← venn-to-topos
Main_venn_topos.tex
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\documentclass[14pt,notheorems,aspectratio=169]{beamer}
\usepackage{array,booktabs}
\usepackage{amsmath,amssymb,mathtools,amsthm}
\usepackage{mathrsfs}
\usepackage{etoolbox}
\usepackage{tikz}
\usepackage{graphicx}
\usetikzlibrary{positioning,arrows.meta,calc,shapes.geometric,fit,backgrounds}
\providecommand{\pgfsyspdfmark}[3]{}
\usepackage[utf8]{inputenc}
\usepackage{bxdpx-beamer}
\usepackage{pxjahyper}
\usepackage[backend=bibtex,style=alphabetic]{biblatex}
\addbibresource{venn-refs.bib}
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\usefonttheme{professionalfonts}
\definecolor{toposblue}{RGB}{0,72,165}
\definecolor{toposorange}{RGB}{236,120,0}
\definecolor{toposgreen}{RGB}{0,130,90}
\definecolor{toposred}{RGB}{190,45,45}
\definecolor{lightblue}{RGB}{225,238,255}
\definecolor{lightorange}{RGB}{255,240,220}
\definecolor{lightgreen}{RGB}{225,248,238}
\definecolor{lightred}{RGB}{255,229,229}
\definecolor{paper}{RGB}{250,250,247}
\definecolor{ink}{RGB}{31,35,38}
\definecolor{muted}{RGB}{112,118,125}
\definecolor{line}{RGB}{214,216,218}
% Color grammar: green = logic, orange = geometry/space. Bridges stay neutral.
\colorlet{logic}{toposgreen}
\colorlet{spacec}{toposorange}
\colorlet{toposc}{spacec}
\colorlet{bridge}{muted}
\colorlet{danger}{toposred}
\colorlet{logicLight}{lightgreen}
\colorlet{spaceLight}{lightorange}
\colorlet{bridgeLight}{line!30}
\colorlet{dangerLight}{lightred}
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\setbeamercolor{block title}{bg=line!45,fg=ink}
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\setbeamercolor{alerted text}{fg=danger}
\newenvironment{definition}[1][]{%
\begingroup
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\newcommand{\Set}{\mathbf{Set}}
\newcommand{\BA}{\mathbf{Bool}}
\newcommand{\FinSet}{\mathbf{FinSet}}
\newcommand{\FinBool}{\mathbf{FinBool}}
\newcommand{\Sub}{\mathrm{Sub}}
\newcommand{\Clop}{\mathrm{Clop}}
\newcommand{\Stone}{\mathrm{Stone}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\Sh}{\mathrm{Sh}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\sem}[1]{[\![#1]\!]}
\newcommand{\true}{\mathsf{true}}
\newcommand{\false}{\mathsf{false}}
\newcommand{\key}[1]{\textcolor{logic}{\textbf{#1}}}
\newcommand{\geom}[1]{\textcolor{spacec}{\textbf{#1}}}
\newcommand{\vocab}[1]{\textbf{#1}}
\AtBeginEnvironment{frame}{\small}
\newcommand{\onemessage}[1]{%
\vfill
\begin{center}
{\large\bfseries\color{ink}#1}
\end{center}
}
\newcommand{\propvar}[1]{%
\mathord{\text{%
\begingroup
\setlength{\fboxsep}{2.2pt}%
\fcolorbox{logic}{logicLight}{\strut #1}%
\endgroup
}}%
}
\newcount\lgtablelevel
\newcommand{\setlogicgeometrylevel}[1]{%
\lgtablelevel=4
\ifstrequal{#1}{intro}{\lgtablelevel=0}{}%
\ifstrequal{#1}{venn}{\lgtablelevel=1}{}%
\ifstrequal{#1}{stone}{\lgtablelevel=2}{}%
\ifstrequal{#1}{locale}{\lgtablelevel=3}{}%
\ifstrequal{#1}{topos}{\lgtablelevel=4}{}%
}
\newcommand{\lgrow}[3]{%
\textcolor{logic}{#2} & \textcolor{spacec}{#3} \\
}
\newcommand{\lgdivider}{\specialrule{.45pt}{.2ex}{.2ex}}
\newcommand{\logicgeometrytable}[1]{%
\begingroup
\setlogicgeometrylevel{#1}%
\vspace{-4mm}%
\normalsize
\renewcommand{\arraystretch}{0.62}
\setlength{\tabcolsep}{4pt}
\setlength{\aboverulesep}{0pt}
\setlength{\belowrulesep}{0pt}
\setlength{\extrarowheight}{0pt}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.52\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}}
\toprule
\textcolor{logic}{\textbf{論理}} & \textcolor{spacec}{\textbf{幾何}} \\
\midrule
\lgrow{1}{\textcolor{logic}{命題}}{(開)部分(集合)}
\lgrow{1}{\textcolor{logic}{論理}積}{共通部分}
\lgrow{1}{\textcolor{logic}{可能世界}}{\textcolor{spacec}{点}}
\lgrow{1}{\textcolor{logic}{論理}包含}{(\textcolor{spacec}{幾何}的)包含}
\lgrow{1}{完全性}{\textcolor{spacec}{点}の存在}
\lgdivider
\lgrow{2}{\textcolor{logic}{ブール代数}}{\textcolor{spacec}{Stone空間}}
\lgdivider
\lgrow{3}{\textcolor{logic}{直観主義}(無限)\textcolor{logic}{命題}\textcolor{logic}{論理}}{\textcolor{spacec}{Locale}(滑らかな空間)}
\lgdivider
\lgrow{4}{\textcolor{logic}{量化子}}{層}
\lgrow{4}{トポス(理論)}{{トポス}(空間)}
\bottomrule
\end{tabular}
\endgroup
}
\tikzset{
>={Latex[length=2.4mm,width=2.0mm]},
world/.style={circle,draw=spacec,fill=spaceLight,very thick,minimum size=9mm,inner sep=0pt,font=\small\bfseries},
worldgood/.style={circle,draw=bridge,fill=bridgeLight,very thick,minimum size=10mm,inner sep=0pt,font=\small\bfseries},
logicbox/.style={rounded corners=2pt,draw=logic,fill=logicLight,very thick,inner sep=7pt,align=center},
spacebox/.style={rounded corners=2pt,draw=spacec,fill=spaceLight,very thick,inner sep=7pt,align=center},
bridgebox/.style={rounded corners=2pt,draw=bridge,fill=bridgeLight,very thick,inner sep=7pt,align=center},
neutralbox/.style={rounded corners=2pt,draw=line,fill=paper,thick,inner sep=7pt,align=center},
dangerbox/.style={rounded corners=2pt,draw=danger,fill=dangerLight,very thick,inner sep=7pt,align=center},
tinylabel/.style={font=\scriptsize\color{muted},align=center},
arrow/.style={->,very thick,draw=bridge},
linearrow/.style={->,thick,draw=ink!70},
dot/.style={circle,fill=ink,inner sep=1.8pt}
}
\title[\texorpdfstring{{ベン図}から{トポス}へ}{ベン図からトポスへ}]{\texorpdfstring{{ベン図}から{トポス}へ}{ベン図からトポスへ}}
\subtitle{\textcolor{logic}{論理}$\leftrightarrow${\textcolor{spacec}{幾何}}}
\author{洞龍弥}
\institute[]{\texorpdfstring{数理{\color{spacec}空間}{トポス} 新歓 2026}{数理空間トポス 新歓 2026}}
\date[2026年5月23日]{2026年5月23日}
\begin{document}
% 1
\begin{frame}{\inserttitle}
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=center,inner sep=0pt] at (current page.center) {%
\includegraphics[width=\paperwidth,height=\paperheight]{figures/title_logic_geometry_world.jpg}%
};
\fill[paper]
(current page.south west) rectangle ([yshift=28mm]current page.south east);
\draw[line width=.35pt,draw=line]
([yshift=28mm]current page.south west) -- ([yshift=28mm]current page.south east);
\node[
anchor=south west,
align=left,
text=ink,
inner sep=0pt
] at ([xshift=9mm,yshift=7mm]current page.south west) {%
{\LARGE\bfseries\inserttitle {\hfill \small \insertsubtitle}}\\[1.3mm]
{\normalsize\insertauthor\quad \insertinstitute\quad \insertdate}%
};
\end{tikzpicture}
\end{frame}
\begin{frame}{オイラーの手紙}
\begin{columns}[T,onlytextwidth]
\begin{column}{0.68\textwidth}
これらの\textcolor{spacec}{丸い図形}，いやむしろ\\
（どのような形を与えようとも\\
かまわないのですから）
\textbf{\textcolor{spacec}{空間}}は，[...]\\
\textbf{\textcolor{logic}{論理学}}の中で自慢げに語られ，\\
多大な苦労をして証明される神秘の全てを，\\
私たちに明らかにしてくれるのです．
\end{column}
\begin{column}{0.32\textwidth}
\centering
\includegraphics[height=5cm]{euler_letters/leonhard_euler_handmann_1753.jpg}
\end{column}
\end{columns}
{\footnotesize ドイツ王女への手紙，引用者訳，強調} \hfill {\footnotesize 煽ってない?}
\end{frame}
% 2
\begin{frame}{目次}
\tableofcontents
\end{frame}
\section[ベン図]{ベン図 $\leftarrow$ 嘘つきパズル}
% 4
\begin{frame}{目次}
\tableofcontents[currentsection]
\end{frame}
% 5
% 6
\begin{frame}{{\color{logic}論理}パズル(1/2) 嘘つきは誰か？}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\foreach \x/\name/\speech in {
-4.1/A/{\color{logic}「B が嘘つき」},
0.0/B/{\color{logic}「C が嘘つき」},
4.1/C/{\color{logic}\mbox{「AもBも正直」}}
}{
\draw[very thick,draw=ink,fill=paper] (\x,1.05) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt] (\x-.52,-0.05) rectangle (\x+.52,.72);
\node[font=\large\bfseries] at (\x,.25) {\name};
\node[font=\large,align=center,text width=4.05cm] at (\x,-1.18) {\speech};
}
\end{tikzpicture}
\end{center}
\end{frame}
% 7
\begin{frame}{{\color{logic}論理}パズル(2/2) {\color{spacec}ベン図}で``{\color{spacec}視覚的}に"理解する}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\foreach \x/\name/\speech in {
-4.1/A/{\color{logic}「B は嘘つき」},
0.0/B/{\color{logic}「C は嘘つき」},
% 4.1/C/{\color{logic}「Bが正直ならAも正直」}
4.1/C/{\color{logic}\mbox{「AもBも正直」}}
}{
\draw[very thick,draw=ink,fill=paper] (\x,1.05) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt] (\x-.52,-0.05) rectangle (\x+.52,.72);
\node[font=\large\bfseries] at (\x,.25) {\name};
\node[font=\large,align=center,text width=4.05cm] at (\x,-1.18) {\speech};
}
\end{tikzpicture}
\begin{tikzpicture}[scale=.88]
\begin{scope}[xshift=-4.15cm]
% \node[font=\bfseries\color{spacec}] at (0,1.22) {A $\leftrightarrow$ Bが嘘};
\draw[very thick,rounded corners=2pt,draw=line,fill=white] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (-.38,.12) circle (.72);
\fill[spaceLight] (.38,.12) circle (.72);
\begin{scope}
\clip (-.38,.12) circle (.72);
\fill[white] (.38,.12) circle (.72);
\end{scope}
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\begin{scope}
% \node[font=\bfseries\color{spacec}] at (0,1.22) {B $\leftrightarrow$ Cが嘘};
\draw[very thick,rounded corners=2pt,draw=line,fill=white] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (.38,.12) circle (.72);
\fill[spaceLight] (0,-.48) circle (.72);
\begin{scope}
\clip (.38,.12) circle (.72);
\fill[white] (0,-.48) circle (.72);
\end{scope}
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\begin{scope}[xshift=4.15cm]
% C <-> (A \land B)
% 背景をまず真として塗る
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
% C の中は一度抜く
\fill[white] (0,-.48) circle (.72);
% A \cap B は C の外では偽にしたいので抜く
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[white] (.38,.12) circle (.72); % A \cap B を白く抜く
\end{scope}
% C の中では A \cap B だけ真なので塗り戻す
\begin{scope}
\clip (0,-.48) circle (.72); % C の中
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[spaceLight] (.38,.12) circle (.72); % A \cap B \cap C
\end{scope}
\end{scope}
% 枠線
\draw[very thick,rounded corners=2pt,draw=line]
(-1.45,-1.18) rectangle (1.45,1.02);
% 円の輪郭
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
% ラベル
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\end{tikzpicture}
\end{center}
% \hfill\vocab{答え: AとCが本当，Bが嘘}
\begin{center}
\vocab{答え: AとCが嘘，Bが正直}
\end{center}
\end{frame}
% 8
\begin{frame}{視点：{\color{logic}可能世界}$\leftrightarrow${\color{spacec}点}}
\textcolor{logic}{可能世界}の\textcolor{spacec}{配置空間}
% \begin{center}
% {\large\bfseries
% \textcolor{logic}{可能世界} $\leftrightarrow$ \textcolor{spacec}{点}
% }
% \end{center}
% \vspace{-3mm}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
% 左：人物と発言
\begin{scope}[shift={(-3.7,0)}]
\foreach \x/\name/\speech in {
-1.55/A/{\color{logic}「B は正直」},
1.55/B/{\color{logic}「A は正直」}
}{
% head
\draw[very thick,draw=ink,fill=paper]
(\x,0.72) circle (.33);
% body
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt]
(\x-.52,-0.4) rectangle (\x+.52,.38);
% name
\node[font=\large\bfseries\color{ink}] at (\x,-0.1) {\name};
% speech
\node[
font=\large\bfseries,
align=center,
text width=3.3cm,
text=logic
] at (\x,-1.05) {\speech};
}
\end{scope}
% 中央：対応の矢印
\node[
font=\Large\bfseries,
text=bridge
] at (0,0.1) {$\leftrightarrow$};
% 右：ベン図
\begin{scope}[shift={(3.7,0)},scale=1.15]
% universe
\fill[spaceLight] (-2.3,-1.25) rectangle (2.3,1.45);
% A, B を一度白で抜く
\fill[white] (-.55,.12) circle (.95);
\fill[white] (.55,.12) circle (.95);
% A \cap B を塗り戻す
\begin{scope}
\clip (-.55,.12) circle (.95);
\fill[spaceLight] (.55,.12) circle (.95);
\end{scope}
% 枠線
\draw[very thick,rounded corners=3pt,draw=line]
(-2.3,-1.25) rectangle (2.3,1.45);
% 円の輪郭
\draw[very thick,spacec] (-.55,.12) circle (.95);
\draw[very thick,spacec] (.55,.12) circle (.95);
% ラベル
\node[font=\bfseries] at (-1.15,.9) {$A$};
\node[font=\bfseries] at (1.15,.9) {$B$};
% model points
\fill[black!65!spacec] (0,.12) circle (3pt);
\node[font=\scriptsize,above] at (0,.22) {$M_1$};
\fill[black!65!spacec] (1.55,-.85) circle (3pt);
\node[font=\scriptsize,above] at (1.55,-.75) {$M_2$};
\end{scope}
\end{tikzpicture}
\end{center}
{\footnotesize(注意: 可能世界とモデルを意図的に混同しているが，そこまで悪くない嘘である．)}
\end{frame}
% 10
\begin{frame}{視点：\textcolor{logic}{矛盾} $\iff$ \textcolor{spacec}{無}}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm,scale=.98]
\foreach \x/\name/\speech in {
-4.0/A/{\color{logic}「B は嘘つき」},
0.0/B/{\color{logic}「C は嘘つき」},
4.0/C/{\color{logic}「A が正直なら B も正直」}
}{
\draw[very thick,draw=ink,fill=paper] (\x,0.72) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt]
(\x-.52,-0.4) rectangle (\x+.52,.38);
\node[font=\large\bfseries] at (\x,-0.1) {\name};
\node[font=\small,align=center,text width=3.35cm] at (\x,-1.15) {\speech};
}
\node[logic] at (7, -1.15) {矛盾};
\end{tikzpicture}
\vspace{-2mm}
\begin{tikzpicture}[scale=.86]
% A <-> not B
\begin{scope}[xshift=-4.15cm]
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (-.38,.12) circle (.72);
\fill[spaceLight] (.38,.12) circle (.72);
\begin{scope}
\clip (-.38,.12) circle (.72);
\fill[white] (.38,.12) circle (.72);
\end{scope}
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
% B <-> not C
\begin{scope}
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (.38,.12) circle (.72);
\fill[spaceLight] (0,-.48) circle (.72);
\begin{scope}
\clip (.38,.12) circle (.72);
\fill[white] (0,-.48) circle (.72);
\end{scope}
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
% C <-> (A => B)
\begin{scope}[xshift=4.15cm]
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
% C の外では反転:
% not(A => B) = A \cap B^c
% したがって C^c \cap A \cap B^c を塗る
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[white] (.38,.12) circle (.72); % B の中を抜く
\fill[white] (0,-.48) circle (.72); % C の中を抜く
\end{scope}
% C の中では A => B:
% C \cap (A^c \cup B) を塗る
\begin{scope}
\clip (0,-.48) circle (.72); % C の中
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
% 反例 A \cap B^c だけ白で抜く
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[white] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (.38,.12) circle (.72); % A \cap B を塗り戻す
\end{scope}
\end{scope}
\draw[very thick,rounded corners=2pt,draw=line]
(-1.45,-1.18) rectangle (1.45,1.02);
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
\node[spacec] at (8, 0) {無};
\end{tikzpicture}
\end{center}
\end{frame}
% 9
\begin{frame}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{venn}
\end{frame}
\section[Stone]{Stone双対$\leftarrow$ 1億円パズル}
% 11
\begin{frame}{目次}
\tableofcontents[currentsection]
\end{frame}
% 12
\begin{frame}{{\color{logic}論理}パズル: この人から1億円もらおう！}
絶対\textcolor{logic}{約束}守るマンがやってきて，言った．
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
% 人
\draw[very thick,draw=ink,fill=paper]
(-5.15,0.65) circle (.42);
\draw[very thick,draw=ink,fill=paper,rounded corners=9pt]
(-5.8,-1.2) rectangle (-4.5,0.15);
\node[font=\large\bfseries] at (-5.15,-.55) {絶};
% 吹き出し本体
\node[
draw=ink,
very thick,
fill=paper,
rounded corners=10pt,
align=center,
text width=9.2cm,
inner sep=9pt,
font=\Large\bfseries
] (speech) at (1.0,0.25)
{\textcolor{logic}{正しい}ことを言ったら1000円あげるよ，\\
けど\textcolor{logic}{間違った}ことを言ったら1000円あげない};
% 吹き出しのしっぽ
% \draw[very thick,draw=ink,fill=paper]
% (-3.55,0.05) -- (-2.95,0.22) -- (-3.45,-0.28) -- cycle;
\end{tikzpicture}
\end{center}
{\Large
どう\textcolor{logic}{発言}すれば\textbf{1億円をもらえるか？}} (メタ可)
\end{frame}
% 13
\begin{frame}{{\color{logic}論理}パズル: 普通の解法}
\textbf{答：「あなたは，私に1000円をくれず，かつ1億円もくれない．」}
\begin{center}
\begin{tikzpicture}[
x=1cm,y=1cm,
nom/.style={
rounded corners=3pt,
draw=ink,
fill=paper,
very thick,
align=center,
font=\small\bfseries,
inner sep=6pt,
minimum width=3.3cm
},
badleaf/.style={
rounded corners=3pt,
draw=danger,
fill=dangerLight,
very thick,
align=center,
font=\small\bfseries,
inner sep=6pt,
minimum width=2.7cm
},
arr/.style={->,very thick,draw=bridge}
]
\node[nom] (root) at (0,1.35) {\textcolor{logic}{発言}の\textcolor{logic}{真偽}};
\node[nom] (truecase) at (-3.9,-2.65)
{\textcolor{logic}{約束}に矛盾};
\node[nom] (falsecase) at (2.0,-0.4)
{1000円か1億円もらえる};
\node[nom] (false1000) at (0.0,-2.65)
{\textcolor{logic}{約束}に矛盾};
\node[nom] (false100m) at (4.0,-2.65)
{$1$億円 GET};
\draw[arr] (root) -- node[left,font=\small] {真} (truecase);
\draw[arr] (root) -- node[right,font=\small] {偽} (falsecase);
\draw[arr] (falsecase) -- node[left,font=\small] {$1000$円} (false1000);
\draw[arr] (falsecase) -- node[right,font=\small] {$1$億円} (false100m);
\end{tikzpicture}
\end{center}
\end{frame}
% 14
\begin{frame}{{\color{logic} 論理}の{\color{logic}代数}(1/2): \textcolor{logic}{ブール代数}の定義}
\begin{columns}
\column{0.5\textwidth}
パズルを方程式で解こう！
\begin{definition}[\textcolor{logic}{ブール代数}(雑)]
\textcolor{logic}{ブール代数}とは，足し算と掛け算ができて\footnote{環}，
\[x^2=x\]を満たすもの．
\end{definition}
\column{0.5\textwidth}
\begin{example}
[\textcolor{logic}{命題}の\textcolor{logic}{ブール代数}]
\begin{center}
\begin{tabular}{c|c}\hline
0 & $\bot$ \\\hline
$p+q$ & $p \not\leftrightarrow q$ \\\hline
1 & $\top$ \\\hline
$\times$ & $p \land q$ \\ \hline
\end{tabular}
\end{center}
\end{example}
\begin{center}
\begin{tabular}{c|c}\hline
$1+p$ & $\lnot p$ \\\hline
$1+p+q$ & $p \leftrightarrow q$ \\\hline
\end{tabular}
\end{center}
\end{columns}
\end{frame}
% 15
\begin{frame}{{{\color{logic} 論理}の{\color{logic}代数}(2/2)}: \textcolor{logic}{ブール代数}は便利}
もらえる額$n$が満たしてほしい\textcolor{logic}{条件}：
\[
\left(
\propvar{$n=10^3$}
\leftrightarrow
\propvar{\textcolor{logic}{発言}}
\right)
=
\propvar{$n=10^8$}
\]
\vspace{-10pt}
\begin{align*}
1+
\propvar{$n=10^3$}
+
\propvar{\textcolor{logic}{発言}}
&=
\propvar{$n=10^8$}
\\
\propvar{\textcolor{logic}{発言}}
&=
\propvar{$n=10^8$}
+
\propvar{$n=10^3$}
+1\\
&=
\propvar{$n=10^8$}
\leftrightarrow
\propvar{$n=10^3$}\\
&= \propvar{$n\neq10^3, 10^8$}
\end{align*}
{\footnotesize(cf. string)}
\end{frame}
% 16
\begin{frame}{\textcolor{spacec}{幾何}側：\textcolor{logic}{発言の無限選択} $\leftrightarrow$ \textcolor{spacec}{Cantor空間の点}}
\begin{center}
\begin{tikzpicture}[xscale=1.02,yscale=1.12,
bit/.style={circle,draw=ink!65,fill=paper,inner sep=0pt,minimum size=3.6mm},
chosen/.style={circle,draw=logic,fill=logicLight,very thick,inner sep=0pt,minimum size=5.0mm},
edgelabel/.style={font=\scriptsize\color{ink},inner sep=1.2pt,fill=paper,fill opacity=.94,text opacity=1}
]
\node[chosen] (r) at (0,2.45) {};
\node[chosen] (a0) at (-3.20,1.30) {};
\node[bit] (a1) at (3.20,1.30) {};
\node[chosen] (b00) at (-4.80,0.15) {};
\node[bit] (b01) at (-1.60,0.15) {};
\node[bit] (b10) at (1.60,0.15) {};
\node[bit] (b11) at (4.80,0.15) {};
\node[bit] (c000) at (-5.60,-1.00) {};
\node[chosen] (c001) at (-4.00,-1.00) {};
\node[bit] (c010) at (-2.40,-1.00) {};
\node[bit] (c011) at (-0.80,-1.00) {};
\node[bit] (c100) at (0.80,-1.00) {};
\node[bit] (c101) at (2.40,-1.00) {};
\node[bit] (c110) at (4.00,-1.00) {};
\node[bit] (c111) at (5.60,-1.00) {};
\foreach \u/\v in {r/a0,r/a1,a0/b00,a0/b01,a1/b10,a1/b11,b00/c000,b00/c001,b01/c010,b01/c011,b10/c100,b10/c101,b11/c110,b11/c111}
\draw[thick,line] (\u) -- (\v);
\draw[very thick,logic] (r) -- (a0) -- (b00) -- (c001);
% the whole infinite binary tree continues downward
\foreach \x in {-5.60,-4.00,-2.40,-0.80,0.80,2.40,4.00,5.60}{
\draw[thick,line,densely dashed] (\x,-1.00) -- (\x,-1.45);
}
% chosen cylinder continues inside the whole boundary
\draw[very thick,spacec,densely dashed] (c001) -- (-4.00,-1.45);
% chosen cylinder continues inside the whole boundary
\draw[very thick,logic,densely dashed] (c001) -- (-4.00,-1.54);
% Cantor space as the boundary/limit of the whole infinite binary tree
% upper stages match the leaves of the binary tree
% % outer guide box
% \draw[very thick,spacec,rounded corners=6pt, fill=spacec]
% (-6.10,-1.64) rectangle (6.10,-1.52);
% label on the right
\node[
anchor=west,
font=\Large\bfseries,
text=spacec
] at (6.28,-1.93) {$2^{\mathbb N}$};
% stage 3: closest to the finite leaves of the binary tree
% each small interval sits below one leaf
\draw[very thick,spacec] (-5.82,-1.66-0.5) -- (-5.38,-1.66-0.5);
\draw[very thick,spacec] (-4.22,-1.66-0.5) -- (-3.78,-1.66-0.5);
\draw[very thick,spacec] (-2.62,-1.66-0.5) -- (-2.18,-1.66-0.5);
\draw[very thick,spacec] (-1.02,-1.66-0.5) -- (-0.58,-1.66-0.5);
\draw[very thick,spacec] (0.58,-1.66-0.5) -- (1.02,-1.66-0.5);
\draw[very thick,spacec] (2.18,-1.66-0.5) -- (2.62,-1.66-0.5);
\draw[very thick,spacec] (3.78,-1.66-0.5) -- (4.22,-1.66-0.5);
\draw[very thick,spacec] (5.38,-1.66-0.5) -- (5.82,-1.66-0.5);
% stage 2
\draw[very thick,spacec] (-5.82,-1.84-0.5) -- (-3.78,-1.84-0.5);
\draw[very thick,spacec] (-2.62,-1.84-0.5) -- (-0.58,-1.84-0.5);
\draw[very thick,spacec] (0.58,-1.84-0.5) -- (2.62,-1.84-0.5);
\draw[very thick,spacec] (3.78,-1.84-0.5) -- (5.82,-1.84-0.5);
% stage 1
\draw[very thick,spacec] (-5.82,-2.02-0.5) -- (-0.58,-2.02-0.5);
\draw[very thick,spacec] (0.58,-2.02-0.5) -- (5.82,-2.02-0.5);
% stage 0: the original interval
\draw[very thick,spacec] (-5.82,-2.20-0.5) -- (5.82,-2.20-0.5);
% the point determined by the chosen infinite path
\fill[logic] (-4.00,-1.66-0.5) circle (2.4pt);
\draw[very thick,logic] (-4.00,-1.54) -- (-4.00,-1.66-0.5);
\end{tikzpicture}
\end{center}
\end{frame}
% 17
\begin{frame}{定理: Stone双対}
(\textcolor{spacec}{$2^\mathbb{N}$}のように)バラバラでパンパンな空間\footnote{totally disconnected compact Hausdorff topological spaceです，すみません}を\textcolor{spacec}{Stone空間}と呼ぶ．
\begin{center}
{\large\bfseries \textcolor{logic}{ブール代数} $\longleftrightarrow$ \textcolor{spacec}{Stone空間}}
\end{center}
\vspace{-2mm}
\begin{theorem}[Stone双対]
\textcolor{logic}{ブール代数}と\textcolor{spacec}{Stone空間}は等価\footnote{反変圏同値}．
\end{theorem}
{\footnotesize \cite{stone1936theory}, \cite{johnstone1982stone}}
\end{frame}
% 18
\begin{frame}{視点： {\color{logic}可能世界} = {\color{spacec}点}，再び}
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{.5\textwidth}
\centering
\begin{tikzpicture}[scale=1]
\node[logicbox,minimum width=3.1cm] (b) at (0,2.35) {\textcolor{logic}{ブール代数} $B$};
\node[logicbox,minimum width=3.1cm] (two) at (0,-2.3) {$2=\{\bot,\top\}$};
\draw[arrow] (b) -- node[right,font=\small\bfseries\color{logic}] {命題に真偽値を与える} (two);
\end{tikzpicture}
\end{column}
\begin{column}{.5\textwidth}
\centering
\begin{tikzpicture}[scale=1,
bit/.style={circle,draw=ink!65,fill=white,inner sep=0pt,minimum size=3.0mm},
chosen/.style={circle,draw=spacec,fill=spaceLight,very thick,inner sep=0pt,minimum size=4.2mm}
]
\node[spacebox,minimum width=2.4cm] (cantor) at (4.40,2.35) {$2^{\mathbb N}$};
\node[spacebox,minimum width=2.4cm] (one) at (4.40,-2.30) {$1$};
\draw[arrow] (one) -- node[right,font=\small\bfseries\color{spacec}] {点を選ぶ} (cantor);
\end{tikzpicture}
\end{column}
\end{columns}
\end{frame}
% 19
\begin{frame}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{stone}
\end{frame}
\section[Locale]{Locale $\leftarrow$ 繋がった空間を扱いたい }
% 20
\begin{frame}{目次}
\tableofcontents[currentsection]
\end{frame}
\begin{frame}{{\color{logic}論理}の不満，{\color{spacec}幾何}の不満}
\begin{center}
{\large\bfseries Stone双対では{\color{logic}論理}も{\color{spacec}幾何}も，全然足りない!}
\end{center}
\vspace{-2mm}
\begin{columns}[T,totalwidth=.98\textwidth]
% left: logic
\begin{column}{.48\textwidth}
\centering
\setlength{\fboxsep}{6pt}%
\fcolorbox{logic}{logicLight}{%
\begin{minipage}{.90\linewidth}
\centering
\vspace{2mm}
{\Large\bfseries\color{logic}論理の不満}
\vspace{3mm}
{\large\bfseries
量化子$\exists, \forall$がない！
% \color{logic} $\exists, \forall$
}
\vspace{2mm}
\end{minipage}}
\vspace{4mm}
{\bfseries\color{logic} $\forall x, \exists y, x=y^2$}$\leftarrow$欲しい
\end{column}
% right: geometry
\begin{column}{.48\textwidth}
\centering
\setlength{\fboxsep}{6pt}%
\fcolorbox{spacec}{spaceLight}{%
\begin{minipage}{.90\linewidth}
\centering
\vspace{2mm}
{\Large\bfseries\color{spacec}幾何の不満}
\vspace{3mm}
{\large\bfseries
空間がバラバラ！
}
\vspace{2mm}
\end{minipage}}
\vspace{4mm}
\begin{tikzpicture}[x=1cm,y=1cm,line cap=round,line join=round,scale=.95]
% outer body
\draw[very thick,spacec,fill=spaceLight]
(0,0) ellipse (1.55 and .78);
% hole
\draw[very thick,spacec,fill=paper]
(0,0) ellipse (.62 and .28);
% front outer edge
\draw[very thick,spacec]
(-1.55,0) arc[start angle=180,end angle=360,x radius=1.55,y radius=.78];
% back inner edge
\draw[thick,spacec!70]
(-.62,0) arc[start angle=180,end angle=360,x radius=.62,y radius=.28];
% % meridian curves
% \draw[thick,spacec!75]
% (-.78,.54) .. controls (-.28,.20) and (-.28,-.20) .. (-.78,-.54);
% \draw[thick,spacec!75]
% (.78,.54) .. controls (.28,.20) and (.28,-.20) .. (.78,-.54);
\end{tikzpicture}
$\leftarrow$欲しい
\end{column}
\end{columns}
\end{frame}
% 22
\begin{frame}{{\color{spacec}幾何}の不満: 空間がバラバラすぎる！}
\begin{center}
\begin{tikzpicture}[scale=.98]
\node[logicbox,minimum width=3.8cm,minimum height=1.35cm,
font=\Large\bfseries\color{logic}] (logic) at (-3.9,0.35)
{\textcolor{logic}{直観主義}\textcolor{logic}{論理}};
\node[spacebox,minimum width=4.6cm,minimum height=1.35cm,
font=\Large\bfseries\color{spacec}] (smooth) at (3.8,0.35)
{繋がった空間};
\draw[arrow] (smooth.west) -- node[above,font=\small\bfseries\color{bridge}]{} (logic.east);
\end{tikzpicture}
\end{center}
\begin{center}
{\large\bfseries \textcolor{logic}{排中律$p\lor \lnot p$}が，\textcolor{spacec}{空間}をバラバラにする}
\end{center}
\vspace{-1mm}
\begin{center}
\begin{tikzpicture}[scale=1.0]
\draw[->,very thick,spacec] (-4.8,0) -- (4.8,0) node[right] {$\mathbb{R}$};
\draw[very thick,spacec] (0.15,0) -- (4.2,0);
\draw[very thick,spacec] (-4.2,0) -- (-0.15,0);
\fill[paper,draw=danger,very thick] (0,0) circle (4pt);
\node[font=\large\color{logic}] at (2.2,0.55) {$\lnot U$};
\node[font=\large\color{logic}] at (-2.3,0.55) {$U$};
\node[font=\large\bfseries] at (0,-1.45) {$U\lor\neg U=\mathbb{R}\setminus\{0\}$};
\end{tikzpicture}
\end{center}
\end{frame}
% 23
\begin{frame}{空間を，\textcolor{spacec}{開集合}=\textcolor{logic}{命題}で再定義する}
\vspace{-5mm}
\begin{columns}[T,onlytextwidth]
\begin{column}{.58\textwidth}
\begin{definition}[\textcolor{spacec}{Locale}]
\textcolor{spacec}{Locale}とは，(点ではなく)開集合で定義される空間
\footnote{厳密には任意 join と有限 meet を持ち，分配法則
\[
a\wedge\bigvee_i b_i=\bigvee_i(a\wedge b_i)
\]
を満たす順序構造の形式的双対．
}
\end{definition}
\textcolor{spacec}{繋がった空間}にも\textcolor{logic}{論理}が対応する
\end{column}
\begin{column}{.38\textwidth}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{顕微鏡.jpg}
\end{figure}
\end{column}
\end{columns}
\end{frame}
% 26
\begin{frame}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{locale}
\end{frame}
\section[トポス]{トポス$\leftarrow$ 量化子$\exists, \forall$を扱いたい}
% 27
\begin{frame}{目次}
\tableofcontents[currentsection]
\end{frame}
% 28
\begin{frame}{{\color{logic}量化子}と {\color{spacec}層}：コト$(2)$からモノ$(\Set)$へ!}
\begin{table}
\centering
\begin{tabular}{l|c|c|}
&\textcolor{logic}{論理}& \textcolor{spacec}{幾何}\\\hline
コト&\textcolor{logic}{命題}& \textcolor{spacec}{開集合} \\\hline
モノ&\textcolor{logic}{量化子}& \textcolor{spacec}{層} \\\hline
\end{tabular}
\end{table}
\begin{columns}[T,totalwidth=.98\textwidth]
\begin{column}{.46\textwidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\path[use as bounding box] (-2.35,-2.00) rectangle (2.35,2.00);
% \node[font=\Large\bfseries\color{logic}] at (0,1.46) {論理};
\node[font=\Large\bfseries\color{logic}] at (0,-.58)
{$\exists x\,\varphi(x), \forall x\,\varphi(x)$};
\end{tikzpicture}
\end{column}
\begin{column}{.50\textwidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm,line cap=round,line join=round]
\path[use as bounding box] (-2.55,-2.00) rectangle (2.55,2.00);
% covering space
\draw[
very thick,
spacec,
smooth,
domain=-540:540,
samples=210,
variable=\t
]
plot ({1+.9*cos(\t)}, {.85 + .0016*\t + .12*sin(\t)});
% base space: tilted S^1
\draw[very thick,spacec]
(0,-1.05) ellipse (1.15 and .42);
% % projection arrow: covering -> S^1
\draw[->,spacec]
(1.0,.32) -- (.50,-.82);
% open subset U placed upper-left of S^1
% a right-half-like open arc, drawn as a separate small curved piece
\draw[very thick,spacec]
(-1.45-0.92,-.35+1.02) .. controls (-1.05-0.92,-.08+1.02) and (-.52-0.92,-.10+1.02) .. (-.12-0.92,-.42+1.02);
% .. controls (-.52,-.72) and (-1.05,-.73) .. (-1.45,-.48);
% embedding arrow: U -> right half of S^1
\draw[->, spacec]
(-1,0.5) -- (-0.5,-.92);
% % image of U: right-half open subset of S^1
% \draw[very thick,spacec]
% (.72,-.92)
% arc[start angle=20,end angle=-70,x radius=1.15,y radius=.42];
% % small endpoint gaps to suggest openness
% \fill[paper] (.99,-.91) circle (2.4pt);
% \fill[paper] (.39,-1.45) circle (2.4pt);
\end{tikzpicture}
\end{column}
\end{columns}
\end{frame}
% % 29
% \begin{frame}{トポス}
% \begin{center}
% {\Large\bfseries \textcolor{logic}{理論} $\longleftrightarrow$ \textcolor{spacec}{トポス}}
% \end{center}
% \vspace{-1mm}
% \begin{theorem}[\textcolor{spacec}{Classifying} トポス]
% Geometric \textcolor{logic}{theory} $T$ には \textcolor{spacec}{classifying} トポス $\mathcal E_T$ が対応し，
% $\mathcal E_T$ は $T$ の\textcolor{logic}{可能世界}を分類する．
% \end{theorem}
% \end{frame}
% 29
\begin{frame}{トポスは何者なのか}
\centering
\vspace{-2mm}
\begin{tikzpicture}[x=1cm,y=1cm]
% 左：Lawvere（写真は出典不明のため公開版では外す，2026-08-22）
\node[
anchor=east,
font=\large\bfseries,
text=logic
] at (-2.4,0.95) {F. W. Lawvere};
% 右：Grothendieck（同上）
\node[
anchor=west,
font=\large\bfseries,
text=spacec
] at (2.9,-3.21) {A. Grothendieck};
% 中央上：Lawvere の発言
\node[
draw=logic,
fill=logicLight,
very thick,
rounded corners=7pt,
align=center,
text width=4cm,
inner sep=7pt,
font=\bfseries
] at (0,0.95) {%
\textcolor{logic}{トポス}は\\
\textcolor{logic}{量化子}を\\
扱える\textcolor{logic}{論理}だ
};
% 中央下：Grothendieck の発言
\node[
draw=spacec,
fill=spaceLight,
very thick,
rounded corners=7pt,
align=center,
text width=4cm,
inner sep=7pt,
font=\bfseries
] at (0+0.51,-1.20-2.01) {%
\textcolor{spacec}{トポス}は\\
繋がった\textcolor{spacec}{空間}を\\
扱える\textcolor{spacec}{幾何}だ
};
\end{tikzpicture}
\end{frame}
% 30
\begin{frame}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{topos}
\end{frame}
% 31
\begin{frame}{視点：論理と幾何の対応は奇跡なのか?}
具体計算を知れば知るほど奇跡に見えてくるが...
\[
\vcenter{\hbox{\includegraphics[height=.3\textheight]{figures/png/yamaguchi_mikan_map_icon.svg.png}}}
\quad
\setminus
\quad
\vcenter{\hbox{\includegraphics[height=.3\textheight]{figures/png/yamaguchi_bear_map_icon.svg.png}}}
\quad
=
\quad
\vcenter{\hbox{\includegraphics[height=.3\textheight]{figures/png/yamaguchi_mikan_without_bear_map_icon.svg.png}}}
\]
\vfill
{\scriptsize 出典: 国土数値情報（国土交通省）・山口県オープンデータ「熊の目撃情報」，いずれも CC BY 4.0，加工して作成}
\end{frame}
\begin{frame}[allowframebreaks]{参考文献}
\printbibliography
\end{frame}
\end{document}