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+\chapter{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness}
+\chaptermark{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness}
+
+\renewcommand*{\thesection}{\arabic{section}.}
+\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
+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
+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
+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.
+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.
+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
+supervaluationism that can tackle the problems that Keefe's
+supervaluationism cannot. I close my essay by teasing out some
+challenges that the proposed theory could face and offering possible
+solutions. I believe that supervaluationism is a very attractive
+approach to vagueness and therefore, it is worth developing further into
+a more robust theory that could tackle its higher orders.
+\end{quote}
+
+\section{Introduction}
+
+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,
+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
+positive and negative cases), and (apparent) \textbf{susceptibility to
+the Sorites paradox}.\footnote{Rosanna Keefe, \emph{Theories of
+  Vagueness}, (Cambridge University Press, 2000), 6-7.}
+
+Vagueness is philosophically relevant because it raises two problems.
+First, the \textbf{semantic problem}: since the vague extension is
+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
+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
+conclude that 120cm is tall. However, this is a contradiction since
+120cm is clearly not tall.\footnote{Fine, \emph{Vagueness}, 3-7.} This
+argument exemplifies the classical form of the Sorites paradox.
+
+Theorizing about vagueness involves accounting for the nature, source
+and meaning of vagueness, providing vague semantics and resolving the
+Sorites. Furthermore, since it is unknowable where the positive
+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,
+  \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,
+  arguing that the vagueness of vague is a real, but unproblematic,
+  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
+  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
+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
+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.
+
+\section{Supervaluationism, a theory of vagueness}
+
+Supervaluationists claim that vagueness is a problem of language, not
+our epistemic capacities. They argue that vague predicates fail to draw
+sharp boundaries, not that these boundaries are unknowable, and that
+they admit borderline cases. The source is semantic indecision. A vague
+predicate admits a range of possible extensions, but it is semantically
+unsettled which one is correct. This is captured through the notion of
+precisification, a way to make a vague term precise.\footnote{Keefe,
+  \emph{Theories of Vagueness}, 154-156.} A precisification must be
+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
+predicate `tall'. We could (reasonably) use precisifications: `tall' is
+true if `\textgreater175cm', `\textgreater180cm' and
+`\textgreater190cm', each of which would precisely divide objects into
+positive and negative extensions. Vague terms do not `choose' between
+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.
+$\mathbf{Fa}$ is super-true (-false) iff $\mathbf{F}$ is true (false) of
+$\mathbf{a}$ under all complete and admissible precisifications.
+$\mathbf{Fa}$ is neither true nor false iff $\mathbf{F}$ 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
+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
+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:
+  Supervaluationism,'' \emph{Philosophy Compass} 3, no. 2 (2008):
+  162-164.}
+
+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
+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
+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
+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
+  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
+  169cm is short, then 170cm is short'. In each case, the antecedent is
+  true and the consequent false (relative to precisification). Since the
+  inductive premise turns out false for some pair under each complete
+  precisification, it is super-false.} for further clarification.
+
+\section{Supervaluationism and higher-order vagueness}
+
+The above metalanguage (talk of truth conditions) expresses the
+vagueness of the object language by dividing cases into three sharply
+bounded sets (true, false, borderline). This can be captured by adding a
+`definitely' D operator to the object language, which functions akin to
+modal necessity.
+
+The FOV of F is expressed as:
+
+\begin{enumerate}
+\def\labelenumi{(\arabic{enumi})}
+\item
+  $DFx$ for definite positive cases (true under all complete and
+  admissible precisifications)
+\item
+  ${\sim} DFx \, \& \, {\sim} D {\sim} Fx$ for borderline cases (true/false
+  under some)
+\item
+  $D {\sim} Fx$ for negative cases (false under all)
+\end{enumerate}
+
+This division is problematic since all cases are sharply categorized,
+allowing no borderlines between the definite and borderline cases,
+leaving no scope for HOV. Supervaluationists argue that this can be
+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:
+
+\begin{enumerate}
+\def\labelenumi{(\arabic{enumi})}
+\item
+  $DDFx$, i.e., definitely definitely positive cases
+\item
+  ${\sim} DDFx \; \& \; {\sim} D {\sim} DFx$, i.e., borderline between positive
+  and borderline
+\item
+  $D {\sim} DFx \; \& \; D{\sim} D{\sim} Fx$, i.e., definitely borderline cases
+\item
+  ${\sim}DD{\sim} Fx \; \& \; {\sim}D{\sim}D{\sim}Fx$, i.e., borderline between
+  negative and borderline
+\item
+  $DD{\sim}Fx$, i.e., definitely definitely negative cases
+\end{enumerate}
+
+The general idea is that for  level vagueness of F, we need
+to show that $\mathbf{n}$ categories are vague. Thus, we need borderlines
+between those, in effect, drawing $2^n+1$ categories.\footnote{Mark
+  Sainsbury, ``Concepts without Boundaries,'' in \emph{Departing From
+  Frege} (Routledge, 1990), 74.}
+
+\subsection{Williamson's challenge}
+
+Williamson argues that for this formalization to work, the D operator
+should not obey these two schemas:
+\begin{enumerate}
+\def\labelenumi{(\arabic{enumi})}
+\item{The S5 principle: If ${\sim}DF$, then $D{\sim}DF$.}
+\item{The S4 principle: If $DF$, then $DDF$.}
+\end{enumerate}
+
+If (1) and (2) hold, then whether a category is definite or indefinite,
+it will also be definitely so at higher levels. The supervaluationist
+cannot accept this since each category must be vague, otherwise it would
+draw sharp boundaries. Thus, Williamson recommends adopting a weaker
+modal logic, like T, with relative admissibility and no transitivity so
+that both S4 and S5 principles fail.\footnote{Williamson,
+  \emph{Vagueness}, 156-159.} See the appendix for a more formal
+explanation.
+
+However, Williamson argues that this is not sufficient to solve the
+problem via the D* argument. He defines \(D^{*}F\) 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
+$F, \; DF, \; DDF, \; \ldots, \; D_{n}F$ 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
+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).
+
+%\includegraphics[width=3.97674in,height=4.66543in]{media/image1.png}
+\begin{center}
+  \begin{tikzpicture}
+    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
+    \end{tikzpicture}
+  \end{center}
+  
+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.\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.
+
+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
+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
+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
+ensures that a non-vague metalanguage cannot be generated.\footnote{Greenough,
+  ``Higher-Order Vagueness,'' 184-186.}
+
+\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
+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
+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
+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
+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
+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
+contradiction. To resolve the second-order paradox, Keefe reuses her
+earlier strategy: for any complete way of making `admissible' precise
+(or making `definitely' definite), there will be a pair such that the
+first precisification is admissible and the second is not. This could be
+run for any level of metalanguage.
+
+Thus, Keefe's account of HOV fulfils all the demands of a theory of
+vagueness. Each metalanguage is vague since it (1) admits borderline
+cases, (2) draws no sharp boundaries and (3) is Sorites susceptible. The
+fact that she achieves this for each order while maintaining her initial
+commitments (using the same technique at each order, characterising all
+levels of vagueness as semantic indecision, and so on) makes her
+strategy simple and elegant.
+
+Even though this iteration neatly maintains the supervaluationist
+method, iterating to infinity is problematic. Keefe boldly claims that
+`if there is no general objection to the claim that the sequence of
+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
+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
+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}.}
+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.
+
+This objection points towards a more general issue with such Tarskian
+metalanguage hierarchies. Namely, that languages in such hierarchies
+cannot be globally quantified over.\footnote{Greenough, ``Higher-Order
+  Vagueness,'' 187.} Keefe could respond that even though the infinite
+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
+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.
+
+A further problem with the structure is that it is highly detached from
+how language functions. Competent speakers would find making sense of
+iterated uses of `definitely' difficult, whether it is indexed or not.
+For example, saying someone is `definitely definitely definitely tall'
+has little meaning apart from emphasis. Keefe might respond by pointing
+out that we do not use expressions like `a googol of a googol of a
+googol' in ordinary conversation either, yet this does not mean the
+concept of `googol' is not a meaningful mathematical concept. However,
+the issue goes deeper. As Saul Kripke pointed out, we cannot
+consistently assign levels to truth. Thus, even if we index the levels
+of `definitely', it is difficult to assign them consistently. Consider
+the following statements: Jan says, `Everything Alfred said is
+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}.}
+
+Keefe might counter these natural language intuitions by arguing that
+her model is only an idealization which is not meant to exactly
+replicate how ordinary language works. While iterating `definitely'
+(e.g., \(D_{3}D_{2}D_{1}F\)) may make little sense in casual
+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
+above), they can still function as distinct metalanguages in a formal
+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
+explanation of the relationship between her model and natural
+language.\footnote{A full discussion of this issue is beyond the scope
+  of this essay, though the problem would require further explanation to
+  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
+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
+an infinite number of categories and a finite number of objects in the
+series: the finite series paradox. Consider a simple series with 5
+objects. To account for 1\textsuperscript{st} level, we divide them into
+3 categories. To account for 2\textsuperscript{nd} level, we divide them
+into 5 categories. At 3\textsuperscript{rd} level there are 9 categories
+to be filled but only 5 objects. This means that at some level we will
+run out of objects with which to fill the categories. As a result, there
+will be no borderline cases between the categories - providing a sharp
+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.
+
+% \includegraphics[width=6.30006in,height=2.89763in]{media/image2.png}
+\begin{center}
+  \begin{tikzpicture}
+    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
+    \end{tikzpicture}
+  \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
+way to accommodate the problem without making strong alterations to the
+model.
+
+\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
+  \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
+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
+we do not know what is admissible. However, this knowledge is
+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
+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
+next paragraphs is my own development of the idea.
+
+Fu applies the AGM theory\footnote{AGM refers to the
+  Alchourrón--Gärdenfors--Makinson model of belief revision, which
+  accounts for rational change in epistemic states represented as belief
+  sets. The theory outlines how agents should expand, contract, or
+  revise their beliefs while preserving logical coherence. For more
+  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}.}
+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
+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
+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
+is made that conflicts with it.
+
+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
+temporally indexed D operators.
+
+More precisely, at any time, t, cases divide into
+\(D_{t}F,D_{t}{\sim}F\), and \({{\sim}D}_{t}F\ \; \& \; {{\sim}D}_{t}{\sim}F\):
+that is true, false, and borderline. However, unlike in Keefe's view,
+HOV arises not from undecided admissibility of a precisification but
+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
+\(t_{2}\), 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
+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.
+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
+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.
+
+\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
+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.
+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
+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.
+There is no rigid hierarchy of metalanguages since each discusses
+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
+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
+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
+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,
+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
+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
+sense hierarchical, none has a clear priority in determining the truth
+of one classification over another. Thus, the objections, such as the
+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
+objects \textbf{a}, \textbf{b}, \textbf{c}, \textbf{d}, and \textbf{e}.
+Suppose that Alfred's initial categorizations are:
+
+\[D_{1}F = \{ a,b\}\]
+
+\[{\sim}D_{1}F\; \& \; {\sim}D_{1}{\sim}F = \{ c\}\]
+
+\[D_{1}{\sim}F = \{ d,e\}\]
+
+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 \(t_{2}\)
+p-set.
+
+\[D_{2}F = \{ a\}\]
+
+\[{\sim}D_{2}F\ \&\ {\sim}D_{2}{\sim}F = \{ b,c\}\]
+
+\[D_{2}{\sim}F = \{ d,e\}\]
+
+% \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}
+ 
+The \(t_{1}\) division, from the perspective of \(t_{2}\) becomes:
+
+\emph{\hfill\break
+}\[{D_{2}D}_{1}F = \{ a\}\]
+
+\[{\sim}D_{2}D_{1}F\ \; \& \; {\sim}D_{2}{{\sim}D}_{1}F = \{ b\}\]
+
+\[D_{2}{\sim}D_{1}F\ \&\ D_{2}{\sim}D_{1}{\sim}F = \{ c\}\]
+
+% \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}
+Hence, in this part of the series, the vagueness of \(D_{1}\) is fully
+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.
+
+\[D_{3}F = \{ a,b\}\]
+
+\[{\sim}D_{3}F \; \& \; {\sim}D_{3}{\sim}F = \{ c\}\]
+
+\[D_{3}{\sim} F = \{ d,e\}\]
+
+% \includegraphics[width=4.46286in,height=2.2071in]{media/image5.emf}
+\begin{center}
+  \begin{tikzpicture}
+    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
+    \end{tikzpicture}
+  \end{center}
+Since $\mathbf{b}$ changed its category membership, from the perspective
+of \(t_{3}\), $\mathbf{b}$ was not a definite borderline case at
+\(t_{2}\). Thus, the \(t_{2}\) division, from the \(t_{3}\) perspective,
+is:
+
+\[{D_{3}D}_{2}F = \{ a\}\]
+
+\[\sim D_{3}D_{2}F\ \&\ \sim D_{3}{\sim D}_{2}F = \{ b\}\]
+
+\[D_{3}\sim D_{2}F\ \&\ D_{3}\sim D_{2}\sim F = \{ c\}\]
+
+% \includegraphics[width=4.875in,height=2.1236in]{media/image6.emf}
+\begin{center}
+  \begin{tikzpicture}
+    \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}};
+    \end{tikzpicture}
+  \end{center}
+Thus, vagueness of \(D_{2}\) is accounted for.
+
+In general, any bordering pair will exhibit change when reassessed.
+Thus, any categorization at $t$ can prove indefinite at $t+1$.
+In effect, you will never reach a point where there are more categories
+than members in the series since the fluid categories will always be
+filled. An object can fill different categories at different times. This
+also does not mean that the \(t_{1}\) categories are definite at
+\(t_{3}\), only that their vagueness cannot be expressed from the
+\(t_{3}\) perspective.
+
+\section{Addressing possible objections}
+
+Dynamizing supervaluationism provides new methods to tackle the
+paradoxes of HOV and other problems, for which standard
+supervaluationism struggles to account. However, it also presents new
+worries, which I will explore and sketch responses to in this section of
+the essay.
+
+\subsection{Fixed time worry}
+
+The first possible objection to the view is that it breaks down when
+time is fixed. This is because the account of HOV relies on shifty
+p-sets, which in turn rely on the progress in time. More precisely, the
+vagueness of some set of categories drawn in period $t$ can only be
+expressed in period $t+1$. Thus, if we hold the time fixed, the
+view breaks down: the categories drawn in period $t$ appear to be
+sharply bounded, which contradicts the foundational claim that there are
+no sharp boundaries.
+
+Although this might seem like a critical blow to the view, there are two
+possible lines of response. First, we could simply reject the inference
+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
+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.
+
+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
+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
+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.
+
+\subsection{Collapse to contextualism worry}
+
+There is a second and more dangerous worry: one could argue that the
+supervaluationist aspect of the dynamic view seems unimportant. By this,
+I mean the use of supervaluationist semantics and classification of
+vagueness through indecision between precisifications. It is only
+directly applied to resolve FOV, and one could argue that the relativity
+of classifications over time, which accounts for HOV, could be equally
+applied to FOV. In effect, the supervaluationist method would disappear.
+If this argument is accepted, and if we further assume that the
+functioning of p-sets is sufficiently similar to that of contexts, then
+the dynamic view risks collapsing into a contextualist one. This could
+have some benefits, such as the preservation of bivalence (which
+contextualists keep) and making the view more parsimonious by unifying
+the approaches to vagueness at different orders.
+
+In what follows, I will defend the dynamic view from this objection. See
+footnotes for background on contextualism\footnote{Contextualism rests
+  on the claim that vagueness is a species of context-sensitivity. This
+  roughly means that, in its application across different contextual
+  circumstances, a vague term maintains a constant \emph{character} but
+  shifts in \emph{content}. Therefore, vague terms function like
+  indexical terms. The relationship of vagueness and indexicality is a
+  contested matter for contextualists. Some hold that vague terms behave
+  \emph{like} indexicals, while others claim they \emph{are} indexicals.
+  However, this distinction is not directly relevant to the discussion,
+  and the objections raised here apply equally to both views. Consider
+  the word \emph{now}. It adheres to the same grammatical rules (i.e.,
+  has the same \emph{character}) when uttered today and tomorrow.
+  However, when said today, it picks out a different time than it does
+  when used tomorrow (i.e., has different \emph{content}). Similarly, a
+  vague predicate like \emph{tall} is used in the same way when applied
+  to members of a group of pygmy peoples, as when applied to a group of
+  Dutch people. However, it would pick out radically different people.
+  In the first case, the extension of \emph{tall} likely includes some
+  of the world's shortest people; in the second, some of the tallest.
+  See Roy Sorensen, ``Vagueness,'' \emph{The Stanford Encyclopedia of
+  Philosophy} (Winter 2023 Edition), ed. Edward N. Zalta and Uri
+  Nodelman,
+  \url{https://plato.stanford.edu/archives/win2023/entries/vagueness/}.}
+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.
+  Similarly to the supervaluationists, the contextualists target the
+  inductive premise (2). The contextualist is committed to the claim of
+  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
+  different extension. See Jonas Åkerman and Patrick Greenough, "Hold
+  the Context Fixed---Vagueness Still Remains," in \emph{Relative
+  Truth}, ed. Manuel García-Carpintero and Max Kölbel (Oxford University
+  Press, 2010), 275--76,
+  \href{https://doi.org/10.1093/acprof:oso/9780199570386.003.0016}.
+
+  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
+  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 n is short, then n+1 is
+  short', fails since the meaning of `short' is not the same for every
+  member \emph{n}. This is because, the shift of context C into C',
+  enables cases where `n is short' is true (in C) but `n+1 is short' is
+  false (in C'). See J. Åkerman, "Contextualist Theories of Vagueness,"
+  \emph{Philosophy Compass} 7 (2012): 470--75,
+  \href{https://doi.org/10.1111/j.1747-9991.2012.00495.x}.}.
+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
+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
+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
+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
+cannot account for any order of vagueness.
+
+A stronger claim could be made that the p-sets are entirely purposeless
+if we do not allow for supervaluation. To see the point, imagine that
+you have some set of precisifications of tall $\{>170\text{cm},
+>180\text{cm}, >190\text{cm}\}$ and you use them to categorize a
+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
+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
+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.
+You now have eight extensions, yet still no explanation for either
+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.
+
+Therefore, the case for the contextual collapse breaks down in the very
+beginning. We simply cannot make the p-sets shifty without maintaining
+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
+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
+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
+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
+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
+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
+and contexts differ significantly. Thus, the threat of the `collapse'
+does not seem to be so imminent.
+
+As a final point to strengthen my argument, I will provisionally assume
+that the dynamic approach could collapse into contextualism. Even in
+such a scenario, there remain independent reasons to prefer the former
+view over the latter. One significant reason is that contextualism
+undermines some of our most basic approaches to reasoning. Contextualism
+requires extensions of vague terms to be unstable, which is precisely
+what enables it to defeat the Sorites. However, these shifty contexts
+become deeply problematic when applied outside of the paradoxical
+setting.
+
+To see this, consider the following example. Saul and Jan are borderline
+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
+
+However, if the extension of the vague predicate \emph{tall} is
+unstable, we can easily imagine a situation in which both premises are
+individually true, yet the conclusion turns out false. This would happen
+if the context shifted midway through the argument. Thus, although
+context sensitivity is useful for solving the Sorites, it is dangerous
+when applied to everyday reasoning. Specifically, how can contexts
+remain sufficiently stable to ensure our logic does not fail even in
+such simple cases?\footnote{J. Åkerman, "Contextualist Theories of
+  Vagueness," \emph{Philosophy Compass} 7 (2012): 475--76,
+  \href{https://doi.org/10.1111/j.1747-9991.2012.00495.x}.}
+
+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
+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
+the premises true, the other false, and the conclusion false. This will
+work for any possible precisification. Consequently, it applies to every
+p-set.\footnote{This follows the exact same reasoning as that applied to
+  the failure of the inductive premise or the truth of the law of
+  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
+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
+along with the conclusion. Therefore, the validity of conjunction
+introduction would remain intact.
+
+Why is this strategy not available to the contextualist? The
+contextualist could simply deny that contexts can shift in such ways,
+insisting instead that we always evaluate the premises and the
+conclusion within a single context. However, this directly contradicts
+the contextualist's equivocation strategy to the Sorites paradox. That
+is, the strategy according to which bordering cases may differ in truth
+value because their evaluation contexts differ. Hence the contextualists
+need contexts to shift. In effect, they cannot deny that the above
+scenario is possible. Instead, their strongest response would likely be
+to argue that such cases rarely happen.
+
+I do not intend to argue that supervaluationism, or its dynamic version,
+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.
+
+Taking stock of these considerations, the collapse argument fails not
+only at its initial stage but also on all subsequent fronts. Dynamic
+supervaluationism is by no means contextualism in disguise; rather it is
+its own theory, deeply grounded in Keefe's original supervaluationist
+framework.
+
+\section{Conclusion}
+
+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
+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
+paradox. I argued that Keefe's account could be dynamized by
+incorporating ideas from Fu, thereby resolving the finite series paradox
+and avoiding issues associated with a rigid hierarchy. Yet, the dynamic
+model itself introduces new difficulties, notably the `fixed time' and
+`collapse to contextualism' problems. To defend the view, I briefly
+outlined potential replies to these issues, showing that they are not
+fatal. Dynamizing supervaluationism may not resolve all problems, but it
+is a promising development of the supervaluationist theory and would be
+worth elaborating on and defending in future enquiries.
+
+\subsection*{Appendix}
+
+\subsubsection*{Why must Keefe deny the S4 and S5 principles?}
+
+\begin{enumerate}
+\def\labelenumi{(\arabic{enumi})}
+\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
+classification:
+\begin{enumerate}
+  \def\labelenumi{(\roman{enumi})}
+\item{$DF$ for definite positive cases.}
+\item{${\sim}DF \; \& \; {\sim}D{\sim}F$ for borderline cases.}
+  \item{$D{\sim}F$ for negative cases.}
+  \end{enumerate}
+
+If (1) holds, it implies that at the second level, $DF$ and
+$D{\sim}F$ transform into $DDF$ and $DD{\sim}F$ (see proofs a and
+b). That is, the definite positive and definite negative case is
+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
+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,
+sharp boundaries are drawn between the three categories since there are
+no cases between them.
+
+% \includegraphics[width=3.125in,height=1in]{media/image7.png}
+
+%\includegraphics[width=2.87025in,height=0.94231in]{media/image8.png}
+\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}
+    \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}
+    \end{center}
+  \end{minipage}
+\bigskip
+
+\noindent \textbf{Proof c:}
+\begin{center}
+  \begin{tikzpicture}
+    \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}};
+    \end{tikzpicture}
+  \end{center}
+
+%\includegraphics[width=6.26806in,height=1.61181in]{media/image9.png}
+
+\refsection
+  
+\begin{hangparas}{\hangingindent}{1}
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+Williamson, Timothy. \emph{Vagueness.} London: Routledge, 1994.
+\end{hangparas}
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