From 8c62aa64a63dfbcde88f8c796a49dee14599a25f Mon Sep 17 00:00:00 2001 From: Jacob Walchuk Date: Tue, 8 Jul 2025 02:55:30 +0100 Subject: All graphics done --- papers/2.tex | 104 ++++++++++++++++++++++++------------------------- papers/figures/2-3.pdf | Bin 0 -> 130978 bytes papers/figures/2-4.pdf | Bin 0 -> 132636 bytes papers/figures/2-5.pdf | Bin 0 -> 131474 bytes papers/figures/2-6.pdf | Bin 0 -> 132443 bytes 5 files changed, 52 insertions(+), 52 deletions(-) create mode 100644 papers/figures/2-3.pdf create mode 100644 papers/figures/2-4.pdf create mode 100644 papers/figures/2-5.pdf create mode 100644 papers/figures/2-6.pdf (limited to 'papers') 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} diff --git a/papers/figures/2-3.pdf b/papers/figures/2-3.pdf new file mode 100644 index 0000000..048f933 Binary files /dev/null and b/papers/figures/2-3.pdf differ diff --git a/papers/figures/2-4.pdf b/papers/figures/2-4.pdf new file mode 100644 index 0000000..825e86a Binary files /dev/null and b/papers/figures/2-4.pdf differ diff --git a/papers/figures/2-5.pdf b/papers/figures/2-5.pdf new file mode 100644 index 0000000..cbe7eec Binary files /dev/null and b/papers/figures/2-5.pdf differ diff --git a/papers/figures/2-6.pdf b/papers/figures/2-6.pdf new file mode 100644 index 0000000..a50acf7 Binary files /dev/null and b/papers/figures/2-6.pdf differ -- cgit v1.2.3