From 8c62aa64a63dfbcde88f8c796a49dee14599a25f Mon Sep 17 00:00:00 2001
From: Jacob Walchuk <jpw24@st-andrews.ac.uk>
Date: Tue, 8 Jul 2025 02:55:30 +0100
Subject: All graphics done

---
 papers/2.tex | 104 +++++++++++++++++++++++++++++------------------------------
 1 file changed, 52 insertions(+), 52 deletions(-)

(limited to 'papers/2.tex')

diff --git a/papers/2.tex b/papers/2.tex
index ddd3dc8..d480e1c 100644
--- a/papers/2.tex
+++ b/papers/2.tex
@@ -275,10 +275,10 @@ In effect, ${{\sim}DD}^{*}F \; \& \; {\sim}D{\sim}D^{*}F$ reduces to ${D{\sim}D}
 Since there are no borderlines to $D^{*}F$, it is not vague.
 
 Williamson offers supervaluationists a way out: to give up semantic
-closure. D* can be vague but its vagueness cannot be expressed using D
-or D*. Instead, we need a meta-language for D*, enriched with a distinct
-operator, D!. Then, to express vagueness of D!, we need a
-meta-metalanguage with D!!. Williamson remarks that the process could
+closure. $D*$ can be vague but its vagueness cannot be expressed using D
+or $D*$. Instead, we need a meta-language for $D*$, enriched with a distinct
+operator, $D!$. Then, to express vagueness of $D!$, we need a
+meta-metalanguage with $D!!$. Williamson remarks that the process could
 continue infinitely.\footnote{Williamson, \emph{Vagueness}, 160-161.}
 
 Keefe takes up this proposal and advocates adopting an infinite,
@@ -421,8 +421,9 @@ boundary, as pictured below.\footnote{Greenough, ``Higher-Order
   Vagueness,'' 180; 185-186.} Whether or not Keefe indexes her D
 operators makes no difference, there will always be an insufficient
 number of objects in the series to fill all categories.
-
+\begin{center}
 \includegraphics[width=0.925\textwidth]{papers/figures/2-2.pdf}
+\end{center}
 In conclusion, even though the rigid hierarchy in Keefe's structure
 might be defended to some extent, her appeal to an infinite hierarchy is
 fundamentally in conflict with the finite Sorites. There seems to be no
@@ -542,14 +543,14 @@ categorizations in another metalanguage, and no pair can be clearly
 ranked as `prior'.
 
 This lack of priority arises because it would be impossible to assign it
-to any particular metalanguage. Surely, the metalanguage at \emph{t}+1
-must be a metalanguage of the metalanguage at \emph{t}, since it is able
-to express facts about \emph{t}. Therefore, it is more `privileged' in
+to any particular metalanguage. Surely, the metalanguage at $t+1$
+must be a metalanguage of the metalanguage at $t$, since it is able
+to express facts about $t$. Therefore, it is more `privileged' in
 this sense. However, suppose that the p-sets evolve over time such that,
-when moving from \emph{t}+1 to \emph{t}+2, we go back to the original
-p-set from \emph{t}. Then, the \emph{t} and \emph{t}+2 metalanguages
+when moving from $t+1$ to $t+2$, we go back to the original
+p-set from $t$. Then, the $t$ and $t+2$ metalanguages
 gain their truth conditions from the same p-set. Therefore, in a sense,
-the t metalanguage becomes `prior' to the \emph{t}+1 metalanguage. This
+the t metalanguage becomes `prior' to the $t+1$ metalanguage. This
 would undermine the strict, unidirectional Tarskian hierarchy.
 
 One could further argue that we could suppose a scenario in which two
@@ -587,13 +588,9 @@ p-set.
 
 \[D_{2}{\sim}F = \{ d,e\}\]
 
-% \includegraphics[width=4.50937in,height=2.12793in]{media/image3.emf}
 \begin{center}
-  \begin{tikzpicture}
-    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
-    \end{tikzpicture}
-  \end{center}
- 
+\includegraphics[width=4.50937in,height=2.12793in]{papers/figures/2-3.pdf}
+ \end{center}
 The \(t_{1}\) division, from the perspective of \(t_{2}\) becomes:
 
 \emph{\hfill\break
@@ -602,13 +599,10 @@ The \(t_{1}\) division, from the perspective of \(t_{2}\) becomes:
 \[{\sim}D_{2}D_{1}F\ \; \& \; {\sim}D_{2}{{\sim}D}_{1}F = \{ b\}\]
 
 \[D_{2}{\sim}D_{1}F\ \&\ D_{2}{\sim}D_{1}{\sim}F = \{ c\}\]
-
-% \includegraphics[width=4.79722in,height=2.19101in]{media/image4.emf}
+\\
 \begin{center}
-  \begin{tikzpicture}
-    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
-    \end{tikzpicture}
-  \end{center}
+\includegraphics[width=4.79722in,height=2.19101in]{papers/figures/2-4.pdf}
+\end{center}
 Hence, in this part of the series, the vagueness of \(D_{1}\) is fully
 accounted for since all \(D_{1}\) categories have borderline cases.
 
@@ -622,11 +616,8 @@ also a definite case, adjusting the p-set again.
 
 \[D_{3}{\sim} F = \{ d,e\}\]
 
-% \includegraphics[width=4.46286in,height=2.2071in]{media/image5.emf}
 \begin{center}
-  \begin{tikzpicture}
-    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
-    \end{tikzpicture}
+    \includegraphics[width=4.46286in,height=2.2071in]{papers/figures/2-5.pdf}
   \end{center}
 Since $\mathbf{b}$ changed its category membership, from the perspective
 of \(t_{3}\), $\mathbf{b}$ was not a definite borderline case at
@@ -639,11 +630,10 @@ is:
 
 \[D_{3}\sim D_{2}F\ \&\ D_{3}\sim D_{2}\sim F = \{ c\}\]
 
-% \includegraphics[width=4.875in,height=2.1236in]{media/image6.emf}
+\\
+
 \begin{center}
-  \begin{tikzpicture}
-    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
-    \end{tikzpicture}
+    \includegraphics[width=4.875in,height=2.1236in]{papers/figures/2-6.pdf}
   \end{center}
 Thus, vagueness of \(D_{2}\) is accounted for.
 
@@ -861,14 +851,9 @@ cases of tall. The former is 176.1cm, and the latter is 176cm. Suppose
 you judge both of them to be tall. Now consider applying the following
 instance of conjunction introduction:
 
-%Saul is tall Jan is tall
-
-%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \(\land I\)
-
-%Saul and Jan are tall
-
 \begin{center}
-  \texttt{[TABLEAUX FORTHCOMING]}
+  $\cfrac{\text{Saul is tall} \hspace{2em} \text{Jan is tall}}
+  {\text{Saul and Jan are tall}} {\land I}$
 \end{center}
 However, if the extension of the vague predicate \emph{tall} is
 unstable, we can easily imagine a situation in which both premises are
@@ -987,29 +972,44 @@ no cases between them.
 \bigskip
 \noindent
 \begin{minipage}[t]{0.48\textwidth}
-  \noindent \textbf{Proof a:} \\
-  \begin{center}
-    \begin{tikzpicture}
-      \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}};
-      \end{tikzpicture}
+    \noindent \textbf{Proof a:} \\
+    \begin{center}
+    $\cfrac{
+        DF \hspace{2em} DF \rightarrow DDF
+    }{DDF} {\ \rightarrow \hspace{-0.25em} E}$
     \end{center}
 \end{minipage}
 \begin{minipage}[t]{0.48\textwidth}
   \noindent \textbf{Proof a:} \\
-  \begin{center}
-    \begin{tikzpicture}
-      \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}};
-      \end{tikzpicture}
+    \begin{center}
+    $\cfrac{
+        D \neg F \hspace{2em} D\neg F \rightarrow DD \neg F
+    }{DD\neg F} {\ \rightarrow \hspace{-0.25em} E}$
     \end{center}
   \end{minipage}
 \bigskip
 
 \noindent \textbf{Proof c:}
-\begin{center}
-  \begin{tikzpicture}
-    \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}};
-    \end{tikzpicture}
-  \end{center}
+  \begin{center}
+    $\cfrac{
+        \cfrac{
+        \begin{array}{c}
+        \\
+        \neg DF \rightarrow D \neg DF
+        \end{array}
+        \cfrac{
+            \neg DF \neg D \neg F
+        }{\neg DF} {\land E}
+        }{D \neg DF} {\ \rightarrow \hspace{-0.25em}E} \hspace{1em}
+        \cfrac{
+            \cfrac{\neg DF \land \neg D \neg F}{\neg D \neg F} {\land E} \hspace{1em}
+            \begin{array}{c}
+            \\
+            \neg D \neg F \rightarrow D \neg D \neg F
+            \end{array}
+        }{D \neg D \neg F}{\ \rightarrow \hspace{-0.25em}E}
+    }{D\neg D F \land D\neg D\neg F} {\land I}$
+    \end{center}
 
 %\includegraphics[width=6.26806in,height=1.61181in]{media/image9.png}
 
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cgit v1.2.3

