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}} +%Classes and relations +%Class signs +\newcommand{\pmcls}[2]{\pmhat{#1}(#2)} %20.01 +\newcommand{\pmcin}{\mathop{\boldsymbol{\epsilon}}} %20.02 +\newcommand{\pmCls}{\text{Cls}} %20.03 +\newcommand{\pmClsn}[1]{\text{Cls}^{#1}} +\newcommand{\pmcinn}{\pmnot\pmcin} %20.06 +\newcommand{\pmcinc}{\mathop{\ooalign{$\boldsymbol{\subset}$\cr\hidewidth$\hspace{.1em}\boldsymbol{\subset}$\cr\hidewidth$\hspace{.15em}\boldsymbol{\subset}$\cr\hidewidth$\hspace{.2em}\boldsymbol{\subset}$}}} %22.01 +\newcommand{\pmccap}{\mathop{\ooalign{\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.1em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.2em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.3em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.4em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.5em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}\cr\hidewidth\hspace{.6em}\scalebox{1.3}[1.75]{$\put(3, 2){\oval(4,1)[t]}\phantom{\circ}$}}}} %22.02 +\newcommand{\pmccup}{\mathop{\ooalign{\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.1em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.2em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.3em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.4em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.5em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}\cr\hidewidth\hspace{.6em}\scalebox{1.3}[1.75]{$\put(3, 2.5){\oval(4,4)[b]}\phantom{\circ}$}}}} %22.03 +\newcommand{\pmccmp}[1]{\boldsymbol{-}#1} %22.04 +\newcommand{\pmcmin}[2]{#1\boldsymbol{-}#2} %22.05 +\newcommand{\pmcuni}{\text{\rotatebox[origin=c]{180}{$\Lambda$}}} %24.01 +\newcommand{\pmcnull}{\Lambda} %24.02 +\newcommand{\pmcexists}{\text{\raisebox{.5em}{\rotatebox{180}{\textbf{E}}}}\hspace{-.1em}\mathop{\pmshr}} %24.03 + +%Relation signs +\newcommand{\pmrel}[3]{\pmhat{#1}\pmhat{#2}#3} %21.01 +\newcommand{\pmrele}[5]{#1\{\pmhat{#2}\pmhat{#3}#4(#2, #3)\}#5} %21.02 +\newcommand{\pmrelep}[3]{#1\{#2\}#3} %21.08, 21.081, 21.082, etc. +\newcommand{\pmrcmp}[1]{\ooalign{$\hidewidth\raisebox{.25em}{$\boldsymbol{\cdot}$}\hidewidth$\cr$\boldsymbol{\pmccmp}$}#1} %23.04 +\newcommand{\pmrmin}[2]{#1\mathrel{\ooalign{$\hidewidth\raisebox{.25em}{$\boldsymbol{\cdot}$}\hidewidth$\cr$\boldsymbol{\pmccmp}$}}#2} %23.05 +\newcommand{\pmruni}{\pmcirc{\text{\rotatebox[origin=c]{180}{$\Lambda$}}}} %25.01 +\newcommand{\pmrnull}{\pmcirc{\Lambda}} %25.02 +\newcommand{\pmrexists}{\pmcirc{\mathop{\text{\raisebox{.5em}{\rotatebox{180}{E}}}}}\mathop{\pmshr}} %25.03 +\newcommand{\pmrinc}{\mathrel{\ooalign{$\hidewidth\boldsymbol{\cdot}\hidewidth$\cr$\boldsymbol{\pmcinc}$}}} %23.01 +\newcommand{\pmrcap}{\mathrel{\ooalign{$\hidewidth\raisebox{.3em}{$\boldsymbol{\cdot}$}\hidewidth$\cr$\boldsymbol{\pmccap}$}}} %23.02 +\newcommand{\pmrcup}{\mathrel{\ooalign{$\hidewidth\raisebox{.1em}{$\boldsymbol{\cdot}$}\hidewidth$\cr$\boldsymbol{\pmccup}$}}} %23.03 + +%Logic of Relations +\newcommand{\pmdscf}[2]{#1\textbf{`}#2} %30.01 +\newcommand{\pmcnv}[1]{\text{Cnv}\textbf{`}#1} %31.01 +\newcommand{\pmCnv}{\text{Cnv}} +\newcommand{\pmcrel}[1]{\pmbreve{#1}} %31.02 +\newcommand{\pmrrf}[2]{\overset{\boldsymbol{\rightarrow}}{#1\textbf{`}}#2} %32.01 +\newcommand{\pmRrf}[1]{\overset{\boldsymbol{\rightarrow}}{#1}} +\newcommand{\pmrrl}[2]{\overset{\boldsymbol{\leftarrow}}{#1\textbf{`}}#2} %32.02 +\newcommand{\pmRrl}[1]{\overset{\boldsymbol{\leftarrow}}{#1}} +\newcommand{\pmsg}[1]{\text{sg}\textbf{`}#1} %32.03 +\newcommand{\pmSg}{\text{sg}} +\newcommand{\pmgs}[1]{\text{gs}\textbf{`}#1} %32.04 +\newcommand{\pmGs}{\text{gs}} +\newcommand{\pmdm}[1]{\text{D}\textbf{`}#1} %33.01 +\newcommand{\pmDm}{\text{D}} +\newcommand{\pmcdm}[1]{\text{\rotatebox[origin=c]{180}{D}}\textbf{`}#1} %33.02 +\newcommand{\pmCdm}{\text{\rotatebox[origin=c]{180}{D}}} +\newcommand{\pmcmp}[1]{C\textbf{`}#1} %33.03 +\newcommand{\pmCmp}{C} +\newcommand{\pmfld}[1]{F\textbf{`}#1} %33.04 +\newcommand{\pmFld}{F} +\newcommand{\pmrprd}[2]{{#1}\mathop{|}{#2}} %34.01 +\newcommand{\pmRprd}{\mathop{|}} +\newcommand{\pmrprdn}[2]{#1^{#2}} %34.02, 34.03, etc. +\newcommand{\pmrld}[2]{#1 \boldsymbol{\upharpoonleft} #2} %35.01 +\newcommand{\pmrlcd}[2]{#1 \boldsymbol{\upharpoonright} #2} %35.02 +\newcommand{\pmrlf}[3]{#1 \boldsymbol{\upharpoonleft} #2 \boldsymbol{\upharpoonright} #3} %35.03 +\newcommand{\pmrl}[2]{#1 \boldsymbol{\uparrow} #2} %35.04 +\newcommand{\pmrlF}[2]{#1 \mathbin{\ooalign{$\upharpoonright$\cr\hidewidth\rotatebox[origin=c]{180}{\text{$\upharpoonleft$}}\hidewidth\cr}} #2} %36.01 +\newcommand{\pmdscff}[2]{#1\textbf{`}\textbf{`}#2} %37.01 +\newcommand{\pmdscfr}[2]{#1_{\pmcin}\textbf{`}#2} %37.02 +\newcommand{\pmdscfR}[1]{#1_{\pmcin}} +\newcommand{\pmdscfcr}[2]{\pmbreve{#1}_{\pmcin}\textbf{`}#2} %37.03 +\newcommand{\pmdscfcR}[1]{\pmbreve{#1}_{\pmcin}} +\newcommand{\pmdscfff}[2]{#1\textbf{`}\textbf{`}\textbf{`}#2} %37.04 +\newcommand{\pmdscfe}[2]{\mathop{\text{E}}\mathop{\pmshr\pmshr}\pmdscff{#1}{#2}} %37.05 +\newcommand{\Female}{{\usefont{U}{mvs}{m}{n}\symbol{126}}} %from the Marvosym package +\newcommand{\pmop}{\mathop{\text{\Female}}} %38.01, 38.02 +\newcommand{\pmopc}[2]{#1 \mathop{\underset{\textbf{''}}{\text{\Female}}} #2} %38.03 + +%Products and sums of classes of classes or relations +\newcommand{\pmccsum}[1]{p\textbf{`}#1} %40.01 +\newcommand{\pmccprd}[1]{s\textbf{`}#1} %40.02 +\newcommand{\pmcrsum}[1]{\pmcirc{p}\textbf{`}#1} %41.01 +\newcommand{\pmcrprd}[1]{\pmcirc{s}\textbf{`}#1} %41.02 +\newcommand{\pmrprdd}[2]{{#1}\mathop{||}{#2}} %43.01 +\newcommand{\pmRprdd}{\mathop{||}} + +%Prolegomena to Cardinal Arithmetic +%Unit Classes and Couples +%Identity and Diversity +\newcommand{\pmrid}{I} %50.01 +\newcommand{\pmrdiv}{J} %50.02 +\newcommand{\pmcunit}[1]{\iota\textbf{`}#1} %51.01 +\newcommand{\pmcUnit}{\iota} +\newcommand{\pmcunits}[1]{\pmbreve{\iota}\textbf{`}#1} %52.01 + +%Cardinal numbers +\newcommand{\pmcn}[1]{#1} %52.01, 54.01, 54.02, etc. + +%Ordinal numbers +\newcommand{\pmoc}[2]{#1 \boldsymbol{\downarrow} #2} %55.01, 55.02, etc. +\newcommand{\pmdn}[1]{\pmcirc{#1}} %56.01 +\newcommand{\pmorn}[1]{#1_r} %56.02, 56.03, etc. + +%Sub-classes, Sub-relations, and Relative Types +%Sub-classes +\newcommand{\pmscl}[1]{\text{Cl}\textbf{`}#1} %60.01 +\newcommand{\pmsCl}{\text{Cl}} +\newcommand{\pmscle}[1]{\text{Cl ex}\textbf{`}#1} %60.02 +\newcommand{\pmsCle}{\text{Cl ex}} +\newcommand{\pmscls}[1]{\text{Cls}\textbf{`}#1} %60.03 +\newcommand{\pmsrl}[1]{\text{Rl}\textbf{`}#1} %61.01 +\newcommand{\pmsRl}{\text{Rl}} +\newcommand{\pmsrle}[1]{\text{Rl ex}\textbf{`}#1} %61.02 +\newcommand{\pmsRle}{\text{Rl ex}} +\newcommand{\pmsrel}[1]{\text{Rel}\textbf{`}#1} %61.03 +\newcommand{\pmRel}{\text{Rel}} +\newcommand{\pmReln}[1]{\text{Rel}^{#1}} %61.04 +\newcommand{\pmrin}{\mathop{\boldsymbol{\epsilon}}} %62.01 + +%Relative type symbols +\newcommand{\pmrt}[1]{t\textbf{`}#1} %63.01 +\newcommand{\pmrti}[2]{t^{#1}\textbf{`}#2} %63.011 +\newcommand{\pmrtc}[2]{t_{#1}\textbf{`}#2} %63.02, 63.03, etc. +\newcommand{\pmrtri}[2]{t^{#1}\textbf{`}#2} %63.04 +\newcommand{\pmrtrc}[2]{t_{#1}\textbf{`}#2} %64.02, 64.021, 64.022, etc. +\newcommand{\pmrtrci}[3]{t_{#1}^{\text{ }#2}\textbf{`}#3} %64.03, 64.031, etc. +\newcommand{\pmrtric}[3]{{}^{#1}t_{#2}\textbf{`}#3} %64.04, 64.041, etc. +\newcommand{\pmrtdi}[2]{#1_{#2}} %65.01 +\newcommand{\pmrtdc}[2]{#1(#2)} %65.02 +\newcommand{\pmrtdri}[2]{#1_{#2}} %65.03 +\newcommand{\pmrtdrc}[2]{#1(#2)} %65.04 + +%One-many, Many-one, and One-one relations +%Similarity relation signs +\newcommand{\pmrdc}[2]{#1\boldsymbol{\to}#2} %70.01 +\newcommand{\pmsmbar}{\mathrel{\overline{\text{sm}}}} %73.01 +\newcommand{\pmsm}{\mathrel{\text{sm}}} %73.02 +\newcommand{\pmSM}{\text{sm}} +\newcommand{\pmsmarr}{\overrightarrow{{\pmsm}}} +\newcommand{\pmonemany}{1\boldsymbol{\to}\pmCls} +\newcommand{\pmmanyone}{\pmCls\boldsymbol{\to}1} +\newcommand{\pmoneone}{1\boldsymbol{\to}1} + +%Selections +\newcommand{\pmselp}[1]{P_{\small\Delta}\boldsymbol{`}#1} %80.01 +\newcommand{\pmSelp}{P_{\Delta}} +\newcommand{\pmsele}[1]{\pmcin_{\small\Delta}\boldsymbol{`}#1} +\newcommand{\pmSele}{\pmcin_{\Delta}} +\newcommand{\pmself}[1]{F_{\small\Delta}\boldsymbol{`}#1} +\newcommand{\pmSelf}{F_{\Delta}} +\newcommand{\pmexc}{\text{Cls}^2 \mathop{\text{excl}}} %84.01 +\newcommand{\pmexcc}[1]{\text{Cl} \mathop{\text{excl}}\textbf{`}#1} %84.02 +\newcommand{\pmex}{\text{Cls excl}} +\newcommand{\pmexcn}{\text{Cls} \mathop{\text{ex}^2} \mathop{\text{excl}}} %84.03 +\newcommand{\pmselc}[2]{#1 \mathrel{\ooalign{\rotatebox[origin=c]{270}{$\boldsymbol{\mapsto}$}}} #2} +\newcommand{\pmmultr}{\mathop{\text{Rel}} \mathop{\text{Mult}}} %88.01 +\newcommand{\pmmultc}{\mathop{\text{Cls}^2} \mathop{\text{Mult}}} %88.02 +\newcommand{\pmmultax}{\mathop{\text{Mult}} \mathop{\text{ax}}} %88.03 + +%Inductive relations +\newcommand{\pmanc}[1]{#1_\pmast} %90.01 +\newcommand{\pmancc}[1]{\pmcrel{#1}_\pmast} %90.02 +\newcommand{\pmrst}[1]{#1_\text{st}} %91.01 +\newcommand{\pmrts}[1]{#1_\text{ts}} %91.02 +\newcommand{\pmpot}[1]{\text{Pot}\boldsymbol{`}#1} %91.03 +\newcommand{\pmpotid}[1]{\text{Potid}\boldsymbol{`}#1} %91.04 +\newcommand{\pmpo}[1]{#1_\text{po}} %91.05 +\newcommand{\pmB}{B} %93.01 +\newcommand{\pmmin}[1]{\text{min}_{#1}} %93.02 +\newcommand{\pmMin}{\text{min}} +\newcommand{\pmmax}[1]{\text{max}_{#1}} %93.021 +\newcommand{\pmMax}{\text{max}} +\newcommand{\pmgen}[1]{\text{gen}\boldsymbol{`}#1} %93.03 +\newcommand{\pmGen}{\text{gen}} +\newcommand{\pmefr}[2]{#1\pmast#2} %95.05 +\newcommand{\pmipr}[2]{I_{#1}\textbf{`}#2} %96.01 +\newcommand{\pmjpr}[2]{J_{#1}\textbf{`}#2} %96.02 +\newcommand{\pmfr}[2]{\overset{\boldsymbol{\leftrightarrow}}{#1}\textbf{`}#2} %97.01 + +%Volume II +%Cardinal arithmetic +%Definition and Logical Properties of Cardinal Numbers +\newcommand{\pmnc}[1]{\text{Nc}\textbf{`}#1} %100.01 +\newcommand{\pmNc}{\text{Nc}} +\newcommand{\pmNC}{\text{NC}} %100.02 +\newcommand{\pmNCat}[2]{\text{NC}^{#1}({#2})} %102.01 +\newcommand{\pmnoc}[1]{\text{N}_0\text{c}\textbf{`}#1} %103.01 +\newcommand{\pmNoc}{\text{N}_0\text{c}} +\newcommand{\pmNoC}{\text{N}_0\text{C}} %103.02 +\newcommand{\pmnca}[2]{\text{N}^{#1}\text{c}\textbf{`}#2} %104.01, 104.011, etc. +\newcommand{\pmNca}[1]{\text{N}^{#1}\text{C}} %104.02, 104.021, etc. +\newcommand{\pmch}[2]{#1^{(#2)}} %104.03, 104.031, etc. +\newcommand{\pmncd}[2]{\text{N}_{#1}\text{c}\textbf{`}#2} %105.01 +\newcommand{\pmNcd}[1]{\text{N}_{#1}\text{C}} %105.02, 105.021, etc. +\newcommand{\pmcl}[2]{#1_{(#2)}} %105.03, 105.031, etc. +\newcommand{\pmncll}[3]{\text{N}_{#1#2}\text{c}\textbf{`}#3} %106.01, 106.012, etc. +\newcommand{\pmnchh}[3]{\text{N}^{#1#2}\text{c}\textbf{`}#3} %106.011 +\newcommand{\pmncaa}[3]{\text{N}_{#1}{}^{#2}\text{c}\textbf{`}#3} %106.02 +\newcommand{\pmncdd}[3]{{}^{#1}\text{N}_{#2}\text{c}\textbf{`}#3} %106.021 +\newcommand{\pmNCll}[2]{\text{N}_{#1#2}\text{C}} %106.03 +\newcommand{\pmNChh}[2]{\text{N}^{#1#2}\text{C}} +\newcommand{\pmcll}[3]{#1_{(#2#3)}} %106.04 +\newcommand{\pmchh}[3]{#1^{(#2#3)}} %106.041 +\newcommand{\pmncr}[1]{\text{N}_{00}\text{c}\textbf{`}#1} %106.01 + +%Addition, Multiplication, Exponentiation +\newcommand{\pmarsumc}{\mathrel{+}} %110.01 +\newcommand{\pmarsumnc}{\mathrel{{+}_{\text{c}}}} %110.02 +\newcommand{\pmsmsmb}{\mathrel{\overline{\text{sm}}\;\overline{\text{sm}}}} %111.01 +\newcommand{\pmcrp}[2]{\text{Crp}(#1)\textbf{`}#2} %111.02 +\newcommand{\pmsmsm}{\mathrel{\text{sm}\;\text{sm}}} %111.03 +\newcommand{\pmarsumcc}[1]{\Sigma\textbf{`}#1} %112.01 +\newcommand{\pmarsumcnc}[1]{\Sigma\pmNc\textbf{`}#1} %112.02 +\newcommand{\pmarprodc}{\times} %113.02 +\newcommand{\pmarprodnc}{\times_\text{c}} %113.03 +\newcommand{\pmarprodcnc}[1]{\Pi\pmNc\textbf{`}#1} %114.01 +\newcommand{\pmarprodcc}[1]{\text{Prod}\textbf{`}#1} %115.01 +\newcommand{\pmarcls}{\pmClsn{3}\text{arithm}} %115.02 +\newcommand{\pmarexp}[2]{#1 \mathrel{\text{exp}} #2} %116.01 +\newcommand{\pmArexp}{\text{exp}} +\newcommand{\pmarncexp}[2]{#1^{#2}} %116.02 +\newcommand{\pmarg}{\mathrel{\boldsymbol{>}}} %117.01 +\newcommand{\pmarl}{\mathrel{\boldsymbol{<}}} %117.04 +\newcommand{\pmargeq}{\mathrel{\ooalign{$\boldsymbol{>}$\hidewidth\cr${\hspace{-.4ex}\raise-.75ex\hbox{\rotatebox[origin=c]{-155}{$\scalebox{1.1}{$\boldsymbol{-}$}$}}}$}}} %117.05 +\newcommand{\pmarleq}{\mathrel{\ooalign{$\boldsymbol{<}$\cr\hidewidth${\raise-.75ex\hbox{\rotatebox[origin=c]{155}{$\scalebox{1.1}{$\boldsymbol{-}$}$}}}\hspace{-.375ex}$}}} %117.06 + +%Finite and infinite +\newcommand{\pmarsubt}[2]{#1 \mathrel{{-}_\text{c}} #2} %119.01 +\newcommand{\pmArsubt}{{-}_\text{c}} +\newcommand{\pmNCinduct}{\text{NC}\,\text{induct}} %120.01 +\newcommand{\pmncinduct}[1]{\text{N}_#1\text{C}\,\text{induct}} %120.011 +\newcommand{\pmClsinduct}{\text{Cls}\,\text{induct}} %120.02 +\newcommand{\pmclsinduct}[1]{\text{Cls}_{#1}\,\text{induct}} %120.021 +\newcommand{\pmInfinax}{\text{Infin}\,\text{ax}} %120.03 +\newcommand{\pminfinax}[1]{\text{Infin}\,\text{ax}(#1)} %120.04 +\newcommand{\pmspec}[1]{\text{spec}\textbf{`}#1} %120.43 +\newcommand{\pmintoo}[2]{P(#1\mathbin{\boldsymbol{-}}#2)} %121.01 +\newcommand{\pmintoc}[2]{P({#1}\mathbin{\scalebox{1.2}[.7]{$\boldsymbol{\dashv}$}}{#2})} %121.011 +\newcommand{\pmintco}[2]{P({#1}\mathbin{\scalebox{1.2}[.7]{$\boldsymbol{\vdash}$}}{#2})} %121.012 +\newcommand{\pmintcc}[2]{P({#1} \mathbin{\ooalign{$\scalebox{1.2}[.7]{$\boldsymbol{\dashv}$}$\hidewidth\cr$\scalebox{1.2}[.7]{$\boldsymbol{\vdash}$}$}} {#2})} %121.013 +\newcommand{\pmintnc}[1]{P_{#1}} %121.02 +\newcommand{\pmfinid}[1]{\text{finid}\textbf{`}#1} %121.03 +\newcommand{\pmfin}[1]{\text{fin}\textbf{`}#1} %121.031 +\newcommand{\pmintt}[2]{#1_{#2}} %121.04 +\newcommand{\pmprog}{\text{Prog}} %122.01 +\newcommand{\pmaleph}{\boldsymbol{\aleph}} %123.01 +\newcommand{\pmsucc}{\text{N}} %123.02 +\newcommand{\pmclsrefl}{\text{Cls}\;\text{refl}} %124.01 +\newcommand{\pmncrefl}{\text{NC}\;\text{refl}} %124.02 +\newcommand{\pmncmult}{\text{NC}\;\text{mult}} %124.03 +\newcommand{\pmncind}{\text{NC}\;\text{ind}} %126.01 +\newcommand{\pmnocind}[1]{\text{N}_0\text{Cinduct}\textbf{`}#1} +\newcommand{\pmNocind}{\text{N}_0\text{Cinduct}} + +%Relation-arithmetic +%Ordinal similarity and relation-numbers +\newcommand{\pmrnsm}[2]{{#1}{\raise.4ex\hbox{\textbf{\large;}}}{#2}} %150.01 +\newcommand{\pmrnsmd}[2]{#1 \mathop{\boldsymbol{\dagger}} #2} %150.02 +\newcommand{\pmrnsmdf}[1]{#1\boldsymbol{\dagger}} +\newcommand{\pmopsc}[2]{#1 \mathrel{\ooalign{${\raise-.7ex\hbox{$\pmdot$}}$\hidewidth\cr$\text{\Female}$\hidewidth\cr${\raise-.8ex\hbox{\hspace{.15cm}\textbf{,}}}$}} #2} %150.03 +\newcommand{\pmsmorb}[2]{#1 \mathrel{\overline{\text{smor}}} #2} %151.01 +\newcommand{\pmSmorb}{\overline{\text{smor}}} %151.01 +\newcommand{\pmsmor}[2]{#1 \mathrel{\text{smor}} #2} %151.02 +\newcommand{\pmSmor}{\text{smor}} +\newcommand{\pmnr}[1]{\text{Nr}\textbf{`}#1} %152.01 +\newcommand{\pmNr}{\text{Nr}} +\newcommand{\pmNR}{\text{NR}} %152.02 +\newcommand{\pmsrrn}[1]{{#1}_{s}} %153.01 +\newcommand{\pmNRat}[2]{\text{NR}^{#1}({#2})} %154.01 +\newcommand{\pmnor}[1]{\text{N}_0\text{r}\textbf{`}#1} %155.01 +\newcommand{\pmNor}{\text{N}_0\text{r}} +\newcommand{\pmNoR}{\text{N}_0\text{R}} %155.02 + +%Addition of Relations, and the Product of Two Relations +\newcommand{\pmrsum}[2]{#1\mathrel{\ooalign{${\raise-.21ex\hbox{$\boldsymbol{-}$}}$\cr\hidewidth$\boldsymbol{\uparrow}$\hidewidth\cr${\raise-.19ex\hbox{$\boldsymbol{-}$}}$}} #2} %160.01 +\newcommand{\pmRsum}{\mathrel{\ooalign{${\raise-.21ex\hbox{$\boldsymbol{-}$}}$\cr\hidewidth$\boldsymbol{\uparrow}$\hidewidth\cr${\raise-.19ex\hbox{$\boldsymbol{-}$}}$}}} +\newcommand{\pmrsume}[2]{#1 \mathrel{\rotatebox[origin=c]{90}{$\pmRsum$}} #2} %161.01 +\newcommand{\pmRsume}{\rotatebox[origin=c]{90}{$\pmRsum$} } +\newcommand{\pmrsumb}[2]{#1 \mathrel{\rotatebox[origin=c]{270}{$\pmRsum$}} #2} %161.02 +\newcommand{\pmRsumb}{\rotatebox[origin=c]{270}{$\pmRsum$}} +\newcommand{\pmrsumr}[1]{\Sigma\textbf{`}#1} %162.01 +\newcommand{\pmRsumr}{\Sigma} +\newcommand{\pmrsumrex}[1]{\mathrel{\text{Rel}^{#1}\text{excl}}} %163.01 +\newcommand{\pmsmorsmorb}[2]{#1 \mathrel{\overline{\text{smor}}\,\overline{\text{smor}}} #2} %164.01 +\newcommand{\pmSmorsmorb}{\overline{\text{smor}}\,\overline{\text{smor}}} +\newcommand{\pmsmorsmor}[2]{#1 \mathrel{\pmSmor\,\pmSmor} #2} %164.02 +\newcommand{\pmSmorsmor}{\pmSmor\,\pmSmor} +\newcommand{\pmrprod}[2]{#1 \times #2} %166.01 + +%First differences and the multiplication and exponentiation of relations +%On the relation of first differences among the sub-classes of a given class +\newcommand{\pmrfdcl}[3]{#2 \mathrel{#1_{\text{cl}}} #3} %170.01 +\newcommand{\pmRfdcl}[1]{#1_{\text{cl}}} +\newcommand{\pmrfdlc}[3]{#2 \mathrel{#1_{\text{lc}}} #3} %170.02 +\newcommand{\pmRfdlc}[1]{#1_{\text{lc}}} +\newcommand{\pmrfddf}[3]{#2 \mathrel{#1_{\text{df}}} #3} %171.01 +\newcommand{\pmRfddf}[1]{#1_{\text{df}}} +\newcommand{\pmrfdfd}[3]{#2 \mathrel{#1_{\text{fd}}} #3} %171.02 +\newcommand{\pmRfdfd}[1]{#1_{\text{fd}}} +\newcommand{\pmrfprod}[1]{\Pi\textbf{`}#1} %172.01 +\newcommand{\pmRfprod}[1]{\text{Prod}\textbf{`}#1} %173.01 +\newcommand{\pmrarrel}[1]{\mathrel{\text{Rel}^{#1}\text{arithm}}} %174.01 +\newcommand{\pmrexp}{\mathrel{\text{exp}}} %176.01 +\newcommand{\pmRexp}[2]{{#1}^{#2}} %176.02 +\newcommand{\pmrnsum}[2]{{#1} + {#2}} %180.01 +\newcommand{\pmRnsum}{+} +\newcommand{\pmrndsum}[2]{{#1} \mathrel{\pmcirc{+}} {#2}} %180.02 +\newcommand{\pmRndsum}{\pmcirc{+}} +\newcommand{\pmrnsumru}[2]{#1 \mathrel{\pmcirc{\pmRsumb}} #2} %181.01 +\newcommand{\pmRnsumru}{\pmcirc{\pmRsumb}} +\newcommand{\pmrnsumur}[2]{#1 \mathrel{\pmcirc{\pmRsume}} #2} %181.011 +\newcommand{\pmRnsumur}{\pmcirc{\pmRsume}} +\newcommand{\pmrn}[1]{\pmcirc{#1}} %181.02 +\newcommand{\pmrsep}[1]{\ooalign{${\raise1.5ex\hbox{\rotatebox[origin=c]{180}{\scalebox{1.4}[1.4]{$\pmbreve{\phantom{.}}$}}}}$\cr\hidewidth$#1$\hidewidth}} %182.01 +\newcommand{\pmrnsumf}[1]{\Sigma\pmNr\textbf{`}#1} %183.01 +\newcommand{\pmrnprod}[2]{#1 \mathrel{\pmcirc{\times}} #2} %184.01 +\newcommand{\pmRnprod}{\pmcirc{\times}} +\newcommand{\pmrnprodf}[1]{\Pi\pmNr\textbf{`}#1} %185.01 +\newcommand{\pmrnexp}[3]{#2 \mathrel{\pmArexp_{#1}} #3} %186.01 +\newcommand{\pmRnexp}[1]{\pmArexp_{#1}} + +%Series +%General theory of series +\newcommand{\pmtrans}{\text{trans}} %201.01 +\newcommand{\pmconnex}{\text{connex}} %202.01 +\newcommand{\pmser}{\text{Ser}} %204.01 +\newcommand{\pmseq}[3]{#1 \mathrel{\text{seq}_{#1}} #2} %206.01 +\newcommand{\pmSeq}[1]{\text{seq}_{#1}} +\newcommand{\pmprec}[3]{#1 \mathrel{\text{prec}_{#1}} #2} %206.02 +\newcommand{\pmPrec}[1]{\text{prec}_{#1}} +\newcommand{\pmlt}[1]{\text{lt}_{#1}} %207.01 +\newcommand{\pmtl}[1]{\text{tl}_{#1}} %207.01 +\newcommand{\pmlimax}[2]{\text{limax}_{#1}\textbf{`}#2} %207.03 +\newcommand{\pmLimax}[1]{\text{limax}_{#1}} +\newcommand{\pmlimin}[2]{\text{limin}_{#1}\textbf{`}#2} %207.04 +\newcommand{\pmLimin}[1]{\text{limin}_{#1}} +\newcommand{\pmcr}[1]{\text{cr}\textbf{`}{#1}} +\newcommand{\pmCr}{\text{cr}} +\newcommand{\pmcror}[1]{\text{cror}\textbf{`}{#1}} %208.01 +\newcommand{\pmCror}{\text{cror}} + +%On sections, segments, stretches, and derivatives +\newcommand{\pmsect}[1]{\text{sect}\textbf{`}{#1}} %211.01 +\newcommand{\pmSect}{\text{sect}} +\newcommand{\pmseg}[1]{\boldsymbol{\varsigma}\textbf{`}{#1}} %212.01 +\newcommand{\pmSeg}{\boldsymbol{\varsigma}} +\newcommand{\pmsym}[1]{\text{sym}\textbf{`}{#1}} %212.02 +\newcommand{\pmSym}{\text{sym}} +\newcommand{\pmsectr}[1]{{#1}_{\pmSeg}} %213.01 +\newcommand{\pmded}{\mathrel{\text{Ded}}} %214.01 +\newcommand{\pmsded}{\mathrel{\text{semi}\;\text{Ded}}} %214.02 +\newcommand{\pmstr}[1]{\text{str}\textbf{`}{#1}} %215.01 +\newcommand{\pmStr}{\text{str}} +\newcommand{\pmder}[2]{\delta_{#1}\textbf{`}#2} %216.01 +\newcommand{\pmDer}[1]{\delta_{#1}} +\newcommand{\pmdern}[3]{\delta_{#1}^{#2}\textbf{`}#3} +\newcommand{\pmden}[1]{\text{dense}\textbf{`}{#1}} %216.02 +\newcommand{\pmDen}{\text{dense}} +\newcommand{\pmclsd}[1]{\text{closed}\textbf{`}{#1}} %216.03 +\newcommand{\pmClsd}{\text{closed}} +\newcommand{\pmperf}[1]{\text{perf}\textbf{`}{#1}} %216.04 +\newcommand{\pmPerf}{\text{perf}} +\newcommand{\pmders}[1]{\rotatebox[origin=c]{180}{$\Delta$}\textbf{`}#1} %216.05 +\newcommand{\pmDers}{\rotatebox[origin=c]{180}{$\Delta$}} + +%On convergence, and the limits of functions +\newcommand{\pmconv}[3]{#1\bar{#2}_{\text{cn}}#3} %230.01 +\newcommand{\pmConv}[1]{{#1}_{\text{cn}}} %230.02 +\newcommand{\pmconvg}[3]{#1\bar{#2}_{\text{cng}}#3} +\newcommand{\pmConvg}[1]{{#1}_{\text{cng}}} +\newcommand{\pmlsc}[3]{#1\bar{#2}_{\text{sc}}#3} %231.01 +\newcommand{\pmosc}[3]{#1\bar{#2}_{\text{os}}#3} %231.02 +\newcommand{\pmlscl}[4]{(#1\bar{#2}#3)_{\text{sc}}\textbf{`}#4} %232.01 +\newcommand{\pmoscl}[4]{(#1\bar{#2}#3)_{\text{os}}\textbf{`}#4} %232.02 +\newcommand{\pmlmx}[4]{(#1\bar{#2}#3)_{\text{lmx}}\textbf{`}#4} %233.01 +\newcommand{\pmLmx}[3]{(#1\bar{#2}#3)_{\text{lmx}}} +\newcommand{\pmlimf}[4]{#1(#2#3)\textbf{`}#4} %233.02 +\newcommand{\pmLimf}[3]{#1(#2#3)} +\newcommand{\pmscf}[3]{\text{sc}(#1, #2)\boldsymbol{`}#3} %234.01 +\newcommand{\pmosf}[3]{\text{os}(#1, #2)\boldsymbol{`}#3} %234.02 +\newcommand{\pmctf}[3]{\text{ct}(#1#2)\boldsymbol{`}#3} %234.03 +\newcommand{\pmcontinf}[3]{\text{contin}(#1#2)\boldsymbol{`}#3} %234.04 +\newcommand{\pmcontin}[2]{#1 \mathrel{\text{contin}} #2} %234.05 +\newcommand{\pmContin}{\text{contin}} + +%Volume III +%Well-Ordered Series +\newcommand{\pmbord}{\text{Bord}} %250.01 +\newcommand{\pmword}{\Omega} %250.02 +\newcommand{\pmordn}{\text{NO}} %251.01 +\newcommand{\pmless}{\mathrel{\text{less}}} %254.01 +\newcommand{\pmLess}{\text{less}} +\newcommand{\pmpsc}[2]{#1 \mathrel{P_{\text{sm}}} #2} %254.02 +\newcommand{\pmPsc}{P_{\text{sm}}} +\newcommand{\pmorle}{\mathrel{\ooalign{$\boldsymbol{<}$\cr\hidewidth$\boldsymbol{\cdot}$}}} %255.01 +\newcommand{\pmorgr}{\mathrel{\ooalign{$\boldsymbol{>}$\hidewidth\cr$\boldsymbol{\cdot}$\hidewidth}}} %255.02 +\newcommand{\pmnoo}{\text{N}_0\text{O}} %255.03 +\newcommand{\pmorleq}{\mathrel{\ooalign{$\boldsymbol{<}$\cr\hidewidth$\boldsymbol{\cdot}$\cr\hidewidth${\raise-.75ex\hbox{\rotatebox[origin=c]{155}{$\scalebox{1.1}{$\boldsymbol{-}$}$}}}\hspace{-.375ex}$}}} %255.04 +\newcommand{\pmorgrq}{\mathrel{\ooalign{$\boldsymbol{>}$\hidewidth\cr$\boldsymbol{\cdot}$\hidewidth\cr${\hspace{-.4ex}\raise-.75ex\hbox{\rotatebox[origin=c]{-155}{$\scalebox{1.1}{$\boldsymbol{-}$}$}}}$\hidewidth}}} %255.05 +\newcommand{\pmm}{\emph{M}} %256.01 +\newcommand{\pmn}{\emph{N}} %256.02, 263.02 +\newcommand{\pmtranc}[3]{(#1\pmast#2)\textbf{`}#3} %257.01 +\newcommand{\pmTranc}[2]{(#1\pmast#2)} %257.01 +\newcommand{\pmtrpot}[3]{#1_{#2#3}} %257.02 +\newcommand{\pma}{\emph{A}} %259.01 +\newcommand{\pmaw}{\emph{A}_{\emph{W}}} %259.02 +\newcommand{\pmwa}{\emph{W}_{\emph{A}}} %259.03 + +%Finite and Infinite Series and Ordinals +\newcommand{\pmintf}{P_{\text{fn}}} %260.01 +\newcommand{\pmserinf}{\text{Ser infin}} %261.01 +\newcommand{\pmwordinf}{\pmword\text{ infin}} %261.02 +\newcommand{\pmserfin}{\text{Ser fin}} %261.03 +\newcommand{\pmwordfin}{\pmword\text{ fin}} %261.04 +\newcommand{\pmwordind}{\pmword\text{ induct}} %261.04 +\newcommand{\pmordnfin}{\text{NO fin}} %262.01 +\newcommand{\pmordninf}{\text{NO infin}} %262.02 +\newcommand{\pmfinord}[1]{#1_r} %262.03 +\newcommand{\pmom}{\boldsymbol{\omega}} %263.01 +\newcommand{\pmpr}[1]{#1_{\text{pr}}} %264.01 +\newcommand{\pmomn}[1]{\pmom_{#1}} %265.01, 265.03, etc. +\newcommand{\pmalephn}[1]{\pmaleph_{#1}} %265.02, 265.04, etc. + +%Compact series, rational series, and continuous series +\newcommand{\pmcomp}{\mathrel{\text{comp}}} %270.01 +\newcommand{\pmComp}{\text{Comp}} +\newcommand{\pmmed}{\mathrel{\text{med}}} %271.01 +\newcommand{\pmMed}{\text{med}} +\newcommand{\pmsimp}[3]{\mathrel{#1_{#2#3}}} %272.01 +\newcommand{\pmsimps}[3]{{#1}_{#2}\textbf{`}{#3}} %273.02 +\newcommand{\pmSimp}[3]{({#1}{#2})_{#3}} %273.03 +\newcommand{\pmSimps}[2]{{#1}_{#2}} %273.04 +\newcommand{\pmrats}{\eta} %273.01 +\newcommand{\pmsfcls}[1]{#1_\pmrats} %274.01 +\newcommand{\pmsfclsm}[2]{#1_m\textbf{`}#2} %274.02 +\newcommand{\pmsfclsp}[2]{\pmbreve{#1}_P\textbf{`}{#2}} %274.03 +\newcommand{\pmsfclsmp}[1]{M_P\textbf{`}{#1}} %274.04 +\newcommand{\pmcser}{\theta} %275.01 +\newcommand{\pmcsercl}[1]{#1_\pmcser} %276.01 +\newcommand{\pmcsercls}[2]{{#1}_{#2}} %276.04 +\newcommand{\pmCsercls}[2]{{#1}_{\text{tl}}\textbf{`}{#2}} %264.05 +%Skipped some temprary definitions as repetitious + +%Quantity +%Generalization of Number +\newcommand{\pmu}{\textit{U}} %300.01 +\newcommand{\pmrnum}{\text{Rel num}} %300.02 +\newcommand{\pmrnumid}{\text{Rel num id}} %300.03 +\newcommand{\pmrpwr}[2]{#1^#2} %301.03 +\newcommand{\pmPrm}{\text{Prm}} %302.01 +\newcommand{\pmrprm}[4]{(#1,#2)\mathbin{\pmPrm_\tau}(#3,#4)} %302.02 +\newcommand{\pmprm}[4]{(#1,#2)\mathbin{\pmPrm}(#3,#4)} %302.03 +\newcommand{\pmhcf}[2]{\text{hcf}(#1,#2)} %302.04 +\newcommand{\pmHcf}{\text{hcf}} +\newcommand{\pmlcm}[2]{\text{lcm}(#1,#2)} %302.05 +\newcommand{\pmLcm}{\text{lcm}} +\newcommand{\pmrat}[2]{#1 \rotatebox[origin=c]{10}{$\boldsymbol{/}$} #2} %303.01 +\newcommand{\pmqn}[1]{#1_q} %303.02 +\newcommand{\pmqnil}{\infty_q} %303.03 +\newcommand{\pmRat}{\text{Rat}} %303.04 +\newcommand{\pmRatdef}{\text{Rat def}} %303.05 +\newcommand{\pmqnle}[2]{#1 \mathrel{\boldsymbol{<}_r} #2} %304.01 +\newcommand{\pmQnle}{\boldsymbol{<}_r} +\newcommand{\pmqnLe}{H} %304.02 +\newcommand{\pmqnlez}{H'} %304.03 +\newcommand{\pmprodsr}[2]{#1 \times_s #2} %305.01 +\newcommand{\pmProdsr}{\times_s} +\newcommand{\pmsumsr}[2]{#1 +_s #2} %306.01 +\newcommand{\pmSumsr}{+_s} +\newcommand{\pmratn}{\text{Rat}_n} %307.01 +\newcommand{\pmratg}{\text{Rat}_g} %307.011 +\newcommand{\pmratnle}[2]{#1 \mathrel{\boldsymbol{<}_n} #2} %307.02 +\newcommand{\pmRatnle}{\boldsymbol{<}_n} +\newcommand{\pmatngr}[2]{#1 \mathrel{\boldsymbol{>}_n} #2} %307.021 +\newcommand{\pmRatngr}{\boldsymbol{>}_n} +\newcommand{\pmratgle}[2]{#1 \mathrel{\boldsymbol{<}_g} #2} %307.03 +\newcommand{\pmRatgle}{\boldsymbol{<}_g} +\newcommand{\pmratggr}[2]{#1 \mathrel{\boldsymbol{>}_g} #2} %307.031 +\newcommand{\pmRatggr}{\boldsymbol{>}_g} +\newcommand{\pmratnLe}{H_n} %307.04 +\newcommand{\pmratgLe}{H_g} %307.05 +\newcommand{\pmratssub}[2]{#1 \boldsymbol{-}_s #2} %308.01 +\newcommand{\pmsumgr}[2]{#1 +_g #2} %308.02 +\newcommand{\pmprodgr}[2]{#1 \times_g #2} %309.01 +\newcommand{\pmrenp}{\Theta} %310.01 +\newcommand{\pmrenpz}{\Theta'} %310.011 +\newcommand{\pmrenn}{\Theta_n} %310.02 +\newcommand{\pmrennz}{\Theta_n'} %310.021 +\newcommand{\pmreng}{\Theta_g} %310.03 +\newcommand{\pmconc}[1]{\text{concord}(#1)} %311.01 +\newcommand{\pmConc}{\text{concord}} +\newcommand{\pmrensumc}[2]{#1 +_p #2} %311.02 +\newcommand{\pmrensub}[2]{#1 -_p #2} %312.01 +\newcommand{\pmrensuma}[2]{#1 +_a #2} %312.02 +\newcommand{\pmrenproda}[2]{#1 \times_a #2} %313.01 +\newcommand{\pmrenrsum}[2]{#1 +_r #2} %314.01 +\newcommand{\pmrenrprod}[2]{#1 \times_r #2} %314.02 +\newcommand{\Male}{{\usefont{U}{mvs}{m}{n}\symbol{124}}} %from the Marvosym package +\newcommand{\pmrenr}{\mathop{\text{\Male}}} %314.03 +\newcommand{\pmrenrssum}[2]{#1 +_\sigma #2} %314.04 +\newcommand{\pmrenrsprod}[2]{#1 \times_\sigma #2} %313.05 + +%Vector Families +\newcommand{\pmcorr}[1]{\text{cr}\textbf{`}#1} %330.01 +\newcommand{\pmabel}{\text{Abel}} %330.02 +\newcommand{\pmvfm}[1]{\text{fm}\textbf{`}#1} %330.03 +\newcommand{\pmVfm}{\text{fm}} +\newcommand{\pmvfmcl}{\textit{FM}} %330.04 +\newcommand{\pmvffb}[1]{#1_\iota} %330.05 +\newcommand{\pmconx}[1]{\text{conx}\textbf{`}#1} %331.01 +\newcommand{\pmconxfm}{\textit{FM}\text{ conx}} %331.02 +\newcommand{\pmfrep}[2]{\text{rep}_#1\textbf{`}#2} %332.01 +\newcommand{\pmfopen}[1]{#1_\partial} %333.01 +\newcommand{\pmfopennid}[1]{#1_{\iota\partial}} %333.011 +\newcommand{\pmfmap}{\textit{FM}\text{ ap}} %333.02 +\newcommand{\pmfmapconx}{\textit{FM}\text{ ap conx}} %333.03 +\newcommand{\pmtrsp}[1]{\text{trs}\textbf{`}#1} %334.01 +\newcommand{\pmfmtrs}{\textit{FM}\text{ trs}} %334.02 +\newcommand{\pmfmconnex}{\textit{FM}\text{ connex}} %334.03 +\newcommand{\pmfmsr}{\textit{FM}\text{ sr}} %334.02 +\newcommand{\pmfmasym}{\textit{FM}\text{ asym}} %334.05 +\newcommand{\pminit}[1]{\text{init}\textbf{`}#1} %335.01 +\newcommand{\pmfminit}{\textit{FM}\text{ init}} %335.02 +\newcommand{\pmvr}[1]{\textit{V}_#1} %336.01 +\newcommand{\pmvrnid}[1]{\textit{U}_#1} %336.011 +\newcommand{\pmarvs}[1]{A_{#1}} %336.02 + +%Measurement +\newcommand{\pmfmsubm}{\textit{FM}\text{ subm}} %351.01 +\newcommand{\pmvrm}[2]{#1_#2} %352.01 +\newcommand{\pmvrmg}[2]{#1_{#2\iota}} %352.02 +\newcommand{\pmfmrt}{\textit{FM}\text{ rt}} %353.01 +\newcommand{\pmfmcx}{\textit{FM}\text{ cx}} %353.02 +\newcommand{\pmfmrtcx}{\textit{FM}\text{ rt cx}} %353.03 +\newcommand{\pmfmg}[1]{#1_g} %354.01 +\newcommand{\pmrtnet}[2]{\text{cx}_#1\textbf{`}#2} %354.02 +\newcommand{\pmfmgrp}{\textit{FM}\text{ grp}} %354.03 +\newcommand{\pmrems}[2]{#1_#2} %356.01 + +%Cyclic Families +\newcommand{\pmfmcycl}{\textit{FM}\text{ cycl}} %370.01 +\newcommand{\pmcycl}[2]{#1_#2} %370.02 +\newcommand{\pmcycli}[2]{#1_#2} %370.03 +\newcommand{\pmvser}[2]{#1_#2} %371.01 +\newcommand{\pmintsecvser}[2]{#1_#2} %372.01 +\newcommand{\pmprime}{\text{Prime}} %373.01 +\newcommand{\pmsfmid}[3]{#1_{#2#3}} %373.02 +\newcommand{\pmsmltid}[2]{(#1, #2)} %373.03 +\newcommand{\pmprrt}[3]{(#1 \rotatebox[origin=c]{10}{$\boldsymbol{/}$} #2)_{#3}} %375.01 + +\title{\texttt{principia.sty}\\ A \LaTeXe \space Package for Typesetting Whitehead and Russell's \textit{Principia Mathematica} (Version 2.0)} +\author{Landon D. C. Elkind \texttt{landon.elkind@wku.edu}} \date{\today} \begin{document} @@ -110,7 +682,7 @@ 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 1.3 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. +Version 2.0 of \texttt{principia} is adequate to typeset all notations throughout Volumes I-III of \textit{Principia} and includes some minor fixes. Below are commands for Volume I. \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. @@ -125,7 +697,7 @@ To load \texttt{principia}, type \texttt{$\backslash$usepackage\{principia\}} in $\pmiddf$ & \verb|\pmiddf| & Identity for definitions (`$=$' differs in spacing). \\ $\pmdf$ & \verb|\pmdf| & Definition. Note the indentation. \\ $\pmdem$ & \verb|\pmdem| & This symbol begins a proof. \\ - $\pmsub{p}{q}$, $\pmsubb{p}{q}{r}{s}$, $\pmsubbb{p}{q}{r}{s}{t}{u}$, ... $\pmSub{\text{Add}}{p}{q}$, ... & \verb|\pmsub{p}{q}|, \verb|\pmsubb{p}{q}{r}{s}|, \verb|\pmsubbb{p}{q}| \par \hfill \verb|{r}{s}{t}{u}|, ... \verb|\pmSub{\text{Add}{p}{q}| & Substitution into theorems. Add `b's to the end of \verb|\pmsub| to increase the number of substitutions (up to four `b's). Each extra `b' adds two arguments. To substitute and specify the theorem as well, capitalize the `s' in \verb|\pmsub|. \\ + $\pmsub{p}{q}$, $\pmsubb{p}{q}{r}{s}$, $\pmsubbb{p}{q}{r}{s}{t}{u}$, ... $\pmSub{\text{Add}}{p}{q}$, ... & \verb|\pmsub{p}{q}|, \verb|\pmsubb{p}{q}{r}{s}|, \verb|\pmsubbb{p}{q}| \par \hfill \verb|{r}{s}{t}{u}|, ... \verb|\pmSub{\text{Add}}{p}{q}| & Substitution into theorems. Add `b's to the end of \verb|\pmsub| to increase the number of substitutions (up to four `b's). Each extra `b' adds two arguments. To substitute and specify the theorem as well, capitalize the `s' in \verb|\pmsub|. \\ $\pmdot$, $\pmdott$, $\pmdottt$, $\pmdotttt$, $\pmdottttt$, $\pmdotttttt$ & \verb|\pmdot|, \verb|\pmdott|, \verb|\pmdottt|, ... & Add `t's to the end of \verb|\pmdot| to increase the number of dots (up to six `t's). \\ $\pmand$, $\pmandd$, $\pmanddd$, $\pmandddd$, $\pmanddddd$, $\pmandddddd$ & \verb|\pmand|, \verb|\pmandd|, \verb|\pmanddd|, ...& Add `d's to the end of \verb|\pmand| command to increase the number of dots (up to six `d's). \\ $\pmor$ & \verb|\pmor| & Disjunction. \\ @@ -149,7 +721,105 @@ To load \texttt{principia}, type \texttt{$\backslash$usepackage\{principia\}} in \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. + $\pmexists$ & \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 $\kappa$. \end{tabular} \end{document}
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