From b35d5a19031af2548facb80e2ad19dfb9107077b Mon Sep 17 00:00:00 2001
From: Hanaa <hanaak001@gmail.com>
Date: Tue, 15 Jul 2025 00:58:49 +0100
Subject: fixed em and en dashes, and some formatting inc adding 1919 signs for
 some equations

---
 papers/2.tex | 208 +++++++++++++++++++++++++++++------------------------------
 1 file changed, 102 insertions(+), 106 deletions(-)

(limited to 'papers/2.tex')

diff --git a/papers/2.tex b/papers/2.tex
index a55ee73..0ec12ed 100644
--- a/papers/2.tex
+++ b/papers/2.tex
@@ -7,13 +7,13 @@ University of St Andrews
 \renewcommand*{\thesubsection}{\arabic{section}.\arabic{subsection}.}
 
 \begin{quote}
-The fact that the phenomenon of vagueness can itself be vague
-- and its vagueness be vague as well - seems impossible to make sense of
-without getting a headache. This so-called higher-order vagueness makes
+The fact that the phenomenon of vagueness can itself be vague---and 
+its vagueness be vague as well---seems impossible to make sense of
+without getting a headache. This so--called higher--order vagueness makes
 theorising about vagueness a notoriously difficult task for philosophers
 of logic and language. This difficulty manifests itself in that, even if
 a theory can convincingly explain what vagueness is and how we can
-reason about it, when faced with the vagueness of the just-tamed
+reason about it, when faced with the vagueness of the just--tamed
 vagueness, it gets flooded with paradoxes and makes the initial theory
 seem implausible. In this paper, I argue that Rosanna Keefe's
 supervaluationism is one such theory. Even though it elegantly accounts
@@ -24,7 +24,7 @@ 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
-be `argued out' of these paradoxes - `finite series' in particular.
+be `argued out' of these paradoxes---`finite series' in particular.
 Instead, I argue that the theory must be substantially modified if it is
 to be salvaged, and one way to do so is by making the proposed structure
 more dynamic. I attempt to do so by sketching an outline of dynamic
@@ -54,7 +54,7 @@ unclear, classical semantics (where meaning is derived from extension),
 and hence classical logic, may not apply. Second, the \textbf{Soritical
 problem}. Consider a series of people of descending heights by 1cm. The
 first is clearly tall (200cm) and the last is clearly not (120cm). Since
-no known boundaries exist, vague predicates are tolerant - a small
+no known boundaries exist, vague predicates are tolerant---a small
 change will not alter the application. Thus, by inductive step, for any
 case $\mathbf{n}$, `if $\mathbf{n}$ is tall then $\mathbf{n}+1$ is tall'.
 Starting at 200cm is tall, via a series of conditionals, you validly
@@ -69,34 +69,33 @@ extension changes to negative, it is equally unknowable where the
 positive changes to borderline. Thus, borderline cases themselves should
 be unbounded; hence there should be borderlines to borderlines. The
 process could be iterated to establish a possibly infinite hierarchy of
-borderline cases: the higher-order vagueness (HOV).\footnote{Keefe,
+borderline cases: the higher--order vagueness (HOV).\footnote{Keefe,
   \emph{Theories of Vagueness}, 31-32.}
 
 Throughout this paper, I will follow Rosanna Keefe and other major
 supervaluationists in assuming that HOV is a genuine problem, that needs
 to be accounted for. However, it is worth pointing out that this is a
 debated matter in the field.\footnote{Some philosophers, such as Dominic
-  Hyde, claim that higher-order vagueness (HOV) is a pseudo-problem,
+  Hyde, claim that higher--order vagueness (HOV) is a pseudo--problem,
   arguing that the vagueness of vague is a real, but unproblematic,
-  phenomenon. Others, including Hao-Cheng Fu and Susanne Bobzien counter
+  phenomenon. Others, including Hao--Cheng Fu and Susanne Bobzien counter
   that this stance fails to adequately address the complexity of the
   issue, maintaining that HOV is indeed a genuine problem. While an
   extensive discussion is beyond the scope of this essay, see Hyde, "Why
-  Higher-Order Vagueness Is a Pseudo-Problem"; Fu, "Saving
+  Higher--Order Vagueness Is a Pseudo-Problem"; Fu, "Saving
   Supervaluationism from the Challenge of Higher-Order Vagueness
   Argument''; and Bobzien, "In Defense of True Higher-Order Vagueness"
   for further details.} Nevertheless, under this assumption a successful
 theory of vagueness, given its commitments, must also account for HOV.
 
-In this essay, I explore how one theory of vagueness --
-supervaluationism, advocated by Rosanna Keefe -- does so. First, I
-outline her account of first-order vagueness (FOV). Then, I explain the
+In this essay, I explore how one theory of vagueness---supervaluationism, advocated by Rosanna Keefe---does so. First, I
+outline her account of first--order vagueness (FOV). Then, I explain the
 problems posed by HOV, examining Tim Williamson's criticisms of the
 theory and how Keefe accommodates them. I will argue that although the
 Williamson problems are solved, the resulting view does not reflect how
 language actually functions and is paradoxical, making the HOV account
 unsatisfactory. I then attempt to modify the view by dynamizing it,
-developing the ideas of Hao-Cheng Fu. I defend the model by showing how
+developing the ideas of Hao--Cheng Fu. I defend the model by showing how
 it solves some of the critical issues faced by Keefe. Lastly, I raise a
 few possible issues endemic to the dynamic view and sketch responses to
 defend it.
@@ -114,7 +113,7 @@ precisification, a way to make a vague term precise.\footnote{Keefe,
 admissible, reasonable in not licensing a misuse of language.\footnote{Timothy
   Williamson, \emph{Vagueness}, (Routledge, 1994), 158.} It also must be
 complete, it categorizes objects into positive and negative extensions,
-leaving nothing in-between. For illustration, consider the vague
+leaving nothing in--between. For illustration, consider the vague
 predicate `tall'. We could (reasonably) use precisifications: `tall' is
 true if `\textgreater175cm', `\textgreater180cm' and
 `\textgreater190cm', each of which would precisely divide objects into
@@ -123,21 +122,20 @@ these; instead, all precisifications are equally good.\footnote{Keefe,
   \emph{Theories of Vagueness}, 154-156.}
 
 Supervaluationists provide semantics for vague predicates, identifying
-truth with super-truth by considering all possible precisifications.
-$F\mathbf{a}$ is super-true (-false) iff $F$ is true (false) of
+truth with super--truth by considering all possible precisifications.
+$F\mathbf{a}$ is super--true (-false) iff $F$ is true (false) of
 $\mathbf{a}$ under all complete and admissible precisifications.
 $F\mathbf{a}$ is neither true nor false iff $F$ is true of
 $\mathbf{a}$ under some precisifications and false of $\mathbf{a}$ under
 others.\footnote{Keefe, \emph{Theories of Vagueness}, 154.}
 
-Thus, vague predicates divide objects in a three-fold manner, where
+Thus, vague predicates divide objects in a three--fold manner, where
 borderline cases are not assigned a definite truth value. Hence,
 supervaluationists give up bivalence, departing from classical
 semantics, by admitting truth value gaps. On the other hand, classical
 logic is mostly preserved because if a sentence is classically true,
 then it is true on all complete and admissible precisifications.
-Consider the law of excluded middle. Using any precisification of tall
--- every object will be either tall or not-tall, since every
+Consider the law of excluded middle. Using any precisification of tall---every object will be either tall or not-tall, since every
 precisification divides objects into two sharp sets. Similarly, all
 classical theorems are retained, thus we can use classical logic to
 reason about vague predicates.\footnote{Rosanna Keefe, ``Vagueness:
@@ -146,10 +144,10 @@ reason about vague predicates.\footnote{Rosanna Keefe, ``Vagueness:
 
 This idea provides a straightforward solution to the Sorites. Namely,
 the inductive premise `if $F\mathbf{n}$ then $F(\mathbf{n}+1)$' is
-super-false, since the antecedent will be true and the consequent false
+super--false, since the antecedent will be true and the consequent false
 for some $\mathbf{n}$ under any complete and admissible precisification.
 This is because each precisification, being complete, provides a sharp
-cut-off between the true and false -- a bordering pair where the first
+cut--off between the true and false---a bordering pair where the first
 entry is true and second one is false.\footnote{Keefe, ``Vagueness:
   Supervaluationism,'' 315-316.} Thus, the supervaluationist account
 fulfils the initial demands of theorizing about vagueness. Consult the
@@ -157,7 +155,7 @@ footnote\footnote{Consider the series of people of varying heights again
   and suppose some examples of complete precisifications: $\mathbf{x}$ is
   short if (1) `$<160\text{cm}$' or (2) `$<165\text{cm}$' or (3)
   `$<170\text{cm}$'. They are complete since they divide objects into
-  positive (short) and negative (not-short) extensions with nothing
+  positive (short) and negative (not--short) extensions with nothing
   in-between. It is easy to see how the inductive premise turns out
   false on each of these precisifications: (1) `If 159cm is short, then
   160 is short'; (2) `If 164cm is short, then 165cm is short'; (3) `If
@@ -195,9 +193,9 @@ resolved by allowing the concept of `admissibility' itself to be vague,
 thus making the metalanguage vague.\footnote{Keefe, \emph{Theories of
   Vagueness}, 202-204.}
 
-Hence, the second-order vagueness of $F$ is captured in the
-meta-metalanguage by expressing vagueness of $DF$ (the metalanguage). This
-yields the following five-fold classification:
+Hence, the second--order vagueness of $F$ is captured in the
+meta--metalanguage by expressing vagueness of $DF$ (the metalanguage). This
+yields the following five--fold classification:
 
 \begin{enumerate}
 \def\labelenumi{(\arabic{enumi})}
@@ -252,14 +250,14 @@ is true at (b); and so on. Thus, $F, \ DF, \ DDF, \ \ldots, \ D_{n}F$ are
 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
+true at all precisifications---and by the same reasoning, so is
 $D^*\hspace{-0.2em}D^*\hspace{-0.2em}F$. 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}
 \end{center}
-Consequently, Williamson concludes that higher-order vagueness
+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
@@ -274,9 +272,9 @@ 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
+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
+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,
@@ -295,17 +293,17 @@ ensures that a non-vague metalanguage cannot be generated.\footnote{Greenough,
 \section{Evaluation}
 
 Even though the above account might seem abstract, its strength lies in
-its simplicity - Keefe only iterates her account of the first order to
+its simplicity---Keefe only iterates her account of the first order to
 higher orders of vagueness. In effect, the initial solutions to
 vagueness problems equally apply to HOV. Vagueness at higher orders
 remains a matter of semantic indecision: we are undecided over whether a
 precisification counts as admissible. Furthermore, each level $n$
-admits borderline cases and lacks sharp boundaries -- a fact that can be
+admits borderline cases and lacks sharp boundaries---a fact that can be
 expressed in the $n+1$ metalanguage using appropriate D operators.
 
 Moreover, each higher order metalanguage is still Sorites susceptible. I
 will explain this by running the paradox for the metalanguage (second
-order vagueness) in natural language terms for clarity - though the same
+order vagueness) in natural language terms for clarity---though the same
 could be done using D operators. The inductive premise for the
 metalanguage can be restated, in natural language, as: `if there are
 admissible precisifications that draw the boundary to `tall' at height
@@ -313,7 +311,7 @@ h, then there are ones that draw it at one-hundredth of an inch
 lower'.\footnote{Keefe, \emph{Theories of Vagueness}, 207-208.} The
 second order series could start with a clearly admissible
 precisification (e.g., taller than 190cm) and end with a clearly
-inadmissible one (e.g., taller than 110cm). Since one-hundredth of an
+inadmissible one (e.g., taller than 110cm). Since one--hundredth of an
 inch does not make a difference in admissibility, you could run a series
 of conditionals, starting with `taller than 190cm is admissible' to
 reach a conclusion that `taller than 110cm is admissible'. This is a
@@ -338,16 +336,16 @@ metalanguages for metalanguages is infinite, then what is the difficulty
 with adding `and each of those languages is vague'\,'.\footnote{Keefe,
   \emph{Theories of Vagueness}, 208.} However, there is a fundamental
 difficulty in this addition. In Keefe's system, the vagueness of an
-n-level metalanguage can only be expressed via an n+1 level
+n-level metalanguage can only be expressed via an $n+1$ level
 metalanguage. If all metalanguages are vague, then the infinite
 metalanguage would have to be vague. To express the vagueness of the
-infinite metalanguage, we would need to use the infinity + 1
+infinite metalanguage, we would need to use the infinity $+ 1$
 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).}
-Thus, the infinite + 1 metalanguage would be on the same meta-level as
+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
 languages is vague' seems meaningless.
@@ -359,7 +357,7 @@ cannot be globally quantified over.\footnote{Greenough, ``Higher-Order
 metalanguage might not be definable in her structure, it does not mean
 that it does not exist. Her structure ensures that vagueness for any
 finite level can be expressed. Even though we cannot say that `all
-metalanguages are vague', we also cannot identify any non-vague
+metalanguages are vague', we also cannot identify any non--vague
 metalanguage within the structure. Thus, even though the concept of
 infinity proves problematic for Keefe at the outset, I will assume that
 this problem does not threaten the explanatory power of her structure.
@@ -389,9 +387,9 @@ replicate how ordinary language works. While iterating `definitely'
 conversation, the model is primarily defended by its explanatory power
 regarding HOV. She could further argue that even though different levels
 of metalanguages, when expressed in natural language, might not be
-clearly marked and distinguishable (such as in the Jan-Alfred example
+clearly marked and distinguishable (such as in the Jan--Alfred example
 above), they can still function as distinct metalanguages in a formal
-framework. A further worry is that such an approach might over-idealise
+framework. A further worry is that such an approach might over--idealise
 HOV making her account arbitrary. It raises the question over whether
 speakers genuinely use implicitly distinct levels of metalanguages to
 assign levels to truth. Thus, Keefe would need to give a more robust
@@ -426,24 +424,24 @@ fundamentally in conflict with the finite Sorites. There seems to be no
 way to accommodate the problem without making strong alterations to the
 model.
 
-\section{5. Positive proposal -- dynamizing supervaluationism}
+\section{5. Positive proposal --- dynamizing supervaluationism}
 
 \subsection{5.1. Introducing dynamic supervaluationism}
 
 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
+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.} 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
+considering a vague predicate, we are using a well--defined set of
+precisifications (p--sets). Keefe might argue this counterintuitive since
 we do not know what is admissible. However, this knowledge is
-unnecessary: the p-set is created when cases are categorized as true,
+unnecessary: the p--set is created when cases are categorized as true,
 false, or borderline at time \(t_{1}\). For example, if 195cm and 190cm
-are tall, 170cm is not, and 180cm is borderline, the p-set is implicitly
+are tall, 170cm is not, and 180cm is borderline, the p--set is implicitly
 formed dividing cases into three groups, on my reading of Fu. Crucially,
-we judge first; the p-set is constructed afterward. What follows in the
+we judge first; the p--set is constructed afterward. What follows in the
 next paragraphs is my own development of the idea.
 
 Fu applies the AGM theory\footnote{AGM refers to the
@@ -455,32 +453,32 @@ Fu applies the AGM theory\footnote{AGM refers to the
   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.}
-to give a complex account of the dynamics of p-sets; however, offers
+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
   the Challenge of Higher-Order Vagueness Argument,'' 149-152.}.
 Moreover, Fu does not address the paradoxes of HOV, and it is difficult
 to see how his account could solve them. In my view, we do not need such
-an elaborate account. I propose that a p-set is dynamic solely in virtue
+an elaborate account. I propose that a p--set is dynamic solely in virtue
 of changing when a case is judged inconsistently with it. For the sake
 of clarity, consider the above example again. Imagine another person,
 \textbf{x}, who is 168cm. You judge \textbf{x} as tall. This is clearly
-inconsistent with your p-set at \(t_{1}\), since you judged 170cm as not
+inconsistent with your p--set at \(t_{1}\), since you judged 170cm as not
 tall. Thus, adding \textbf{x} to the tall category updates the \(t_{1}\)
 set to the \(t_{2}\) set with revised precisifications. This change
 occurs by either (1) expanding (adding a precisification), (2)
 contracting (removing one), or (3) both. Therefore, I retain the core
-idea of dynamic p-sets and Fu's terminology but limit the scope of the
-mechanism to a minimal principle: a p-set updates only when a judgment
+idea of dynamic p--sets and Fu's terminology but limit the scope of the
+mechanism to a minimal principle: a p--set updates only when a judgment
 is made that conflicts with it.
 
-I will now attempt to formalise the above proposed working of p-sets,
+I will now attempt to formalise the above proposed working of p--sets,
 which I will later apply to the challenges haunting supervaluationism.
 Vagueness, on the dynamic view, remains semantic indecision. At the
 first level, we follow Keefe's supervaluationism with a slight addition
 of the temporal component. While Fu does not offer a formalisation of
 his view in the spirit of Keefe's system with D operators, the following
-temporal framework develops my own way of modelling dynamic p-sets using
+temporal framework develops my own way of modelling dynamic p--sets using
 temporally indexed D operators.
 
 More precisely, at any time, t, cases divide into
@@ -491,13 +489,13 @@ from the instability of precisifications. Suppose that you make some
 categorisations at \(t_{1}\). According to the p-set that you just
 formed; some arbitrary case is classified as \(D_{1}F\). Now suppose
 that you consider the series again, but you are no longer sure about the
-definiteness of your classification. Thus, your p-set is adjusted at
+definiteness of your classification. Thus, your p--set is adjusted at
 \(t_{2}\), and according to it, the case is borderline. Therefore, from
 \(t_{2}\)'s perspective it was a borderline definite case at \(t_{1}\)
 (\({{\sim}D_{2}D}_{1}F\)).
 
 In general, when considering a borderline case after categorisation at
-\emph{t}, tolerance ensures a mis-categorisation. To see this, remember
+\emph{t}, tolerance ensures a mis--categorisation. To see this, remember
 that the supervaluation technique divides cases sharply into true,
 false, and borderline. However, tolerance guarantees that when viewing
 two neighbouring cases, we will not be able to tell the difference.
@@ -505,7 +503,7 @@ Therefore, there is a clear tension; we divided sharply, enabling a
 border pair where, for instance, one member is true and another
 borderline. However, since we cannot distinguish between neighbouring
 cases, they must be categorised equally. That means that one of the
-cases had to be categorised mistakenly and thus, the p-set must be
+cases had to be categorised mistakenly and thus, the p--set must be
 revised to maintain consistency in our judgments. When we reconsider the
 series at \(t_{2}\), the earlier categorisations from \(t_{1}\) turn out
 to be indefinite, as case memberships shift.
@@ -525,13 +523,13 @@ 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
-be non-vague, this is because we need to move to t + 1 to realize its
+be non--vague, this is because we need to move to $t + 1$ to realize its
 vagueness.
 
 Secondly, Keefe's view faced concerns about rigid hierarchies, but the
 dynamic approach eliminates these. When two speakers disagree over a
 case's definiteness, neither statement must be `prior'. They are simply
-speaking from different p-sets that underwent different evolutions.
+speaking from different p--sets that underwent different evolutions.
 There is no rigid hierarchy of metalanguages since each discusses
 categorizations in another metalanguage, and no pair can be clearly
 ranked as `prior'.
@@ -540,18 +538,18 @@ This lack of priority arises because it would be impossible to assign it
 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,
+this sense. However, suppose that the p--sets evolve over time such that,
 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,
+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 $t+1$ metalanguage. This
 would undermine the strict, unidirectional Tarskian hierarchy.
 
 One could further argue that we could suppose a scenario in which two
-identical people, A and B, undergo identical p-set evolutions. However,
+identical people, A and B, undergo identical p--set evolutions. However,
 A's evolution stops at \emph{t} and B's evolution stops at \emph{t}+1.
 On the one hand, we might be tempted to assign priority to B's
-statements, which would be counter-intuitive on the natural language
+statements, which would be counter--intuitive on the natural language
 objection. However, there is no reason to suppose that A's evolution
 should go the same way; she might consider a different part of the
 Sorites spectrum. Therefore, although the metalanguages are in some
@@ -561,7 +559,7 @@ ones made by Kripke, do not apply here.
 
 Thirdly, the dynamic view can help tackle the finite series paradox,
 which was a critical blow to Keefe's account. I will explain how it
-could achieve this through an example. Consider a 5-element Sorites with
+could achieve this through an example. Consider a 5--element Sorites with
 objects \textbf{a}, \textbf{b}, \textbf{c}, \textbf{d}, and \textbf{e}.
 Suppose that Alfred's initial categorizations are:
 
@@ -574,7 +572,7 @@ $D_{1}{\sim}F$ & = & $\{ \mathbf{d}, \mathbf{e}\}$ \\
 \end{center}
 Alfred considers the pair \textbf{b} and \textbf{c} again. He realizes
 that he cannot tell the difference, concluding that \textbf{b} is also
-borderline. He adjusts his p-set accordingly, forming a new \(t_{2}\)
+borderline. He adjusts his p--set accordingly, forming a new \(t_{2}\)
 p-set.
 
 \begin{center}
@@ -606,7 +604,7 @@ 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
 $\mathbf{b}$. Since he cannot tell the difference, he decides that b is
-also a definite case, adjusting the p-set again.
+also a definite case, adjusting the p--set again.
 
 \begin{center}
 \begin{tabulary}{\textwidth}{RCL}
@@ -671,22 +669,22 @@ from our inability to express the vagueness of some order when time is
 fixed, to the claim that there are sharp boundaries. After all, the fact
 that we cannot express it does not imply that it does not exist. This,
 however, demands further explanation of why we cannot express it. One
-response is that at a certain time, we are just using a well-defined but
+response is that at a certain time, we are just using a well--defined but
 arbitrary set of precisifications. However, this division is surely
 wrong; it is made under one of many sets of equally good
 precisifications. Thus, there is no reason to believe that the term was
-made precise -- we just have not realized our mistake yet.
+made precise---we just have not realized our mistake yet.
 
 A second and more powerful response is to deny the possibility of fixing
 time in this sense. This could supplement the above argument. Suppose
 that the critic of the view wants to prove to us that there are sharp
 boundaries. However, in order to show that there are sharp boundaries,
 they would have to find them in the series. Suppose that you manage to
-find the extension-switching pair. Even if you do this, you will
+find the extension--switching pair. Even if you do this, you will
 realize, per tolerance, that you cannot tell the difference between the
 two cases. In effect, you must conclude that one of the cases was
 falsely classified when you made the division in the previous period.
-Thus, your p-set changes. Therefore, the very considering of the sharp
+Thus, your p--set changes. Therefore, the very considering of the sharp
 distinction would automatically progress us to $t+1$, ensuring that there
 was no sharp boundary. In conclusion, the fixed time objection is not a
 significant worry to the dynamic view.
@@ -757,20 +755,20 @@ and their solution to the Sorites.\footnote{Contextualists exploit this
   \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
-p-sets account for HOV, why not apply them to FOV and get rid of
+the context--reminiscent p--sets. The intuitive idea is that, since shifty
+p--sets account for HOV, why not apply them to FOV and get rid of
 additional semantic claims and concessions altogether? However, this
 intuition is misguided, since the supervaluationist solution to FOV is
-required to make the shifty p-set account of HOV work. This is because
-the first-order divisions allow for the p-sets to shift in the first
+required to make the shifty p--set account of HOV work. This is because
+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
+determined by the p--set, which sets out the supervaluationist truth
 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
-beginning, the p-sets cannot shift. And if they cannot shift, they
+truly exist. This tension allows for future revisions of p--sets, and
+thus for p--sets to shift. Hence, without supervaluation in the
+beginning, the p--sets cannot shift. And if they cannot shift, they
 cannot account for any order of vagueness.
 
 A stronger claim could be made that the p-sets are entirely purposeless
@@ -780,21 +778,20 @@ you have some set of precisifications of tall $\{>170\text{cm},
 group of people in the series. Without supervaluation, you end up with
 six extensions, i.e., three positive and three negative extensions, one
 per precisification. There are no borderline cases, since without
-supervaluationist truth conditions --- where borderlines are true under
-some precisifications and false under others --- such cases are not
+supervaluationist truth conditions---where borderlines are true under
+some precisifications and false under others---such cases are not
 defined. Since this is a key symptom of vagueness, as stressed in the
 beginning, this result would require further explanation of why we think
 there are borderlines at all.
 
-An enemy of the view could argue that this response misses the point ---
-vagueness did not fail to arise due to the absence of supervaluation,
-but rather because the p-sets did not shift. After all, on the dynamic
-account, it is the shiftiness of p-sets that allows for HOV. To address
+An enemy of the view could argue that this response misses the point---vagueness did not fail to arise due to the absence of supervaluation,
+but rather because the p--sets did not shift. After all, on the dynamic
+account, it is the shiftiness of p--sets that allows for HOV. To address
 this, let us suppose, for the sake of the argument, that the p-set can
 somehow shift without supervaluation. Imagine, for instance, that the
-p-set expands by incorporating an additional precisification to the set.
+p--set expands by incorporating an additional precisification to the set.
 You now have eight extensions, yet still no explanation for either
-first-order or higher-order vagueness. Thus, even with shifty p-sets,
+first--order or higher-order vagueness. Thus, even with shifty p--sets,
 the dynamic view cannot function without supervaluation, showing it to
 be an essential, not merely supportive, component of the account.
 
@@ -804,30 +801,29 @@ the baseline supervaluationist aspects of the theory. If we cannot make
 the p-sets shifty, they cannot resolve FOV, let alone HOV. Hence,
 supervaluation is by no means obsolete. However, to strengthen the
 defense, I will demonstrate that the next step needed for the
-contextualist collapse fails. That is, I will show that p-sets and
+contextualist collapse fails. That is, I will show that p--sets and
 contexts behave very differently.
 
 Although they might appear similar, the former crucially relies on the
 characterization of vagueness as semantic indecision, while the latter
 depend on context sensitivity. We might express this difference by
-saying that the p-sets are inward-oriented, while contexts are more
-outward-oriented. This is because the former shifts due to our
+saying that the p--sets are inward--oriented, while contexts are more
+outward--oriented. This is because the former shifts due to our
 indecision among several equally good precisifications at the initial
 stage. This indecision prompts us to make mistakes, which we
-subsequently correct by revising the p-set into another equally
+subsequently correct by revising the p--set into another equally
 acceptable p-set. Thus, the changes directly follow our judgments. By
-contrast, shifts in contexts seem to have an effect on our judgments -
-contexts shift first, and judgments follow. Thus, the machinery appears
+contrast, shifts in contexts seem to have an effect on our judgments---contexts shift first, and judgments follow. Thus, the machinery appears
 to be quite different.
 
 One could even argue that shifty p-sets rest on a firmer theoretical
-ground -- their shiftiness is caused by our inconsistent judgments. On
+ground---their shiftiness is caused by our inconsistent judgments. On
 the other hand, the contexts appear to shift arbitrarily. Thus, the
 contextualist requires some external justification for this instability.
 Additionally, the contextualist needs to show how contexts could become
 shifty enough to prevent every instance of the Sorites. In other words,
 enough shiftiness must be generated. I do not intend to digress further,
-but the key takeaway is that despite their apparent similarities, p-sets
+but the key takeaway is that despite their apparent similarities, p--sets
 and contexts differ significantly. Thus, the threat of the `collapse'
 does not seem to be so imminent.
 
@@ -848,7 +844,7 @@ instance of conjunction introduction:
 
 \begin{center}
   $\cfrac{\text{Saul is tall} \hspace{2em} \text{Jan is tall}}
-  {\text{Saul and Jan are tall}} {\land I}$
+  {\text{Saul and Jan are tall}}  { \ \land  } \ \text{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
@@ -863,7 +859,7 @@ such simple cases?\footnote{J. Åkerman, "Contextualist Theories of
 
 In contrast, dynamic supervaluationism does not provoke such worries.
 Under supervaluationism, the rule of a conjunction introduction always
-preserves validity. To illustrate, consider a p-set representing
+preserves validity. To illustrate, consider a p--set representing
 precisifications for \emph{tall}: \{\textgreater170, \textgreater175,
 \textgreater176\}. First two precisifications make both premises true
 and the conclusion true as well. The third precisification makes one of
@@ -874,14 +870,14 @@ p-set.\footnote{This follows the exact same reasoning as that applied to
   excluded middle discussed in more detail at the beginning of the
   essay.}
 
-One might argue that, similarly to a shifting context, the p-set could
+One might argue that, similarly to a shifting context, the p--set could
 shift over the course of an argument. For example, we might initially
 classify both premises as true (e.g., using the set \{\textgreater170,
 \textgreater175\}, but later we classify the conclusion as false (e.g.,
 shifting to the set \{\textgreater177, \textgreater180\}). However, this
 objection reflects a misunderstanding of supervaluationist semantics,
-since arguments must always be evaluated relative to a single p-set. If
-we shifted the p-set to the second one, both premises would become false
+since arguments must always be evaluated relative to a single p--set. If
+we shifted the p--set to the second one, both premises would become false
 along with the conclusion. Therefore, the validity of conjunction
 introduction would remain intact.
 
@@ -901,8 +897,8 @@ is superior to contextualism. Such a claim is clearly beyond the scope
 of this essay and perhaps beyond the scope of any single essay. Rather,
 my point is simply that there are independent reasons to prefer the
 dynamic view over contextualism. Therefore, the claim that contextualism
-explains everything that the dynamic view explains - but more simply,
-and thus more parsimoniously - is clearly not accurate.
+explains everything that the dynamic view explains---but more simply,
+and thus more parsimoniously---is clearly not accurate.
 
 Taking stock of these considerations, the collapse argument fails not
 only at its initial stage but also on all subsequent fronts. Dynamic
@@ -913,7 +909,7 @@ framework.
 \section{Conclusion}
 
 While Keefe's supervaluationism remains an attractive account of
-vagueness, it ultimately struggles to account for higher-order
+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
 theory from natural language. Even if, as I briefly explored, she could
@@ -938,7 +934,7 @@ worth elaborating on and defending in future enquiries.
 \item The S5 principle: If ${\sim}DF$ then $D{\sim}DF$.
 \item The S4 principle: If $DF$ then $DDF$. 
 \end{enumerate}
-Suppose that (1) and (2) hold and that we have the first-order
+Suppose that (1) and (2) hold and that we have the first--order
 classification:
 \begin{enumerate}
   \def\labelenumi{(\roman{enumi})}
@@ -954,7 +950,7 @@ definitely definite positive and definitely definite negative,
 subsequently. If (2) holds, it implies ${\sim}DF\ \&\  {\sim}D {\sim}F$
 ${\sim}DF \ \& \ {\sim}D{\sim}F$ transforms into $D{\sim}DF \ \& \ D{\sim}D{\sim}F$ (see proof c). That is,
 the borderline case is definitely a borderline case. However,
-second-order vagueness would require two more categories -- the
+second--order vagueness would require two more categories---the
 borderline between positive and borderline
 (${\sim}DDF \ \& \ {\sim}D{\sim}DF$) and the borderline between borderline
 and negative (${\sim}DD{\sim}F \ \& \ {\sim}D{\sim}D{\sim}F$). As a result,
-- 
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