Merge branch 'main' of github.com:aporia-sta/aporia-25

11 files changed, 1414 insertions, 1 deletions

diff --git a/main.tex b/main.tex
index 62e128e..cea4e7b 100644
--- a/main.tex
+++ b/main.tex
@@ -17,6 +17,7 @@
 ]{geometry}
 % Fonts
 \usepackage{ebgaramond-maths}
+\usepackage{amsmath}
 \usepackage{amssymb}                    % Math symbols (e.g. therefore)
 \usepackage{enumitem}                   % indenting next line of list
 \usepackage[scale=0.78]{plex-mono}      % Monospace font
@@ -49,7 +50,10 @@
 \usepackage{suffix}
 % for list processing tools
 \usepackage{etoolbox}
-
+% for TikZ figures
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{trees}
 % Gap definitions
 \def \hangingindent {3em}
 \def \credgap {15pt}
@@ -110,6 +114,50 @@
 % Define refsection
 \def \refsection {\newpage\section*{Bibliography}}
 
+% Packages/declarations required by paper 3
+\usepackage{graphicx}
+\usepackage{setspace}
+\usepackage{amssymb}
+\usepackage{amsmath} 
+\usepackage{amsthm}
+\usepackage{xcolor}
+\usepackage{turnstile}
+\usepackage{array}
+\usepackage{stmaryrd}
+\usepackage[normalem]{ulem}
+\usepackage{bussproofs}
+\usepackage[linguistics]{forest}
+\usepackage{tipa}
+\usepackage{tabulary}
+\forestset{
+fairly nice empty nodes/.style={
+delay={where content={}
+{shape=coordinate, for siblings={anchor=north}}{}},
+for tree={s sep=4mm}
+}
+}
+\usepackage{gb4e}
+\usepackage{cgloss4e}
+\usepackage{drs}
+\usepackage[stable]{footmisc}
+\newcommand{\lsv}{\llbracket}
+\newcommand{\rsv}{\rrbracket}
+\newcommand{\la}{\leftarrow}
+\newcommand{\ra}{\rightarrow}
+\newcommand{\La}{\Leftarrow}
+\newcommand{\Ra}{\Rightarrow}
+\newcommand{\ba}{\leftrightarrow}
+\newcommand{\Ba}{\Leftrightarrow}
+\newcommand{\lag}{\langle}
+\newcommand{\rag}{\rangle}
+\newcommand{\lam}{\lambda}
+\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
+\DeclareMathSymbol{\strictif}{\mathrel}{symbolsC}{74}
+\DeclareMathSymbol{\boxright}{\mathrel}{symbolsC}{128}
+\DeclareMathSymbol{\boxRight}{\mathrel}{symbolsC}{136} % Lewis’s stronger ‘would’ counterfactual
+\DeclareMathSymbol{\diamondRight}{\mathrel}{symbolsC}{140} % Lewis’s stronger ‘might’ counterfactual
+\DeclareMathSymbol{\diamonddot}{\mathord}{symbolsC}{144} % Lewis’s inner necessity
+
 % MAIN
 \begin{document}
 \includepdf{COVER.pdf}
@@ -125,6 +173,12 @@
 
 \mainmatter
 \include{papers/1}
+  \begingroup
+  \let\mathit\mathrm
+  \let\mathnormal\mathrm
+\include{papers/2}
+\endgroup
+\include{papers/3}
 \include{papers/4}
 
 \backmatter
@@ -132,3 +186,8 @@
 \cleardoublepage
 \include{99-rear-cover}
 \end{document}
+
+%%% Local Variables:
+%%% mode: LaTeX
+%%% TeX-master: "../main"
+%%% End:
diff --git a/papers/1.tex b/papers/1.tex
index 77aa73b..1984f1b 100644
--- a/papers/1.tex
+++ b/papers/1.tex
@@ -646,3 +646,9 @@ Problem of Silence,'' \emph{Philosophical Perspectives} 5 (1991):
 135-165.
 
 \end{hangparas}
+
+%%% Local Variables:
+%%% mode: LaTeX
+%%% TeX-master: t
+%%% TeX-master: "../main"
+%%% End:
diff --git a/papers/2.tex b/papers/2.tex
new file mode 100644
index 0000000..d480e1c
--- /dev/null
+++ b/papers/2.tex
@@ -0,0 +1,1078 @@
+\chapter{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness}
+\chaptermark{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness}
+\chapterauthor{Wiktor Przybrorwski, \textit{
+University of St Andrews
+}}
+\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).
+
+\begin{center}
+\includegraphics[width=3.97674in,height=4.66543in]{papers/figures/2-1.pdf}
+\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.
+\begin{center}
+\includegraphics[width=0.925\textwidth]{papers/figures/2-2.pdf}
+\end{center}
+In conclusion, even though the rigid hierarchy in Keefe's structure
+might be defended to some extent, her appeal to an infinite hierarchy is
+fundamentally in conflict with the finite Sorites. There seems to be no
+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 $t+1$
+must be a metalanguage of the metalanguage at $t$, since it is able
+to express facts about $t$. Therefore, it is more `privileged' in
+this sense. However, suppose that the p-sets evolve over time such that,
+when moving from $t+1$ to $t+2$, we go back to the original
+p-set from $t$. Then, the $t$ and $t+2$ metalanguages
+gain their truth conditions from the same p-set. Therefore, in a sense,
+the t metalanguage becomes `prior' to the $t+1$ 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.
+
+\[D_{2}F = \{ a\}\]
+
+\[{\sim}D_{2}F\ \&\ {\sim}D_{2}{\sim}F = \{ b,c\}\]
+
+\[D_{2}{\sim}F = \{ d,e\}\]
+
+\begin{center}
+\includegraphics[width=4.50937in,height=2.12793in]{papers/figures/2-3.pdf}
+ \end{center}
+The \(t_{1}\) division, from the perspective of \(t_{2}\) becomes:
+
+\emph{\hfill\break
+}\[{D_{2}D}_{1}F = \{ a\}\]
+
+\[{\sim}D_{2}D_{1}F\ \; \& \; {\sim}D_{2}{{\sim}D}_{1}F = \{ b\}\]
+
+\[D_{2}{\sim}D_{1}F\ \&\ D_{2}{\sim}D_{1}{\sim}F = \{ c\}\]
+\\
+\begin{center}
+\includegraphics[width=4.79722in,height=2.19101in]{papers/figures/2-4.pdf}
+\end{center}
+Hence, in this part of the series, the vagueness of \(D_{1}\) is fully
+accounted for since all \(D_{1}\) categories have borderline cases.
+
+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.
+
+\[D_{3}F = \{ a,b\}\]
+
+\[{\sim}D_{3}F \; \& \; {\sim}D_{3}{\sim}F = \{ c\}\]
+
+\[D_{3}{\sim} F = \{ d,e\}\]
+
+\begin{center}
+    \includegraphics[width=4.46286in,height=2.2071in]{papers/figures/2-5.pdf}
+  \end{center}
+Since $\mathbf{b}$ changed its category membership, from the perspective
+of \(t_{3}\), $\mathbf{b}$ was not a definite borderline case at
+\(t_{2}\). Thus, the \(t_{2}\) division, from the \(t_{3}\) perspective,
+is:
+
+\[{D_{3}D}_{2}F = \{ a\}\]
+
+\[\sim D_{3}D_{2}F\ \&\ \sim D_{3}{\sim D}_{2}F = \{ b\}\]
+
+\[D_{3}\sim D_{2}F\ \&\ D_{3}\sim D_{2}\sim F = \{ c\}\]
+
+\\
+
+\begin{center}
+    \includegraphics[width=4.875in,height=2.1236in]{papers/figures/2-6.pdf}
+  \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:
+
+\begin{center}
+  $\cfrac{\text{Saul is tall} \hspace{2em} \text{Jan is tall}}
+  {\text{Saul and Jan are tall}} {\land I}$
+\end{center}
+However, if the extension of the vague predicate \emph{tall} is
+unstable, we can easily imagine a situation in which both premises are
+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}
+    $\cfrac{
+        DF \hspace{2em} DF \rightarrow DDF
+    }{DDF} {\ \rightarrow \hspace{-0.25em} E}$
+    \end{center}
+\end{minipage}
+\begin{minipage}[t]{0.48\textwidth}
+  \noindent \textbf{Proof a:} \\
+    \begin{center}
+    $\cfrac{
+        D \neg F \hspace{2em} D\neg F \rightarrow DD \neg F
+    }{DD\neg F} {\ \rightarrow \hspace{-0.25em} E}$
+    \end{center}
+  \end{minipage}
+\bigskip
+
+\noindent \textbf{Proof c:}
+  \begin{center}
+    $\cfrac{
+        \cfrac{
+        \begin{array}{c}
+        \\
+        \neg DF \rightarrow D \neg DF
+        \end{array}
+        \cfrac{
+            \neg DF \neg D \neg F
+        }{\neg DF} {\land E}
+        }{D \neg DF} {\ \rightarrow \hspace{-0.25em}E} \hspace{1em}
+        \cfrac{
+            \cfrac{\neg DF \land \neg D \neg F}{\neg D \neg F} {\land E} \hspace{1em}
+            \begin{array}{c}
+            \\
+            \neg D \neg F \rightarrow D \neg D \neg F
+            \end{array}
+        }{D \neg D \neg F}{\ \rightarrow \hspace{-0.25em}E}
+    }{D\neg D F \land D\neg D\neg F} {\land I}$
+    \end{center}
+
+%\includegraphics[width=6.26806in,height=1.61181in]{media/image9.png}
+
+\refsection
+  
+\begin{hangparas}{\hangingindent}{1}
+Åkerman, Jonas. "Contextualist Theories of Vagueness." \emph{Philosophy
+Compass} 7 (2012): 470--80.
+
+Åkerman, Jonas, and Patrick Greenough. ``Hold the Context
+Fixed---Vagueness Still Remains.'' In \emph{Relative Truth}, edited by
+Manuel García-Carpintero and Max Kölbel, 275--288. Oxford: Oxford
+University Press, 2010.
+
+Alchourrón, Carlos E., 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. https://doi.org/10.2307/2274239.
+
+Bobzien, Susanne. "In Defense of True Higher-Order Vagueness."
+\emph{Synthese} 199, no. 3--4 (2021): 10197--10229.
+
+Fine, Kit. \emph{Vagueness: A Global Approach.} Rutgers Lectures in
+Philosophy Series. New York: Oxford Academic, 2020. Online edition, May
+21, 2020. \url{https://doi.org/10.1093/oso/9780197514955.001.0001}.
+Accessed November 15, 2024.
+
+Fu, Hao-Cheng. ``Saving Supervaluationism from the Challenge of
+Higher-Order Vagueness Argument.'' In \emph{Philosophical Logic: Current
+Trends in Asia}, 139--52. 2017.
+
+Greenough, Patrick. ``Higher-Order Vagueness.'' \emph{Proceedings of the
+Aristotelian Society, Supplementary Volumes} 79 (2005): 167--90.
+\url{http://www.jstor.org/stable/4106939}.
+
+Hyde, Dominic. "Why Higher-Order Vagueness Is a Pseudo-Problem."
+\emph{Mind} 103, no. 409 (1994): 35--41.
+
+Keefe, Rosanna. \emph{Theories of Vagueness.} Cambridge: Cambridge
+University Press, 2000.
+
+Keefe, Rosanna. ``Vagueness: Supervaluationism.'' \emph{Philosophy
+Compass} 3, no. 2 (2008): 315--24.
+
+Kripke, Saul. ``Outline of a Theory of Truth.'' \emph{The Journal of
+Philosophy} 72, no. 19 (1975): 690--716.
+\url{https://www.jstor.org/stable/2024634}. Accessed November 30, 2024.
+
+MIT OpenCourseWare. \emph{Session 11: Mathematics for Computer Science.}
+\emph{6.042J: Mathematics for Computer Science, Spring 2015.}
+Massachusetts Institute of Technology, 2015. Accessed December 8, 2024.
+
+Sainsbury, Mark. ``Concepts without Boundaries.'' Chapter three of
+\emph{Departing From Frege}. London: Routledge, 1990.
+
+Sorensen, Roy. ``Vagueness.'' \emph{The Stanford Encyclopedia of
+Philosophy} (Winter 2023 Edition), edited by Edward N. Zalta and Uri
+Nodelman.
+\href{https://plato.stanford.edu/archives/win2023/entries/vagueness/}.
+
+Williamson, Timothy. \emph{Vagueness.} London: Routledge, 1994.
+\end{hangparas}
+%%% Local Variables:
+%%% mode: LaTeX
+%%% TeX-master: "../main"
+%%% End:
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+\chapter{Defending Williamson's Explanatory Challenge to Contingentism}
+\chaptermark{Defending Williamson's Explanatory Challenge to Contingentism}
+\chapterauthor{Koda Li,
+\textit{Brown University}}
+
+\section{Introduction}
+In his book \emph{Modal Logic as Metaphysics}, Timothy Williamson developed a series of arguments against contingentism and in favor of necessitism. I outline the two theses in the following: 
+\begin{quote}
+(Contingentism) \hspace{\labelsep} $\Diamond \exists x \Diamond \neg \exists y x=y$ \smallskip \\
+Informally, some things could have not existed. \smallskip \\
+``The table could have been destroyed in the making process and therefore does not exist." \bigskip \\
+(Necessitism) \hspace{\labelsep} $\Box \forall x \Box \exists y x=y$.  \smallskip \\
+Informally, everything necessarily exists. \smallskip \\
+``This table, the person John, and all other things exist necessarily."
+\end{quote}
+Williamson's arguments are complex and intricate. This paper will focus on one particular challenge he raised to contingentism in Chapter 6 of his book and various responses toward this challenge. The paper is structured as the following: Section 2 reconstructs Williamson's challenge; Section 3 explains two ``trivialization" worries about this challenge and respond to them on Williamson's behalf; Section 4 develops a substantive response to Williamson's challenge and criticize it on Williamson's behalf. I argue that Williamson's challenge is successful and contingentists have considerable dialectical disadvantages.  
+
+\section{Williamson's Explanation Challenge to contingentism}
+Williamson raised a challenge to contingentists who accept (Comp_M) in high-order modal logic.\footnote{The background logic Williamson assumes is the one developed in his Chapter 5, p. 225. What is of significance is that the underlying modal logic is S5. This paper will not tap into the debate of which modal logic is the correct modal logic. I will assume Williamson's logic and develop challenges and responses.}
+Below is (Comp_M): 
+\begin{quote}
+(Comp_M) \hspace{\labelsep} $\vdash \exists X \Box \forall x (Xx \ba A)$ \smallskip \\ where $A$ is a metalinguistic variable ranging over formulas. 
+\end{quote}
+Informally, (Comp_M) says that for any formula $A$, there is some property that something instantiates just in case $A$ is true.
+
+I will first say something to motivate (Comp_M) before getting into Williamson's challenge using this principle. For one, (Comp_M) is a very attractive higher-order logic principle, for it says roughly that given any formula A, one can define a property such that necessarily, something has it just in case A is true. Intuitively, this seems true. We frequently define complex properties using this way. Given an open formula, for example, ``$x$ is white and $x$ is big", certainly there is some property $P$ such that necessarily, a thing $y$ has $P$ iff $y$ is white and $y$ is big. In other words, it seems that we should be able to use any formula A to give the necessary and sufficient conditions for something having a certain property. 
+
+Further, we need (Comp_M) to capture compelling natural language inferences, for example the following:\footnote{Timothy Williamson, \textit{Modal Logic as Metaphysics} (Oxford University Press, 2013), p. 227.} 
+
+\begin{quote}
+\begin{tabulary}{\textwidth}{lLr}
+P$1$. & Alice doesn't smoke a cigar, but she could have done so. & ($\neg Sa \wedge \Diamond Sa$) \\
+C. & Alice doesn't do something she could have done. & ( $\exists X (\neg Xa \wedge \Diamond Xa)$ )
+\end{tabulary}
+\end{quote}
+This inference is valid. To capture this, we need precisely an instance of (Comp_M): $\exists X \Box \forall x (Xx \ba \neg Sx \wedge \Diamond Sx)$.\footnote{One might argue that this valid inference can be equally captured by adding an existential generalization axiom to the logic. I just want to point out that this EG axiom is in the exact same spirit as (Comp_M) here: they are both saying that we can form complex properties from simpler ones. So they are not in tension: if one accepts one, one should have reasons to accept the other. } 
+Finally, more generally, (Comp_M) is an example of comprehension principles for higher-order logic (even for non-modal logic). Standard second-order non-modal logic usually has comprehension principles of similar form: given any formula $A$, $\exists P \forall x (Px \ba A)$. This ensures that the logic has enough power to prove important theorems  that intuitively needs to be provable. For example, second-order Peano arithmetic typically contain the following Induction axiom: 
+\begin{quote}
+(Induction Axiom) \hspace{\labelsep}  $\vdash \forall P \forall x (P(\mathbf{0}) \wedge (P(x) \ra P(x+1)) \ra \forall x P(x))$ 
+\end{quote}
+Now suppose I have the following formula: $x$ is even or $x$ is odd. Certainly, every natural number has this property: being either even or odd. However, the formula itself cannot instantiate the induction axiom given above, as it is a formula not a predicate. With the comprehension principle, we have: $\exists Q \forall x (Qx \ba x $ is even or $ x $ is odd)$ $. Then we can fix on this property $Q$ and use it to instantiate the induction axiom.\footnote{I am using a very informal argument here to motivate and illustrate the use of comprehension principles. For one, ``even" and ``odd" are not primitive in the formal language of arithmetic, but must be defined. For another, the exact proof does not go the way the informal illustration went. However, these are technical details irrelevant for illustrating the use of comprehension principles, so I will not go into them here.}
+This shows again that (Comp_M) is not some novel/strange principle that Williamson cooked up but a typical example of logical principles in higher-order logic. So to sum up, (Comp_M) is a very natural and useful logical principle that we want to add to our higher-order modal logic. 
+
+Now we can move on to reconstruct Williamson's challenge. Suppose we instantiate $A$ with $x=y$. We will derive the following:
+\begin{quote}
+(Haecceity) \hspace{\labelsep} $\Box \forall y \Box \exists X \Box \forall x (Xx \ba x=y)$ \footnote{Necessitation is: if $\vdash A$, then $\vdash \Box A$; Universal Quantifier Rule is: if $\vdash A$, then $\forall x A$. }
+\end{quote}
+Informally, this says that necessarily everything necessarily has some property such that having this property is necessary and sufficient for being that thing. This property (of necessary and sufficient for being this thing) can be called the haecceity of that individual, following previous literature. 
+
+Next, we can introduce some terminology: let $Haec(X)(y)$ abbreviate $\Box \forall x (Xx \ba x=y)$, informally, ``X is the haecceity of y"; $Tra(X)(y)$ abbreviates $Haec(X)(y) \wedge \neg \Diamond \exists z (Haec(X)(z) \wedge y \neq z)$, informally ``X tracks y". Then, we have: 
+\begin{quote}
+(Tracking)  \hspace{\labelsep} $\vdash Haec(X)(y) \ra \Box Tra(X)(y))$\footnote{I include a full proof in the Appendix (Section 6), and say more about the significance of the proof.} \smallskip \\
+Informally, ``my haecceity necessarily tracks me."
+\end{quote}
+Then consider an individual $o$ (say, John). By the above theorems, we have: 
+\begin{quote}
+($o$-Haecceity) \hspace{\labelsep} $\vdash \Box \exists X Haec(X)(o)$\footnote{We here instantiate (Haecceity) with $o$.}
+\end{quote}
+Putting the above two theorems together, we can derive: 
+\begin{quote}
+($o$-Tracking)  \hspace{\labelsep} $\Box \exists X Tra(X)(o)$
+\end{quote}
+Now the challenge according to Williamson is this: 
+\begin{quote}
+``Even if I had never been, [...], there would still have been a property tracking me (and only me). But how can it lock onto me in my absence? In those circumstances, what makes me rather than something else its target?"\footnote{Williamson, \textit{Modal Logic as Metaphysics}, p. 269.}
+\end{quote}
+In other words, there is a challenge to contingentists who accept (Comp_M) to explain how the haecceity of an individual can track this individual in a situation where that individual does not even exist. More intuitively, one might identify haecceities ostensively: when John is here, I can point to him, and say \textit{the property of being John, that person}. However, in a case where John does not even exists, how can you identify such a property? How can a property in that situation manage to behave like a haecceity of John? Even if some property manages to do that, what can possibly explain why it necessarily targets this non-existing individual but not some existing individual? What would the identity condition be when comparing an existing individual and non-existing one?  
+Put in more formal terms, the contingentist needs to explain why ($o$-tracking) is true while $\Diamond \neg \exists y o=y$. 
+The same challenge can be given for anti-haecceity of individuals, the property $X$ such that $\Box \forall x (Xx \ba x \neq y)$. I will not reiterate the argument here. I call this challenge the Explanation Challenge since it is demanding contingentists to offer an explanation of some sort about consequences of their view. 
+
+I will end this section with a final clarification note on the broader dialectical situation in Williamson' book. The above Explanation Challenge is what Williamson deemed as ``the first horn" in a dilemma for contingentists. The ``other horn" is when contingentists attempt to weaken (Comp_M), which is a natural response if one finds the Explanation Challenge a genuine problem. Williamson in the second half of the chapter argued that this weakening also faces serious problems. Thus, the Explanation Challenge is only a part of a larger argument against contingentism. I do not attempt to survey and evaluate the other horn in this paper. 
+
+\section{Two worries about Williamson's challenge}
+\subsection{The Minimalist Response}
+The minimalist response is motivated by the intuition that there is not really much to explain. In other words, they want to insist that some metaphysical claims do not require substantive explanation. This is a response \emph{on behalf of} contingentists adopting (Comp_M). 
+There are two specific strategies implementing this response: (i) insisting that no explanation is required, and this does not render contingentism in a dialectically weaker position; (ii) insisting that there is a trivial explanation, and so contingentism again does not fair worse. I will develop these two strategies in more detail and respond to them on Williamson's behalf. 
+
+Strategy 1 can be developed in the following, Williamson pointed out that individuals can have contingent existence, but there is always a property tracking them. However, this could be seen just as a brute fact of the modal structure of the world and requires no further explanation. Contingentists do not need to be impressed by this phenomenon at all. This response can be generalized to respond to Williamson's challenge concerning the asymmetry between first-order and higher-order necessitism. The challenge is that given higher-order comprehension principles like (Comp_M), one can prove higher-order analogues of first-order necessitism (which is shown in Section 1), like the following second-order version: $\Box \forall X \Box \exists Y \Box \forall x (Xx \ba Yx)$.\footnote{Williamson, p. 264.} Thus, contigentists will need to endorse this systematic asymmetry between first-order and higher-order claims.  
+A contingentist can just say that this is exactly what they adopt, and the consequent asymmetry is just a fact that does not call for any further explanation. It is worth emphasizing that this minimalist should not be thought of as ``resisting" or ``refusing" to explain ($o$-tracking), but does not see the need to explain in the first place. 
+
+I find this minimalist response unconvincing. My criticism will be different from Williamson's, so I will not reiterate his arguments here.
+First, adopting a minimalist response does not refute or falsify contingentism. It is just that in this dialectical context, contingentism will look much less attractive because there is an alternative theory that has a perfectly simple explanation of ($o$-tracking). More generally, not being able to explain something is no defeat for the theory (probably every theory has something that it has not yet explained), yet in comparing theories, a phenomenon that one theory can readily explain while the other cannot certainly favors one over the other. In this case, necessitism has a very simple explanation of ($o$-tracking): just as the usual case, they can point to $o$ which exists necessarily, and identify the property of \emph{being that thing}. 
+
+Here is an analogy with the Supervenience Challenge to metaethical non-naturalism, the thesis that moral properties are sui generic non-natural properties. The Supervenience Challenge is also an explanatory challenge. The Supervenience Thesis (abbreviated as ``(Supervenience)") of the moral properties on the natural properties claims that two objects cannot differ in their moral properties unless they differ in some natural properties.\footnote{For a more concrete example, we can imagine John and Bill, who are students in the same class. They both arrive at class on time, handed in assignments on time, etc. Now if the teacher start to punish John for alleged moral reasons, he is rightly to object that the teacher's moral assessment is groundless: what could possibly distinguish him from Bill morally? For a more abstract example, one can imagine John in our world and John' in another possible world. Suppose they do exactly the same things and have the same intentions, etc. It seems that they must receive the same moral evaluation (whether that is virtuous or evil): what could possibly distinguish John from John' morally? \\
+Here is (Supervenience) formulated in higher-order logic just to draw out the analogy with the current case more clearly: 
+\begin{quote}
+(ST) \hspace{\labelsep} $\Box \forall X (Moral(X) \ra \forall x (Xx \ra \exists Y (Natural(Y) \wedge Yx \wedge \Box \forall y (Yy \ra Xy))))$
+\end{quote}} 
+Here the challenge for non-naturalism is to explain why (Supervenience) holds. The key is not the first box since that is usually understood to be conceptual necessity but the second box representing metaphysical necessity.\footnote{Note that contingentists cannot appeal to conceptual necessity or facts of meaning to explain ($o$-tracking) since all of the boxes in the theorems refer to metaphysical necessity. At least as Williamson framed the debate, contingentism and necessitism are full-blown metaphysical theories about the world. I suspect that there are ways to think about this debate using conceptual methods, which will be beyond the scope of this paper but interesting to explore.}
+In other words, why the instantiation of natural property necessitate the instantiation of some non-natural property? Just like the minimalist sketched above, some non-naturalists have tried to argue that (Supervenience) does not need an explanation. It is just a fact about the metaphysical structure of the world. This quietist response does not falsify non-naturalism. It just puts non-naturalism in a dialectically weaker position, especially when there are alternative theories which offer an explanation: naturalism does this by identifying moral properties and natural ones. 
+The upshot is that the force of the Explanation Challenge does not derive from posing a counterexample/contradicting contingentism but identifying a source of explanatory weakness.
+ 
+Further, I think when phrased in terms of explanation, the burden will be on to contingentism to say why the phenomenon does not demand explanation. Here is a parallel in the sciences. We encounter some natural phenomenon: water freezes in winter, leaves fall down in the fall, etc. The default is that these all call for explanation. The only exception might be that when we get to the most fundamental level of nature:  only when we get to the fundamental particles can we say: those particles just have those properties they have, by nature. There is nothing more we can say. It is simply bad science if one look at a macroscopic phenomenon and just say it is just there and requires no special explanation at all.\footnote{While for daily life/practical purposes, this attitude is entirely justified, it is not for scientific purposes. Otherwise, it is hard to see how explanatory science can ever get started. Consider vision science. One basic question is, what explains our visual capacity? If a person comes along and says ``Well, I can see those things, and not some other things. That's just how I am evolved. What else is to explain?'' That is just bad vision science.}
+ The same applies for modal metaphysics as long as it aspires to be (explanatory) science. The default is to assume every modal facts about ordinary objects require explanation, and only when we get to the most fundamental level can we resist the explanation demand. The upshot of all these is that a minimalist adopting strategy 1 will simply be doing bad metaphysics. 
+ 
+Strategy 2, which is more interesting, can be developed in the following way. Contingentists can accept that there is an explanatory demand but argue that there is a trivial explanation. Specifically, contingentists can argue that higher-order necessitism and tracking follows \emph{logically} from $Comp_M$ and the background logic, and that explains why there is tracking and ``locking on to individuals." Again, there is nothing further to it. To take an analogy, suppose one was asked why he believes that $A$ and $B$, he can answer: ``I believe $A$, and I believe $B$, so I believe the logical consequence of my beliefs, namely $A$ and $B$.'' He has indeed provided an explanation of his belief in the conjunction, though a trivial one. Further, the contingentists can even use Williamson's argument in the latter half of the chapter: Williamson discusses how (Comp_M) is the superior comprehension principle in higher-order logic and various technical reasons why one wants to adopt this as part of the logic rather than weaker principles: simplicity, elegance, and enough expressive power to serve logical/metaphysical purposes. Thus, contingentists can even maintain that they have independent reasons to adopt (Comp_M) and argue that ($o$-tracking) is a logical consequence. 
+
+I find this strategy to be problematic, too. 
+Firstly, it seems that the logical consequence explanation does not give us a deep-enough explanation. Suppose a contingentist does offer this explanation; a necessitist can just inquire further for an explanation of (Comp_M). Why is that for any condition whatsoever there exists a property such that that condition holds just in the case it is instantiated? Notice now, contingentists cannot use the Williamsonian justification for (Comp_M), since that will answer the wrong question - why we should \emph{believe in/accept} (Comp_M). Thus, the explanatory demand just gets pushed further back. In general, one can always explain $p$ by saying that it is logical entailed by some $q$. But then the explanatory demand just got pushed back to why $q$. Of course, if a theory has to push back infinitely, then it is not a good theory. 
+
+Secondly, I worry that logical consequence is too lax an explanatory basis; that it generates bad explanations. 
+Here is an example that I have in mind:
+
+\begin{quote}
+(Racist Explanation). \hspace{\labelsep} The Racist believes that every member of race A is evil. Consequently, he believes that a member of A $o$ is evil. When asked why he thinks $o$ is evil and consequently refused to offer $o$ equal payment/respect as other employees, the Racist says, ``Well, this is a logical consequence of my belief. What more do you want me to explain?''
+\end{quote}
+
+There is something wrong with both explanations.\footnote{Of course the Racist is subject to my previous challenge as well: I can simply ask the Racist why he believes that every member of race A is evil in the first place. He is not discharged of explanatory demand. But here I am developing a different problem for the Racist.} 
+One, what validates the Racist's universal belief is precisely the character of each individual. Thus, the Racist is not entitled to appeal to this universal belief as an explanation of his specific belief. Two, the Racist's belief attributes some structure to human beings in general and to particular individuals like $o$. He cannot simply ignore this structure by appealing to the general principle that attributes this structure. He needs to look at this specific person and explain why the ``evil" structure is really there, as his theory claims it is. 
+One can now see the same problems hold for contingentism too: (i) ($o$-tracking) is supposed to support (Comp_M), not the other round. Whether contingentism should accept (Comp_M) partially depends on whether they find its consequences compelling. The logical consequence explanation gets the order of explanation wrong; (ii) contingentists by adopting (Comp_m) attribute some metaphysical structure to individuals (like ($o$-tracking)), and they now need to find actual features of the world/individuals that support this structure in order to vindicate their theory. In either case, contingentists are not done.
+To sum up, logical consequence cannot serve as a good explanation. Consequently, Strategy 2 fails.\footnote{An anonymous reviewer asked whether there are previous contingentists literature defending the worry that I criticized in this section. The reviewer asked because my criticism of this contingentist response seems sweeping, and so this response may seem like a ``low hanging fruit objection that contingentist philosophers would, in general, avoid". There are several things to say: \\
+Firstly, the relevant literature does not seem to have focused particularly on the Explanation Challenge and the role of explanation. 
+Second, although my objections can seem comprehensive, I think they only scratch the surface of the relevant problems related to explanation, the need for explanation, etc. I am sure there are insights that contingentists can bring in from the literature about explanation in general (and philosophy of science etc) to resist my arguments. However, my goal in this paper is to articulate more clearly a worry that was raised in the seminar discussing this book and give some compelling response to it. So I did not get into potential larger debates about explanation and related notions.}
+
+\subsection{A worry about the notion of explanation}
+The following worry is not so much directly on behalf of contingentists but trying to undercut Williamson's entitlement to raise the Explanation Challenge in the first place. 
+
+There are two parts to this worry. 
+Firstly, one might be skeptical of the notion of explanation evoked here. At a first pass, the explanation demanded seems metaphysical.\footnote{It certainly is not causal or physical: there is no causal relationship or ``physical" process in place. It is not constitutive either: nothing seems to constitute another. Normative explanation is certainly not what is being demanded: they are descriptive claims through and through.} However, Williamson is skeptical of notions like grounding and truth-making (the often-evoked notion in metaphysical explanation), so they cannot be used to give metaphysical explanation. Thus, how is Williamson's demand for a metaphysical explanation (potentially using these notions) from the contingentists legitimate? To put it in another way, suppose Williamson thinks that the notion of ``metaphysical" explanation (as distinct from physical/causal, social, mathematical, etc) is bogus, and there is no genuine metaphysical explanation at all. Then, trivially, there is no genuine metaphysical explanation for the truth of ($o$-tracking) and (Contingentism). So how can he demand contingentists perform this task? Perhaps more strongly, he should actually be happy with contingentists giving no metaphysical explanation as that is exactly what they should be doing. 
+
+Secondly, one might be skeptical of whether Williamson is entitled to this challenge given his methodology. Williamson is very insistent on evaluating metaphysical theories by appealing to their simplicity, strength, and elegance. One of the main threads of his book argues that necessitism allows us to have much simpler semantics for quantified modal logic by allowing us to treat possible world models realistically, adopt simple elegant axioms, avoid complex restrictions in the proof theory, and have an expressive high-order language. Those notions like simplicity/strength are easily demonstrated by observing the kind of formal apparatus present. In contrast, the notion of explanation is comparatively murky.
+
+I think both worries can be dissolved. 
+Regarding the first worry, Williamson can respond in the following ways. 
+One is that we do not have to fix on the nature of explanation prior to giving one. There is an intuitive grasp of explanation that one can rely on. Consider the Supervenience Challenge again. There, a ``metaphysical" explanation is also demanded. But one does not necessarily have to give a ``metaphysical" explanation in terms of grounding, metaphysical laws, essences, truth-making, etc. Naturalists just say that moral properties are identical to natural ones, so (Supervenience) is a trivial truth; expressivists just say that I cannot explain (Supervenience) as it is about ``inflated" properties, but I can explain why people have good reasons to commit to (Supervenience).\footnote{Gibbard had given one such account. Now whether expressivists are thereby discharged is actually not trivial. See Dreier, 2015. Dreier argues that quasi-realist formulation of expressivism, which one may reject, is not off the hook. However, this is beyond the scope of this paper.} Those explanations are in no way ``metaphysical". Similarly, the explanation of ($o$-tracking) does not need to end up being distinctively metaphysical.\footnote{The necessitists' explanation is not metaphysical at all. One can just imagine the counterfactual circumstance and point to the existing object, concrete or not, and say the property of being that object.} 
+In fact, it is actually the \emph{contingentists} for whom we might have this kind of worry about problematic ``metaphysical" explanations, because they presumably need to give some ``metaphysical" explanations for ($o$-tracking), just like one might be skeptical of non-naturalism when non-naturalists invoke all kinds of metaphysical notions/relations (normative laws, grounding, normative essences, hybrid properties, etc) in their explanation of (Supervenience).\footnote{To illustrate, Stephanie Leary posited hybrid properties that have both normative and natural essences to explain (Supervenience) --- see Leary, 2017. In general, I think that her additional ontology is not independently motivated and appears very ad hoc as it does not handle any other major (explanatory) challenges to non-naturalism like explaining our moral knowledge or making sense of alien communities and their ``moral" language. Of course, these objections need to be further elaborated and defended, which is beyond the scope of this paper. The point I am making here is that one can \emph{easily} come to worry about whether these metaphysical entities/postulates are real, whether they are really motivated, whether they really explain anything, etc. The upshot is that these (potential) worries about non-naturalist evoking heavy-duty metaphysical notions are exactly parallel to worries about contingentists evoking heavy-duty metaphysical notions.} Thus, in fact, having to recourse to inflated metaphysical explanation is a disadvantage for contingentists, not for Williamson. 
+
+Regarding the second worry, I think the best response is to argue that theory comparison cannot \emph{all} be about simplicity, strength, and elegance especially when the theory is about the actual world. This is true for scientific theories. Comparison between physical theories cannot all come down to which one has more simplicity, strength, and elegance. These formal criterion are very important and rules out many strange theories, but they cannot be all there is to it. Physicists look at fit with empirical data, explanatory power, etc. One cannot decide between Newtonian mechanics or Relativisitic physics just on their formal features. 
+Here is a more specific version for contingentism vs. necessitism. One cannot just decide between them based on formal features. Take the argument that necessitism provides a simpler formal semantics for quantified modal logic. Whether this is true depends on what language one is speaking. Suppose one is speaking a contingentist-commiting language, then of course contingentist semantics will be a better fit and simpler.\footnote{To see the challenge: consider religious discourse. There is this word ``God" that appear in the discourse very often. What is its meaning? In some sense, a semantics that assigns it a supernatural being is the simplest as it will validate all the sentences in which the word occur, compared to a semantics that does not. But of course, this cannot be an argument for the existence of ``God".} A similar reverse argument applies to logical strength. Contingentists might say if you adopt all those principles, it will vastly over-generate statements that are false or suspicious so one has to block their derivation. That surely does not look elegant. The upshot is that for empirical theories, we need to look at their empirical content and whether that fits with the world in the right way. 
+Thus, Williamson need not only appeal to formal notions to adjudicate between theories. He is completely entitled to appeal to explanatory power and fit with data to evaluate the two theories. This accords with Williamson's general methodology in any case: modal metaphysics is a science. 
+
+\section{An Anti-Haecceitist response}
+In this section, I explore a substantive contingentist response to Williamson's challenge. I argue that if true, this successfully answers Williamson's challenge, but this response ties contingentism to another controversial metaphysical doctrine. 
+
+The response is to first adopt anti-haecceitism and then use that to explain ($o$-tracking) even when $o$ does not exist. Anti-haecceitism (abbreviated as AH) is roughly the thesis that there are only qualitative properties (consequently all seemingly non-qualitative properties are reducible to qualitative ones). Qualitative properties are everyday familiar properties like volume/size, color, material composition, or relational properties (\emph{being the mother of}, etc).\footnote{Robert Stalnaker, \textit{Mere Possibilities: Metaphysical Foundations of Modal Semantics} (Princeton University Press, 2012).} Non-qualitative properties are harder to describe directly for they are intended to pick out precisely those properties that are not qualitative. Perhaps the most intuitive example is the property of \emph{being this very object}.\footnote{I think it is hard to identify non-qualitative properties in an entirely theory-neutral way, since AH precisely denies their existence! However, generally, non-qualitative properties are motivated by thought experiments like the following: Consider a possible world where there are only two balls. They are identical in every aspect: shape, size, color, material, etc. Suppose one brings in spatial locations or relationships to an observer, then one is ``illegally" bringing in distinct qualitative properties or assuming the existence of an observer that the thought experiment stipulated not to exist. Thus, there seems to be no way of distinguishing between them but through properties like \emph{being this very ball} and \emph{being that very ball}. }
+The point of debate is whether for a particular individual $o$, the property of being $o$ is qualitative or not. AH says yes, haecceitists say no. In other words, AH maintains that this property of \emph{being $o$} can be reduced to a complex qualitative property like the property of \emph{being the individual that occupies this location, has this height/weight/taste/wealth, ...}. 
+Now we can see the AH response to Williamson's challenge. Suppose that $o$'s haecceity $X$ can be reduced to this complex qualitative property built out of simple qualitative properties. Suppose further that $o$ does not exist in some counterfactual world. In this situation, very plausibly those simple qualitative properties still exist and so does this complex property. Then indeed there will be a haecceity of $o$, namely this complex property, and it locks on to $o$ because by stipulation whatever instantiates it will be identical $o$ instead of identical to other objetcs. $o$ is ``defined" by all and only these properties.\footnote{Here $o$ does not have to be identified with those properties. So the AH explanation does not have to commit to a bundle theory of individuals.} Note that $o$ does not have to exist: this complex qualitative property is just not instantiated in this situation. So AH gives contingentism a \emph{substantive} explanation of why $o$ can fail to exist while ($o$-tracking) is still true. 
+
+One might be tempted to say that AH targets the wrong explanandum. The idea is that AH response is phrased in terms of properties and individuals instantiating them, but Williamson's (Comp_M) and ($o$-tracking) are all formulated in higher-order logic. Further, if one thinks that higher-order quantification is not first-order quantification over objects of higher-type, then the AH response has failed to explain the right thing. They have explained why properties of a certain kind track the individual, but they have not explained why $\Box \exists X Tra(X)(o)$ given $\Diamond \neg \exists y o=y$. 
+However, this response is not successful since one can easily formulate AH in higher-order logic: 
+(AH) $\forall x \forall X (Haec(X)(x) \ra Q(X)$ where $Q(X) := $X is qualitative$ $. \\
+Then one can reformulate the AH response above in those terms. 
+In fact, after the formulation in higher-order logic, there can be a very clear path to a successful explanation. Suppose one has discovered that: \begin{quote}
+$\Box \forall x (x=o \ba P_1x \wedge P_2x \cdots \wedge P_nx)$ where $P_i$ is a qualitative property for any $i$. \end{quote}
+Then one can form a complex predicate $P_{total}$ such that: 
+\begin{quote}
+$\Box \forall x(x=o \ba P_{total}x)$ where $P_{total} := \lam x. P_1x \wedge P_2x \cdots \wedge P_nx$. \end{quote}
+Then one can easily, by Existential Generalization, derive:
+\begin{quote}
+$\exists X \Box \forall x(x=o \ba Xx)$ \end{quote}
+
+Williamson has various responses to his anticipated AH response. Against a purely qualitative conception of properties, he said: 
+\begin{quote}
+``Thus the purely qualitative conception of properties may well require a highly contentious form of the identity of indiscernibles for individuals, on which qualitative identity entails numerical identity. That is a far less plausible claim than the trivial form of the identity of indiscernibles that permits non-qualitative properties such as identity with y. We have no serious evidence against the metaphysical possibility of a symmetrical universe in which every individual can be reflected (rotated, translated) onto its qualitative double."\footnote{Williamson, \textit{Modal Logic as Metaphysics}, pp. 271--272.}
+\end{quote}
+However, I think this response targets AH as a self-standing doctrine rather than the AH explanation of the Explanation Challenge, which I will come back to. 
+
+Against the previously developed AH response, he said: 
+\begin{quote}
+``The theory becomes still more elaborate once fitted out with an account of the persistence of individuals across times and possibilities, since an individual typically has many of its purely qualitative properties, such as shape and size, temporarily and contingently. Alternatively, if the theory denies identity through change and through contingency, not only is that yet another implausible consequence, which requires still more theoretical complexity to save the appearances, it also fits badly with the underlying motivation for contingentism, by treating a vast range of apparent contingency as an illusion. Thus the purely qualitative conception drags the contingentist into proliferating complications of metaphysical theory with no independent plausibility.''\footnote{Williamson, p. 272.}
+\end{quote}
+I find the remarks here sketchy, too. 
+For one, I am not sure if AH should regard having contingent qualitative properties as a problem for their theory. Either they can say that the qualitative properties include modal and temporal properties (so \emph{being John} involves \emph{being a possible lawyer}) or they can index haecceities such that one should really say: $X$ is a haecceity of $y$ at $w,t$. In this second way, the original theorems are still in place because, in any world $X$ is still a haecceity of $y$ at a particular indice $w,t$.\footnote{Now I do not want to get into the debate between AH and haecceitis here. These two ways of responding to Williamson's challenge at least seem prima facie available. They could be false in the end, but at least they show AH is not easily defeated. } 
+Further, and more importantly, I do not think his challenge shows that the AH explanation is unsuccessful. Williamson at best shows that AH has various undesirable consequences.
+Thus, overall, I think Williamson's own remarks in the chapter do not constitute the right kind of dialectic challenge to the AH response.  
+
+However, Williamson's skepticism brings us to what I think is a more successful general response to the AH explanation. The idea is to admit that the AH strategy does respond to Williamson's challenge, and to point out that this AH response forces contingentism to accept a controversial metaphysical doctrine, thereby having to accept its dialectical challenges. 
+
+Firstly, adopting the AH strategy forces contingentists to be anti-haecceitists, thereby having to answer challenges to anti-haecceitism itself. 
+Contingentists need to accept AH to give the AH explanation. Granting that AH does offer a successful explanation of ($o$-tracking), we can say that
+if AH is indeed true, then we can solve Williamson's Challenge. Consequently, the plausibility of the theory package consisting of solving Williamson's Challenge and AH - will be the plausibility of AH itself. Assuming that Solving Williamson's Challenge amounts to vindicating contingentism, the plausibility of this version of contingentism (which will be the conjunction of AH and (Contingentism)) will be the plausibility of AH itself.\footnote{Readers familiar with probability theory might think in terms of probabilities assigned to these propositions: the conditional probability P(Solving Williamson's Challenge $|$ AH) is roughly 1. So P(Solving Williamson's Challenge $\wedge$ AH) will be equal to P(AH). So  P(Contingentism $\wedge$ AH) = P(AH). These follow from basic probability axioms and logic. }
+Now there is a problem if AH is not very plausible. If AH is not a compelling metaphysical thesis in the first place, contingentism will not fare well having to accept it.
+This is where many of Williamson's previous charges can be properly incorporated: skepticism about distinction between qualitative and non-qualitative properties, counterexamples from indiscernible objects, etc. The upshot is that now the dialectic cost for contingentism is no longer not being able to explain something but having to accept some controversial/implausible doctrine in order to be able to explain something. This seems to be a cost. Contingentists give themselves a greater burden compared to a necessitist that can remain neutral on this issue. 
+
+I am personally not very worried about combining a theory with another controversial theory in itself. Intuitively, theories should be allowed to appeal to other resources (like other theories) in developing and defending itself even if those resources are controversial. Denying the legitimacy of this appeal would render theorizing very difficult and limited. We want to establish connection between theories across domains and explore how they can inform each other.\footnote{I can give numerous examples. For example, expressivists in metaethics appeals to truth minimalism to recover the legitimacy of ordinary moral talk/thought, even if truth minimalism is controversial; non-naturalists appeals to post-modal/hyperintensional metaphysics in developing their theories, even if notions like grounding/essence invoked in hyperintensional metaphysics are very controversial --- see Bengson, Cuneo, and Shafer-Landau, 2024. I think they can make these appeals. Metaethicists have made a lot of progress by doing this. Their theorizing would just be very limited if they cannot do this. }
+However, the real worry is whether this combination is the only viable combination (because we do not know which theory is ultimately right). That is, if contingentism can only be effectively defended relying on a particular theory of haecceities, then it looks less attractive than a view that is compatible with a variety of theories of haecceities. 
+Contingentism is not supposed to be a global thesis that aims to provide complete answers to all metaphysical questions. It is not even aiming to be a comprehensive theory of metaphysical modality. Necessitism is the same. Thus, one would hope that it can remain local instead of having global consequences. However, if it can only be a good local theory when combining with a particular (global) theory, then one should be more skeptical as this local theory seems to demand too much packaged in along with it. Necessitism in contrast is compatible with both haecceitism and AH.\footnote{This is exactly the same for expressivism and truth minimalism. Expressivism is meant to be a local thesis about moral language. However, to defend it, one would need to reject truth-conditional semantics (which is incompatible with truth minimalism) in general, then it no longer seems very attractive.}
+Thus, overall, while AH response is a good substantive explanation answering the Explanation Challenge, there will be considerable dialectical cost for contingentists to accept it.  
+
+\section{Conclusion}
+In this paper, I have examined three responses to Williamson's Explanation Challenge and argued that each response faces their own problems. While I argue for the stronger conclusion that the first two challenges fail, I argue for the weaker conclusion that the last response succeeds but only with additional dialectical cost to contingentism. I hope this paper has helped to clarify the stake of Williamson's ``first horn" to contingentism in Chapter 6 and strengthen his argument against contingentism. 
+
+\section{Appendix}
+
+\subsection{The proof for (Tracking)}
+First, we can observe the following proof: \\
+\begin{quote}
+\begin{tabulary}{\textwidth}{Lr}
+$\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash \bot \ $ & (Reductio, Cond. proof, Universal generalization)  \\
+\end{tabulary}
+\begin{tabulary}{\textwidth}{Lr}
+$\vdash \Box \forall x (Xx \ba x=y) \ra \Box \forall z \neg (Haec(X)(z) \wedge z \neq y)$ & (K) \\
+$\vdash \Box \forall x (Xx \ba x=y) \ra \neg \Diamond \exists z (Haec(X)(z) \wedge z \neq y)$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & (Equivalence) \\
+$\vdash Haec(X)(y) \ra Tra(X)(y)$& \\
+\end{tabulary} % terrible hack but it fixed the rendering, I am very sorry Mohit
+\end{quote}
+Now I will show the first line.
+\begin{quote}
+$\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash Xy \ba y=y \\
+\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash Xy \ba z=y \\
+\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash y=y \ba z=y \\
+\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash z=y \\
+\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash z \neq y \\
+\forall x (Xx \ba x=y), Haec(X)(z) \wedge z \neq y \vdash \bot $ \\
+\end{quote}
+Then, from $\vdash Haec(X)(y) \ra Tra(X)(y)$, we can observe that: \\
+\begin{quote}
+\begin{tabulary}{\textwidth}{LR}
+$\vdash \Box Haec(X)(y) \ra \Box Tra(X)(y)$ & (K)  \\
+$\vdash \Box \forall x (Xx \ba x=y) \ra \Box \Box \forall x (Xx \ba x=y)$ & (\textbf{4})\\
+$\vdash Haec(X)(y) \ra \Box Haec(X)(y)$ & (Chaining conditionals) \\
+$\vdash Haec(X)(y) \ra \Box Tra(X)(y)$ &
+\end{tabulary}
+\end{quote}
+\subsection{The proof for (o-Tracking)} 
+\begin{quote}
+\begin{tabulary}{\textwidth}{LR}
+$\vdash Haec(X)(o) \ra Tra(X)(o)$ & (Proved above) \\
+$\vdash \exists X Haec(X)(o) \ra \exists X Tra(X)(o)$ & (Derivable from $\forall$ rule) \\
+$\vdash \Box \exists X Haec(X)(o) \ra \Box \exists X Tra(X)(o)$ & (K)\\
+$\vdash \Box \exists X Tra(X)(o)$ & (MP, $o$-Haecceity) 
+\end{tabulary}
+\end{quote}
+
+
+\noindent I include these proofs in detail for two reasons. One, Williamson did not lay out the proof at all in the book. So I think reconstructing it here will help the reader to see clearly how the seemingly strong principles are derived. Second, and more importantly, this proof shows how little background logic is needed to derive the later-shown-to-be-problematic (Tracking). This proof assumes only modal logic principles \textbf{4} and \textbf{K}, and the usual meta-rules like conditional proof, reductio, etc. Thus, it  does not require a strong logic to prove (Tracking). The significance is that, suppose one accepts that (Tracking) has problematic consequences, one thing we can always see is if there is any logical principle we can reject which contributes to the proof. That would be a natural contingentist way out. However, this proof shows that it will not be easy to take this route. K is the least contentious axiom in modal logic; \textbf{4} is somewhat controversial, but not very, since intuitively, modal properties/facts should themselves be necessary and not mere accidental. Further, the controversial B axiom that actually bears on the necessitism and contingentists debates are not used essentially. Thus, there is not much reasonable/non-ad hoc room for contingentists to weaken their background logic to escape from Williamson's challenge. Williamson himself does not make this point, but I think it is important. 
+
+\refsection
+
+\begin{hangparas}{\hangingindent}{1}
+  Bengson, John, Terence Cuneo, and Russ Shafer-Landau. \textit{The Moral Universe.} Oxford University Press, 2024.
+
+  Dreier, James. ``Explaining the Quasi-Real.'' \textit{Oxford Studies in Metaethics} 12 (2017): 273--297.
+
+  Leary, Stephanie. ``Non-Naturalism and Normative Necessities.'' \textit{Oxford Studies in Metaethics} 12 (2017): 76--105.
+
+  Stalnaker, Robert. \textit{Mere Possibilities: Metaphysical Foundations of Modal Semantics}. Princeton University Press, 2012.
+
+  Williamson, Timothy. \textit{Modal Logic as Metaphysics}. Oxford University Press, 2013.
+  \end{hangparas}
+%%% Local Variables:
+%%% mode: LaTeX
+%%% TeX-master: "../main"
+%%% End:
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+    \noindent
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+    $D_1\neg F$ \bigskip \\
+    $D_2D_1F$ \bigskip \\
+    $\neg D_2 D_1 F \; \& \; \neg D_2 \neg D_1 F$ \bigskip \\
+    $D_2 \neg D_1 F \; \& \; D_2 \neg D_1 \neg F$ \bigskip \\
+    $\neg D_2 D_1 \neg F \; \& \; \neg D_2 \neg D_1 \neg F$ \bigskip \\
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