From 4a45aefb431365da5e6e93ceec8e7931630682f7 Mon Sep 17 00:00:00 2001
From: Jacob Walchuk <jpw24@st-andrews.ac.uk>
Date: Mon, 14 Jul 2025 15:56:41 -0700
Subject: paper 2: now doesnt break overleaf, fixed math mode spacing

---
 papers/2.tex | 96 ++++++++++++++++++++++++++++--------------------------------
 1 file changed, 44 insertions(+), 52 deletions(-)

(limited to 'papers/2.tex')

diff --git a/papers/2.tex b/papers/2.tex
index 4fd42b0..a55ee73 100644
--- a/papers/2.tex
+++ b/papers/2.tex
@@ -20,7 +20,7 @@ supervaluationism is one such theory. Even though it elegantly accounts
 for the first order of vagueness, it becomes less elegant when
 questioned about the higher orders. To demonstrate this, I show that
 Keefe's system fails to resolve various paradoxes of higher-order
-vagueness such as the finite series paradox or the D* paradox.
+vagueness such as the finite series paradox or the $D^*$ paradox.
 Furthermore, I argue that in her attempts to accommodate the paradoxes
 by adopting a rigid hierarchy of metalanguages, Keefe invites new
 worries. Given these criticisms, it is unlikely that Keefe's theory can
@@ -40,8 +40,7 @@ a more robust theory that could tackle its higher orders.
 
 Vagueness in language refers to an indeterminate relationship between
 its terms and the world they describe.\footnote{Kit Fine,
-  \emph{Vagueness: A Global Approach} (Oxford Academic, 2020), 2-3,
-  \url{https://doi.org/10.1093/oso/9780197514955.001.0001}.} Minimally,
+  \emph{Vagueness: A Global Approach} (Oxford Academic, 2020), 2-3.} Minimally,
 a predicate is vague if it has three features: \textbf{admission of
 borderline cases} (objects to which its application is unclear),
 \textbf{a lack of known, sharp boundaries} (no clear case separating the
@@ -242,20 +241,20 @@ that both S4 and S5 principles fail.\footnote{Williamson,
 explanation.
 
 However, Williamson argues that this is not sufficient to solve the
-problem via the $D^*$ argument. He defines $D^{*}F$ as an infinite
+problem via the $D^*$ argument. He defines $D^*\hspace{-0.2em}F$ as an infinite
 conjunction $F \ \&  \ DF \ \& \ DDF \ \& \ldots \& \ D_{n}F$. Suppose
 precisifications (a), (b), and (c), where (a) admits (b), and (b) admits
 (c), but (a) does not admit (c), since admissibility is non-transitive.
-Suppose $D^{*}F$ at (a). This means that
+Suppose $D^*\hspace{-0.2em}F$ at (a). This means that
 $F, \ DF, \ DDF, \ \ldots, \ D_{n}F$ are true at (a). If $DF$ is true at
 (a), then $F$ is true at (b); if $DDF$ is true at (a), then $DF$
 is true at (b); and so on. Thus, $F, \ DF, \ DDF, \ \ldots, \ D_{n}F$ are
-all true at (b), and hence $D^{*}F$ is true at (b). The same reasoning
-applies to (c). Thus, if $D^{*}F$ is true at some precisification,
-then $D^{*}F$ is true at all precisifications. Hence, ${DD}^{*}F$ is
+all true at (b), and hence $D^*\hspace{-0.2em}F$ is true at (b). The same reasoning
+applies to (c). Thus, if $D^*\hspace{-0.2em}F$ is true at some precisification,
+then $D^*\hspace{-0.2em}F$ is true at all precisifications. Hence, ${DD}^{*}F$ is
 true at all precisifications - and by the same reasoning, so is
-${D^{*}D}^{*}F$. Therefore, the S4 principle effectively applies to
-$D^{*}$ (see diagram below).
+$D^*\hspace{-0.2em}D^*\hspace{-0.2em}F$. Therefore, the S4 principle effectively applies to
+$D^*$ (see diagram below).
 
 \begin{center}
 \includegraphics[width=3.97674in,height=4.66543in]{papers/figures/2-1.pdf}
@@ -264,33 +263,32 @@ Consequently, Williamson concludes that higher-order vagueness
 disappears.\footnote{Williamson, \emph{Vagueness}, 160.} This is
 because, for supervaluationism to succeed, each metalanguage must be
 vague. Thus, supervaluationists need a borderline case between
-$D^{*}F$ and $D^{*}{\sim}F$, namely
-${{\sim}DD}^{*}F \ \& \ {\sim}D{\sim}D^{*}F$. However, ${{\sim}DD}^{*}F$
-collapses to ${{\sim}D}^{*}F$ by modus tollens on the S4 principle. ${{\sim}D}^{*}F$ then collapses to ${D\sim D}^{*}F$, given closure of
+$D^*\hspace{-0.2em}F$ and $D^*{\sim}F$, namely
+${\sim}DD^*\hspace{-0.2em}F \ \& \ {\sim}D{\sim}D^*\hspace{-0.2em}F$. However, ${\sim}DD^*\hspace{-0.2em}F$
+collapses to ${{\sim}D}^{*}F$ by modus tollens on the S4 principle. ${\sim}D^*\hspace{-0.2em}F$ then collapses to $D {\sim}D^*\hspace{-0.2em}F$, given closure of
 D.\footnote{Patrick Greenough, ``Higher-Order Vagueness,''
   \emph{Proceedings of the Aristotelian Society, Supplementary Volumes}
-  79 (2005): 183,
-  \href{http://www.jstor.org/stable/4106939}.}
-In effect, ${{\sim}DD}^{*}F \ \& \ {\sim}D{\sim}D^{*}F$ reduces to ${D{\sim}D}^{*}F \ \& \ {\sim}D{\sim}D^{*}F$ which is a contradiction.
-Since there are no borderlines to $D^{*}F$, it is not vague.
+  79 (2005): 183.}
+In effect, ${\sim}DD^*\hspace{-0.2em}F \ \& \ {\sim}D{\sim}D^*\hspace{-0.2em}F$ reduces to $D{\sim}D^*\hspace{-0.2em}F \ \& \ {\sim}D{\sim}D^*\hspace{-0.2em}F$ which is a contradiction.
+Since there are no borderlines to $D^*\hspace{-0.2em}F$, 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
+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,
 hierarchical series of metalanguages. In this model, the vagueness of
-the $n$th-level metalanguage can only be expressed in the $(n+1)$th
+the $n^{\text{th}}$-level metalanguage can only be expressed in the $(n+1)^{\text{th}}$
 metalanguage, which is essentially richer than the nth language. She
 argues that, since there is no reason not to adopt such an infinite
 sequence, she can just stipulate that all the languages in the series
 are vague.\footnote{Keefe, \emph{Theories of Vagueness}, 202-208.}
 Greenough sketches a formalization where the object language is enriched
 with indexed D operators where each \(D_{n + 1}\) is used to express the
-vagueness of \(D_{n}\). Such formalization stops the D* paradox and
+vagueness of \(D_{n}\). Such formalization stops the $D^*$ paradox and
 ensures that a non-vague metalanguage cannot be generated.\footnote{Greenough,
   ``Higher-Order Vagueness,'' 184-186.}
 
@@ -348,8 +346,7 @@ metalanguage. However, adding another element to an infinite set would
 not alter the size of this set.\footnote{MIT OpenCourseWare,
   \emph{Session 11: Mathematics for Computer Science}, \emph{6.042J:
   Mathematics for Computer Science, Spring 2015} (Massachusetts
-  Institute of Technology, 2015),
-  \url{https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_session11.pdf}.}
+  Institute of Technology, 2015).}
 Thus, the infinite + 1 metalanguage would be on the same meta-level as
 the infinite metalanguage. Hence, the vagueness of the infinite
 metalanguage cannot be expressed and the statement `each of those
@@ -383,8 +380,7 @@ definitely false', and Saul says, `Everything Jan said is definitely
 false'. To make sense of these, we would need to place one at a higher
 level in the hierarchy. However, this does not happen in natural
 language.\footnote{Saul Kripke, ``Outline of a Theory of Truth,''
-  \emph{The Journal of Philosophy} 72, no. 19 (1975): 694-697,
-  \href{https://www.jstor.org/stable/2024634}.}
+  \emph{The Journal of Philosophy} 72, no. 19 (1975): 694-697.}
 
 Keefe might counter these natural language intuitions by arguing that
 her model is only an idealization which is not meant to exactly
@@ -405,7 +401,7 @@ language.\footnote{A full discussion of this issue is beyond the scope
   defend the account effectively.}
 
 Lastly, even though Keefe's iteration method allows her to respond to
-Williamson's D* paradox and establish that there cannot be a non-vague
+Williamson's $D^*$ paradox and establish that there cannot be a non-vague
 metalanguage, the non-vagueness of each metalanguage requires further
 borderline cases. We need \(2^{n} + 1\) categories to express the
 vagueness of the nth metalanguage. However, there is a tension between
@@ -438,8 +434,7 @@ I believe that Keefe's problems can be addressed by making the
 structure's categories dynamic. My proposal is loosely based on
 Hao-Cheng Fu's model.\footnote{Hao-Cheng Fu, ``Saving Supervaluationism
   from the Challenge of Higher-Order Vagueness Argument,'' in
-  \emph{Philosophical Logic: Current Trends in Asia} (2017), 147-152,
-  \url{https://doi.org/10.1007/978-981-10-6355-8_7}.} Fu rejects Keefe's
+  \emph{Philosophical Logic: Current Trends in Asia} (2017), 147-152.} Fu rejects Keefe's
 claim that admissibility is vague and instead claims that, when
 considering a vague predicate, we are using a well-defined set of
 precisifications (p-sets). Keefe might argue this counterintuitive since
@@ -459,8 +454,7 @@ Fu applies the AGM theory\footnote{AGM refers to the
   detail, see Carlos E. Alchourrón, Peter Gärdenfors, and David
   Makinson, ``On the Logic of Theory Change: Partial Meet Contraction
   and Revision Functions,'' \emph{The Journal of Symbolic Logic} 50, no.
-  2 (1985): 510--30,
-  \href{https://doi.org/10.2307/2274239}.}
+  2 (1985): 510--30.}
 to give a complex account of the dynamics of p-sets; however, offers
 little formalisation and does not explain how this idea could be applied
 to the challenges of HOV\footnote{Fu, ``Saving Supervaluationism from
@@ -519,18 +513,18 @@ to be indefinite, as case memberships shift.
 \subsection{Applying dynamic supervaluationism}
 
 Having formalised the view, I will now apply it to the challenges of
-HOV, starting with Williamson's D* argument. To attack the dynamic
-approach, D* could be restated as the conjunction `DA at \(t_{1}\) \& DA
+HOV, starting with Williamson's $D^*$ argument. To attack the dynamic
+approach, $D^*$ could be restated as the conjunction `DA at \(t_{1}\) \& DA
 at \(t_{2}\) \& DA at \(t_{3}\) \& \ldots{} \& DA at \(t_{n}\)'. As
-discussed in section 3, the D* argument establishes that, if D* is not
-shown to be vague, then the cases where D* is true and the cases where
-D* is false will both be ultimately definite. Hence, there will be no
-borderline cases between D* categories, which provides a sharp boundary.
+discussed in section 3, the $D^*$ argument establishes that, if $D^*$ is not
+shown to be vague, then the cases where $D^*$ is true and the cases where
+$D^*$ is false will both be ultimately definite. Hence, there will be no
+borderline cases between $D^*$ categories, which provides a sharp boundary.
 This contradicts the foundational supervaluationist claim that there are
 no sharp boundaries. However, this argument loses its force under the
 dynamic view. The dynamic framework allows us to easily account for the
-vagueness of D*. Just as in the case of any D, we need to progress in
-time to express D*'s vagueness. Thus, while D* may initially appear to
+vagueness of $D^*$. Just as in the case of any D, we need to progress in
+time to express $D^*$'s vagueness. Thus, while $D^*$ may initially appear to
 be non-vague, this is because we need to move to t + 1 to realize its
 vagueness.
 
@@ -594,7 +588,7 @@ $D_2 {\sim}F$ & = & $\{ \mathbf{d}, \mathbf{e} \}$
 \begin{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:
+The $t_{1}$ division, from the perspective of $t_{2}$ becomes:
 
 \begin{center}
 \begin{tabulary}{\textwidth}{RCL}
@@ -608,9 +602,9 @@ $D_{2}{\sim}D_{1}F\ \&\ D_{2}{\sim}D_{1}{\sim}F $ & = & $\{ \mathbf{c} \}$
 \includegraphics[width=0.77\textwidth]{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.
+accounted for since all $D_{1}$ categories have borderline cases.
 
-Now suppose that at time \(t_{3}\), he looks at the pair $\mathbf{a}$ and
+Now suppose that at time $t_{3}$, he looks at the pair $\mathbf{a}$ and
 $\mathbf{b}$. Since he cannot tell the difference, he decides that b is
 also a definite case, adjusting the p-set again.
 
@@ -735,8 +729,7 @@ footnotes for background on contextualism\footnote{Contextualism rests
   of the world's shortest people; in the second, some of the tallest.
   See Roy Sorensen, ``Vagueness,'' \textit{The Stanford Encyclopedia of
   Philosophy} (Winter 2023 Edition), ed. Edward N. Zalta and Uri
-  Nodelman,
-  \url{https://plato.stanford.edu/archives/win2023/entries/vagueness/}.}
+  Nodelman.}
 and their solution to the Sorites.\footnote{Contextualists exploit this
   idea of unstable extensions over contexts to solve the Sorites by
   accusing it of equivocating different meanings of a vague term.
@@ -745,24 +738,23 @@ and their solution to the Sorites.\footnote{Contextualists exploit this
   weak tolerance (WT), which states that when two members of a bordering
   pair are considered in the same context $C$, they will belong to the
   same extension. However, WT permits that when one member is considered
-  in context $C$ and the other in $C'$, then they might belong to a
+  in context $C$ and the other in $C^\prime$, then they might belong to a
   different extension. See Jonas Åkerman and Patrick Greenough, "Hold
   the Context Fixed---Vagueness Still Remains," in \textit{Relative
   Truth}, ed. Manuel García-Carpintero and Max Kölbel (Oxford University
-  Press, 2010), 275--76,
-  \url{https://doi.org/10.1093/acprof:oso/9780199570386.003.0016}.
+  Press, 2010), 275--76.
 
   The WT explains why the inductive premise seems to hold. If we
   consider any pair in the series, we will conclude that both members
   belong to the same extension. But this is just because we are disposed
-  to view them in the same context C. The contextualist says that, in
+  to view them in the same context $C$. The contextualist says that, in
   fact, the context will gradually change across the series. This means
   that even if we classify neighbouring terms the same at first, this
   classification will not persist throughout the series. Thus, the
   inductive premise of the sorites, such as `if $\mathbf{n}$ is short, then $\mathbf{n}+1$ is short', fails since the meaning of `short' is not the same for every
-  member $\mathbf{n}$. This is because, the shift of context $C$ into $C'$,
-  enables cases where `$\mathbf{n}$ is short' is true (in $C$) but `$\mathbf{n}+1$ is short' is false (in $C'$). See J. Åkerman, "Contextualist Theories of Vagueness,"
-  \textit{Philosophy Compass} 7 (2012): 470--75, \url{https://doi.org/10.1111/j.1747-9991.2012.00495.x}.}
+  member $\mathbf{n}$. This is because, the shift of context $C$ into $C^\prime$,
+  enables cases where `$\mathbf{n}$ is short' is true (in $C$) but `$\mathbf{n}+1$ is short' is false (in $C^\prime$). See J. Åkerman, "Contextualist Theories of Vagueness,"
+  \textit{Philosophy Compass} 7 (2012): 470--75.}
 The first point that I address is the idea that supervaluation is
 obsolete. On this view, its role at the first level could be replaced by
 the context-reminiscent p-sets. The intuitive idea is that, since shifty
@@ -774,7 +766,7 @@ the first-order divisions allow for the p-sets to shift in the first
 place. At the first stage, we implicitly categorize objects into
 positive, negative, and borderline cases. These categories are directly
 determined by the p-set, which sets out the supervaluationist truth
-conditions (i.e., DF iff true for all precisifications and so on). These
+conditions (i.e., $DF$ iff true for all precisifications and so on). These
 categorizations are provisional: they impose sharp boundaries where none
 truly exist. This tension allows for future revisions of p-sets, and
 thus for p-sets to shift. Hence, without supervaluation in the
@@ -923,7 +915,7 @@ framework.
 While Keefe's supervaluationism remains an attractive account of
 vagueness, it ultimately struggles to account for higher-order
 vagueness. Her adoption of a rigid, Tarskian infinite hierarchy may
-block Williamson's D* argument, but at the cost of disconnecting the
+block Williamson's $D^*$ argument, but at the cost of disconnecting the
 theory from natural language. Even if, as I briefly explored, she could
 respond to these problems, adopting an infinite metalanguage hierarchy
 still leaves Keefe subject to a seemingly unresolvable finite series
-- 
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