From e00d4ffcf45fcac8af01946c23c4bf6a7af1c4f9 Mon Sep 17 00:00:00 2001 From: Jacob Walchuk Date: Mon, 30 Jun 2025 22:38:47 +0100 Subject: Paper 2 everything except graphics (forgot to stage) --- papers/2.tex | 1082 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1082 insertions(+) create mode 100644 papers/2.tex (limited to 'papers') diff --git a/papers/2.tex b/papers/2.tex new file mode 100644 index 0000000..37d36e7 --- /dev/null +++ b/papers/2.tex @@ -0,0 +1,1082 @@ +\chapter{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness} +\chaptermark{Supervaluationism, Dynamic Supervaluationism, and Higher-Order Vagueness} + +\renewcommand*{\thesection}{\arabic{section}.} +\renewcommand*{\thesubsection}{\arabic{section}.\arabic{subsection}.} + +\begin{quote} +The fact that the phenomenon of vagueness can itself be vague +- and its vagueness be vague as well - seems impossible to make sense of +without getting a headache. This so-called higher-order vagueness makes +theorising about vagueness a notoriously difficult task for philosophers +of logic and language. This difficulty manifests itself in that, even if +a theory can convincingly explain what vagueness is and how we can +reason about it, when faced with the vagueness of the just-tamed +vagueness, it gets flooded with paradoxes and makes the initial theory +seem implausible. In this paper, I argue that Rosanna Keefe's +supervaluationism is one such theory. Even though it elegantly accounts +for the first order of vagueness, it becomes less elegant when +questioned about the higher orders. To demonstrate this, I show that +Keefe's system fails to resolve various paradoxes of higher-order +vagueness such as the finite series paradox or the D* paradox. +Furthermore, I argue that in her attempts to accommodate the paradoxes +by adopting a rigid hierarchy of metalanguages, Keefe invites new +worries. Given these criticisms, it is unlikely that Keefe's theory can +be `argued out' of these paradoxes - `finite series' in particular. +Instead, I argue that the theory must be substantially modified if it is +to be salvaged, and one way to do so is by making the proposed structure +more dynamic. I attempt to do so by sketching an outline of dynamic +supervaluationism that can tackle the problems that Keefe's +supervaluationism cannot. I close my essay by teasing out some +challenges that the proposed theory could face and offering possible +solutions. I believe that supervaluationism is a very attractive +approach to vagueness and therefore, it is worth developing further into +a more robust theory that could tackle its higher orders. +\end{quote} + +\section{Introduction} + +Vagueness in language refers to an indeterminate relationship between +its terms and the world they describe.\footnote{Kit Fine, + \emph{Vagueness: A Global Approach} (Oxford Academic, 2020), 2-3, + \url{https://doi.org/10.1093/oso/9780197514955.001.0001}.} Minimally, +a predicate is vague if it has three features: \textbf{admission of +borderline cases} (objects to which its application is unclear), +\textbf{a lack of known, sharp boundaries} (no clear case separating the +positive and negative cases), and (apparent) \textbf{susceptibility to +the Sorites paradox}.\footnote{Rosanna Keefe, \emph{Theories of + Vagueness}, (Cambridge University Press, 2000), 6-7.} + +Vagueness is philosophically relevant because it raises two problems. +First, the \textbf{semantic problem}: since the vague extension is +unclear, classical semantics (where meaning is derived from extension), +and hence classical logic, may not apply. Second, the \textbf{Soritical +problem}. Consider a series of people of descending heights by 1cm. The +first is clearly tall (200cm) and the last is clearly not (120cm). Since +no known boundaries exist, vague predicates are tolerant - a small +change will not alter the application. Thus, by inductive step, for any +case $\mathbf{n}$, `if $\mathbf{n}$ is tall then $\mathbf{n+1}$ is tall'. +Starting at 200cm is tall, via a series of conditionals, you validly +conclude that 120cm is tall. However, this is a contradiction since +120cm is clearly not tall.\footnote{Fine, \emph{Vagueness}, 3-7.} This +argument exemplifies the classical form of the Sorites paradox. + +Theorizing about vagueness involves accounting for the nature, source +and meaning of vagueness, providing vague semantics and resolving the +Sorites. Furthermore, since it is unknowable where the positive +extension changes to negative, it is equally unknowable where the +positive changes to borderline. Thus, borderline cases themselves should +be unbounded; hence there should be borderlines to borderlines. The +process could be iterated to establish a possibly infinite hierarchy of +borderline cases: the higher-order vagueness (HOV).\footnote{Keefe, + \emph{Theories of Vagueness}, 31-32.} + +Throughout this paper, I will follow Rosanna Keefe and other major +supervaluationists in assuming that HOV is a genuine problem, that needs +to be accounted for. However, it is worth pointing out that this is a +debated matter in the field.\footnote{Some philosophers, such as Dominic + Hyde, claim that higher-order vagueness (HOV) is a pseudo-problem, + arguing that the vagueness of vague is a real, but unproblematic, + phenomenon. Others, including Hao-Cheng Fu and Susanne Bobzien counter + that this stance fails to adequately address the complexity of the + issue, maintaining that HOV is indeed a genuine problem. While an + extensive discussion is beyond the scope of this essay, see Hyde, "Why + Higher-Order Vagueness Is a Pseudo-Problem"; Fu, "Saving + Supervaluationism from the Challenge of Higher-Order Vagueness + Argument''; and Bobzien, "In Defense of True Higher-Order Vagueness" + for further details.} Nevertheless, under this assumption a successful +theory of vagueness, given its commitments, must also account for HOV. + +In this essay, I explore how one theory of vagueness -- +supervaluationism, advocated by Rosanna Keefe -- does so. First, I +outline her account of first-order vagueness (FOV). Then, I explain the +problems posed by HOV, examining Tim Williamson's criticisms of the +theory and how Keefe accommodates them. I will argue that although the +Williamson problems are solved, the resulting view does not reflect how +language actually functions and is paradoxical, making the HOV account +unsatisfactory. I then attempt to modify the view by dynamizing it, +developing the ideas of Hao-Cheng Fu. I defend the model by showing how +it solves some of the critical issues faced by Keefe. Lastly, I raise a +few possible issues endemic to the dynamic view and sketch responses to +defend it. + +\section{Supervaluationism, a theory of vagueness} + +Supervaluationists claim that vagueness is a problem of language, not +our epistemic capacities. They argue that vague predicates fail to draw +sharp boundaries, not that these boundaries are unknowable, and that +they admit borderline cases. The source is semantic indecision. A vague +predicate admits a range of possible extensions, but it is semantically +unsettled which one is correct. This is captured through the notion of +precisification, a way to make a vague term precise.\footnote{Keefe, + \emph{Theories of Vagueness}, 154-156.} A precisification must be +admissible, reasonable in not licensing a misuse of language.\footnote{Timothy + Williamson, \emph{Vagueness}, (Routledge, 1994), 158.} It also must be +complete, it categorizes objects into positive and negative extensions, +leaving nothing in-between. For illustration, consider the vague +predicate `tall'. We could (reasonably) use precisifications: `tall' is +true if `\textgreater175cm', `\textgreater180cm' and +`\textgreater190cm', each of which would precisely divide objects into +positive and negative extensions. Vague terms do not `choose' between +these; instead, all precisifications are equally good.\footnote{Keefe, + \emph{Theories of Vagueness}, 154-156.} + +Supervaluationists provide semantics for vague predicates, identifying +truth with super-truth by considering all possible precisifications. +$\mathbf{Fa}$ is super-true (-false) iff $\mathbf{F}$ is true (false) of +$\mathbf{a}$ under all complete and admissible precisifications. +$\mathbf{Fa}$ is neither true nor false iff $\mathbf{F}$ is true of +$\mathbf{a}$ under some precisifications and false of $\mathbf{a}$ under +others.\footnote{Keefe, \emph{Theories of Vagueness}, 154.} + +Thus, vague predicates divide objects in a three-fold manner, where +borderline cases are not assigned a definite truth value. Hence, +supervaluationists give up bivalence, departing from classical +semantics, by admitting truth value gaps. On the other hand, classical +logic is mostly preserved because if a sentence is classically true, +then it is true on all complete and admissible precisifications. +Consider the law of excluded middle. Using any precisification of tall +-- every object will be either tall or not-tall, since every +precisification divides objects into two sharp sets. Similarly, all +classical theorems are retained, thus we can use classical logic to +reason about vague predicates.\footnote{Rosanna Keefe, ``Vagueness: + Supervaluationism,'' \emph{Philosophy Compass} 3, no. 2 (2008): + 162-164.} + +This idea provides a straightforward solution to the Sorites. Namely, +the inductive premise `if $F\mathbf{n}$ then $F(\mathbf{n+1})$' is +super-false, since the antecedent will be true and the consequent false +for some $\mathbf{n}$ under any complete and admissible precisification. +This is because each precisification, being complete, provides a sharp +cut-off between the true and false -- a bordering pair where the first +entry is true and second one is false.\footnote{Keefe, ``Vagueness: + Supervaluationism,'' 315-316.} Thus, the supervaluationist account +fulfils the initial demands of theorizing about vagueness. Consult the +footnote\footnote{Consider the series of people of varying heights again + and suppose some examples of complete precisifications: $\mathbf{x}$ is + short if (1) `$<160\text{cm}$' or (2) `$<165\text{cm}$' or (3) + `$<170\text{cm}$'. They are complete since they divide objects into + positive (short) and negative (not-short) extensions with nothing + in-between. It is easy to see how the inductive premise turns out + false on each of these precisifications: (1) `If 159cm is short, then + 160 is short'; (2) `If 164cm is short, then 165cm is short'; (3) `If + 169cm is short, then 170cm is short'. In each case, the antecedent is + true and the consequent false (relative to precisification). Since the + inductive premise turns out false for some pair under each complete + precisification, it is super-false.} for further clarification. + +\section{Supervaluationism and higher-order vagueness} + +The above metalanguage (talk of truth conditions) expresses the +vagueness of the object language by dividing cases into three sharply +bounded sets (true, false, borderline). This can be captured by adding a +`definitely' D operator to the object language, which functions akin to +modal necessity. + +The FOV of F is expressed as: + +\begin{enumerate} +\def\labelenumi{(\arabic{enumi})} +\item + $DFx$ for definite positive cases (true under all complete and + admissible precisifications) +\item + ${\sim} DFx \, \& \, {\sim} D {\sim} Fx$ for borderline cases (true/false + under some) +\item + $D {\sim} Fx$ for negative cases (false under all) +\end{enumerate} + +This division is problematic since all cases are sharply categorized, +allowing no borderlines between the definite and borderline cases, +leaving no scope for HOV. Supervaluationists argue that this can be +resolved by allowing the concept of `admissibility' itself to be vague, +thus making the metalanguage vague.\footnote{Keefe, \emph{Theories of + Vagueness}, 202-204.} + +Hence, the second-order vagueness of F is captured in the +meta-metalanguage by expressing vagueness of DF (the metalanguage). This +yields the following five-fold classification: + +\begin{enumerate} +\def\labelenumi{(\arabic{enumi})} +\item + $DDFx$, i.e., definitely definitely positive cases +\item + ${\sim} DDFx \; \& \; {\sim} D {\sim} DFx$, i.e., borderline between positive + and borderline +\item + $D {\sim} DFx \; \& \; D{\sim} D{\sim} Fx$, i.e., definitely borderline cases +\item + ${\sim}DD{\sim} Fx \; \& \; {\sim}D{\sim}D{\sim}Fx$, i.e., borderline between + negative and borderline +\item + $DD{\sim}Fx$, i.e., definitely definitely negative cases +\end{enumerate} + +The general idea is that for level vagueness of F, we need +to show that $\mathbf{n}$ categories are vague. Thus, we need borderlines +between those, in effect, drawing $2^n+1$ categories.\footnote{Mark + Sainsbury, ``Concepts without Boundaries,'' in \emph{Departing From + Frege} (Routledge, 1990), 74.} + +\subsection{Williamson's challenge} + +Williamson argues that for this formalization to work, the D operator +should not obey these two schemas: +\begin{enumerate} +\def\labelenumi{(\arabic{enumi})} +\item{The S5 principle: If ${\sim}DF$, then $D{\sim}DF$.} +\item{The S4 principle: If $DF$, then $DDF$.} +\end{enumerate} + +If (1) and (2) hold, then whether a category is definite or indefinite, +it will also be definitely so at higher levels. The supervaluationist +cannot accept this since each category must be vague, otherwise it would +draw sharp boundaries. Thus, Williamson recommends adopting a weaker +modal logic, like T, with relative admissibility and no transitivity so +that both S4 and S5 principles fail.\footnote{Williamson, + \emph{Vagueness}, 156-159.} See the appendix for a more formal +explanation. + +However, Williamson argues that this is not sufficient to solve the +problem via the D* argument. He defines \(D^{*}F\) as an infinite +conjunction $F \; \& \; DF \; \& \; DDF \; \& \ldots \& \; D_{n}F$. Suppose +precisifications (a), (b), and (c), where (a) admits (b), and (b) admits +(c), but (a) does not admit (c), since admissibility is non-transitive. +Suppose \(D^{*}F\) at (a). This means that +$F, \; DF, \; DDF, \; \ldots, \; D_{n}F$ are true at (a). If $DF$ is true at +(a), then $F$ is true at (b); if $DDF$ is true at (a), then $DF$ +is true at (b); and so on. Thus, $F, \; DF, \; DDF, \; \ldots, \; D_{n}F$ are +all true at (b), and hence $D^{*}F$ is true at (b). The same reasoning +applies to (c). Thus, if $D^{*}F$ is true at some precisification, +then $D^{*}F$ is true at all precisifications. Hence, ${DD}^{*}F$ is +true at all precisifications - and by the same reasoning, so is +${D^{*}D}^{*}F$. Therefore, the S4 principle effectively applies to +$D^{*}$ (see diagram below). + +%\includegraphics[width=3.97674in,height=4.66543in]{media/image1.png} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} + +Consequently, Williamson concludes that higher-order vagueness +disappears.\footnote{Williamson, \emph{Vagueness}, 160.} This is +because, for supervaluationism to succeed, each metalanguage must be +vague. Thus, supervaluationists need a borderline case between +$D^{*}F$ and $D^{*}{\sim}F$, namely +${{\sim}DD}^{*}F \; \& \; {\sim}D{\sim}D^{*}F$. However, ${{\sim}DD}^{*}F$ +collapses to ${{\sim}D}^{*}F$ by modus tollens on the S4 principle. ${{\sim}D}^{*}F$ then collapses to ${D\sim D}^{*}F$, given closure of +D.\footnote{Patrick Greenough, ``Higher-Order Vagueness,'' + \emph{Proceedings of the Aristotelian Society, Supplementary Volumes} + 79 (2005): 183, + \href{http://www.jstor.org/stable/4106939}.} +In effect, ${{\sim}DD}^{*}F \; \& \; {\sim}D{\sim}D^{*}F$ reduces to ${D{\sim}D}^{*}F \; \& \; {\sim}D{\sim}D^{*}F$ which is a contradiction. +Since there are no borderlines to $D^{*}F$, it is not vague. + +Williamson offers supervaluationists a way out: to give up semantic +closure. D* can be vague but its vagueness cannot be expressed using D +or D*. Instead, we need a meta-language for D*, enriched with a distinct +operator, D!. Then, to express vagueness of D!, we need a +meta-metalanguage with D!!. Williamson remarks that the process could +continue infinitely.\footnote{Williamson, \emph{Vagueness}, 160-161.} + +Keefe takes up this proposal and advocates adopting an infinite, +hierarchical series of metalanguages. In this model, the vagueness of +the $n$th-level metalanguage can only be expressed in the $(n+1)$th +metalanguage, which is essentially richer than the nth language. She +argues that, since there is no reason not to adopt such an infinite +sequence, she can just stipulate that all the languages in the series +are vague.\footnote{Keefe, \emph{Theories of Vagueness}, 202-208.} +Greenough sketches a formalization where the object language is enriched +with indexed D operators where each \(D_{n + 1}\) is used to express the +vagueness of \(D_{n}\). Such formalization stops the D* paradox and +ensures that a non-vague metalanguage cannot be generated.\footnote{Greenough, + ``Higher-Order Vagueness,'' 184-186.} + +\section{Evaluation} + +Even though the above account might seem abstract, its strength lies in +its simplicity - Keefe only iterates her account of the first order to +higher orders of vagueness. In effect, the initial solutions to +vagueness problems equally apply to HOV. Vagueness at higher orders +remains a matter of semantic indecision: we are undecided over whether a +precisification counts as admissible. Furthermore, each level $n$ +admits borderline cases and lacks sharp boundaries -- a fact that can be +expressed in the $n+1$ metalanguage using appropriate D operators. + +Moreover, each higher order metalanguage is still Sorites susceptible. I +will explain this by running the paradox for the metalanguage (second +order vagueness) in natural language terms for clarity - though the same +could be done using D operators. The inductive premise for the +metalanguage can be restated, in natural language, as: `if there are +admissible precisifications that draw the boundary to `tall' at height +h, then there are ones that draw it at one-hundredth of an inch +lower'.\footnote{Keefe, \emph{Theories of Vagueness}, 207-208.} The +second order series could start with a clearly admissible +precisification (e.g., taller than 190cm) and end with a clearly +inadmissible one (e.g., taller than 110cm). Since one-hundredth of an +inch does not make a difference in admissibility, you could run a series +of conditionals, starting with `taller than 190cm is admissible' to +reach a conclusion that `taller than 110cm is admissible'. This is a +contradiction. To resolve the second-order paradox, Keefe reuses her +earlier strategy: for any complete way of making `admissible' precise +(or making `definitely' definite), there will be a pair such that the +first precisification is admissible and the second is not. This could be +run for any level of metalanguage. + +Thus, Keefe's account of HOV fulfils all the demands of a theory of +vagueness. Each metalanguage is vague since it (1) admits borderline +cases, (2) draws no sharp boundaries and (3) is Sorites susceptible. The +fact that she achieves this for each order while maintaining her initial +commitments (using the same technique at each order, characterising all +levels of vagueness as semantic indecision, and so on) makes her +strategy simple and elegant. + +Even though this iteration neatly maintains the supervaluationist +method, iterating to infinity is problematic. Keefe boldly claims that +`if there is no general objection to the claim that the sequence of +metalanguages for metalanguages is infinite, then what is the difficulty +with adding `and each of those languages is vague'\,'.\footnote{Keefe, + \emph{Theories of Vagueness}, 208.} However, there is a fundamental +difficulty in this addition. In Keefe's system, the vagueness of an +n-level metalanguage can only be expressed via an n+1 level +metalanguage. If all metalanguages are vague, then the infinite +metalanguage would have to be vague. To express the vagueness of the +infinite metalanguage, we would need to use the infinity + 1 +metalanguage. However, adding another element to an infinite set would +not alter the size of this set.\footnote{MIT OpenCourseWare, + \emph{Session 11: Mathematics for Computer Science}, \emph{6.042J: + Mathematics for Computer Science, Spring 2015} (Massachusetts + Institute of Technology, 2015), + \url{https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_session11.pdf}.} +Thus, the infinite + 1 metalanguage would be on the same meta-level as +the infinite metalanguage. Hence, the vagueness of the infinite +metalanguage cannot be expressed and the statement `each of those +languages is vague' seems meaningless. + +This objection points towards a more general issue with such Tarskian +metalanguage hierarchies. Namely, that languages in such hierarchies +cannot be globally quantified over.\footnote{Greenough, ``Higher-Order + Vagueness,'' 187.} Keefe could respond that even though the infinite +metalanguage might not be definable in her structure, it does not mean +that it does not exist. Her structure ensures that vagueness for any +finite level can be expressed. Even though we cannot say that `all +metalanguages are vague', we also cannot identify any non-vague +metalanguage within the structure. Thus, even though the concept of +infinity proves problematic for Keefe at the outset, I will assume that +this problem does not threaten the explanatory power of her structure. + +A further problem with the structure is that it is highly detached from +how language functions. Competent speakers would find making sense of +iterated uses of `definitely' difficult, whether it is indexed or not. +For example, saying someone is `definitely definitely definitely tall' +has little meaning apart from emphasis. Keefe might respond by pointing +out that we do not use expressions like `a googol of a googol of a +googol' in ordinary conversation either, yet this does not mean the +concept of `googol' is not a meaningful mathematical concept. However, +the issue goes deeper. As Saul Kripke pointed out, we cannot +consistently assign levels to truth. Thus, even if we index the levels +of `definitely', it is difficult to assign them consistently. Consider +the following statements: Jan says, `Everything Alfred said is +definitely false', and Saul says, `Everything Jan said is definitely +false'. To make sense of these, we would need to place one at a higher +level in the hierarchy. However, this does not happen in natural +language.\footnote{Saul Kripke, ``Outline of a Theory of Truth,'' + \emph{The Journal of Philosophy} 72, no. 19 (1975): 694-697, + \href{https://www.jstor.org/stable/2024634}.} + +Keefe might counter these natural language intuitions by arguing that +her model is only an idealization which is not meant to exactly +replicate how ordinary language works. While iterating `definitely' +(e.g., \(D_{3}D_{2}D_{1}F\)) may make little sense in casual +conversation, the model is primarily defended by its explanatory power +regarding HOV. She could further argue that even though different levels +of metalanguages, when expressed in natural language, might not be +clearly marked and distinguishable (such as in the Jan-Alfred example +above), they can still function as distinct metalanguages in a formal +framework. A further worry is that such an approach might over-idealise +HOV making her account arbitrary. It raises the question over whether +speakers genuinely use implicitly distinct levels of metalanguages to +assign levels to truth. Thus, Keefe would need to give a more robust +explanation of the relationship between her model and natural +language.\footnote{A full discussion of this issue is beyond the scope + of this essay, though the problem would require further explanation to + defend the account effectively.} + +Lastly, even though Keefe's iteration method allows her to respond to +Williamson's D* paradox and establish that there cannot be a non-vague +metalanguage, the non-vagueness of each metalanguage requires further +borderline cases. We need \(2^{n} + 1\) categories to express the +vagueness of the nth metalanguage. However, there is a tension between +an infinite number of categories and a finite number of objects in the +series: the finite series paradox. Consider a simple series with 5 +objects. To account for 1\textsuperscript{st} level, we divide them into +3 categories. To account for 2\textsuperscript{nd} level, we divide them +into 5 categories. At 3\textsuperscript{rd} level there are 9 categories +to be filled but only 5 objects. This means that at some level we will +run out of objects with which to fill the categories. As a result, there +will be no borderline cases between the categories - providing a sharp +boundary, as pictured below.\footnote{Greenough, ``Higher-Order + Vagueness,'' 180; 185-186.} Whether or not Keefe indexes her D +operators makes no difference, there will always be an insufficient +number of objects in the series to fill all categories. + +% \includegraphics[width=6.30006in,height=2.89763in]{media/image2.png} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} +In conclusion, even though the rigid hierarchy in Keefe's structure +might be defended to some extent, her appeal to an infinite hierarchy is +fundamentally in conflict with the finite Sorites. There seems to be no +way to accommodate the problem without making strong alterations to the +model. + +\section{5. Positive proposal -- dynamizing supervaluationism} + +\subsection{5.1. Introducing dynamic supervaluationism} + +I believe that Keefe's problems can be addressed by making the +structure's categories dynamic. My proposal is loosely based on +Hao-Cheng Fu's model.\footnote{Hao-Cheng Fu, ``Saving Supervaluationism + from the Challenge of Higher-Order Vagueness Argument,'' in + \emph{Philosophical Logic: Current Trends in Asia} (2017), 147-152, + \url{https://doi.org/10.1007/978-981-10-6355-8_7}.} Fu rejects Keefe's +claim that admissibility is vague and instead claims that, when +considering a vague predicate, we are using a well-defined set of +precisifications (p-sets). Keefe might argue this counterintuitive since +we do not know what is admissible. However, this knowledge is +unnecessary: the p-set is created when cases are categorized as true, +false, or borderline at time \(t_{1}\). For example, if 195cm and 190cm +are tall, 170cm is not, and 180cm is borderline, the p-set is implicitly +formed dividing cases into three groups, on my reading of Fu. Crucially, +we judge first; the p-set is constructed afterward. What follows in the +next paragraphs is my own development of the idea. + +Fu applies the AGM theory\footnote{AGM refers to the + Alchourrón--Gärdenfors--Makinson model of belief revision, which + accounts for rational change in epistemic states represented as belief + sets. The theory outlines how agents should expand, contract, or + revise their beliefs while preserving logical coherence. For more + detail, see Carlos E. Alchourrón, Peter Gärdenfors, and David + Makinson, ``On the Logic of Theory Change: Partial Meet Contraction + and Revision Functions,'' \emph{The Journal of Symbolic Logic} 50, no. + 2 (1985): 510--30, + \href{https://doi.org/10.2307/2274239}.} +to give a complex account of the dynamics of p-sets; however, offers +little formalisation and does not explain how this idea could be applied +to the challenges of HOV\footnote{Fu, ``Saving Supervaluationism from + the Challenge of Higher-Order Vagueness Argument,'' 149-152.}. +Moreover, Fu does not address the paradoxes of HOV, and it is difficult +to see how his account could solve them. In my view, we do not need such +an elaborate account. I propose that a p-set is dynamic solely in virtue +of changing when a case is judged inconsistently with it. For the sake +of clarity, consider the above example again. Imagine another person, +\textbf{x}, who is 168cm. You judge \textbf{x} as tall. This is clearly +inconsistent with your p-set at \(t_{1}\), since you judged 170cm as not +tall. Thus, adding \textbf{x} to the tall category updates the \(t_{1}\) +set to the \(t_{2}\) set with revised precisifications. This change +occurs by either (1) expanding (adding a precisification), (2) +contracting (removing one), or (3) both. Therefore, I retain the core +idea of dynamic p-sets and Fu's terminology but limit the scope of the +mechanism to a minimal principle: a p-set updates only when a judgment +is made that conflicts with it. + +I will now attempt to formalise the above proposed working of p-sets, +which I will later apply to the challenges haunting supervaluationism. +Vagueness, on the dynamic view, remains semantic indecision. At the +first level, we follow Keefe's supervaluationism with a slight addition +of the temporal component. While Fu does not offer a formalisation of +his view in the spirit of Keefe's system with D operators, the following +temporal framework develops my own way of modelling dynamic p-sets using +temporally indexed D operators. + +More precisely, at any time, t, cases divide into +\(D_{t}F,D_{t}{\sim}F\), and \({{\sim}D}_{t}F\ \; \& \; {{\sim}D}_{t}{\sim}F\): +that is true, false, and borderline. However, unlike in Keefe's view, +HOV arises not from undecided admissibility of a precisification but +from the instability of precisifications. Suppose that you make some +categorisations at \(t_{1}\). According to the p-set that you just +formed; some arbitrary case is classified as \(D_{1}F\). Now suppose +that you consider the series again, but you are no longer sure about the +definiteness of your classification. Thus, your p-set is adjusted at +\(t_{2}\), and according to it, the case is borderline. Therefore, from +\(t_{2}\)'s perspective it was a borderline definite case at \(t_{1}\) +(\({{\sim}D_{2}D}_{1}F\)). + +In general, when considering a borderline case after categorisation at +\emph{t}, tolerance ensures a mis-categorisation. To see this, remember +that the supervaluation technique divides cases sharply into true, +false, and borderline. However, tolerance guarantees that when viewing +two neighbouring cases, we will not be able to tell the difference. +Therefore, there is a clear tension; we divided sharply, enabling a +border pair where, for instance, one member is true and another +borderline. However, since we cannot distinguish between neighbouring +cases, they must be categorised equally. That means that one of the +cases had to be categorised mistakenly and thus, the p-set must be +revised to maintain consistency in our judgments. When we reconsider the +series at \(t_{2}\), the earlier categorisations from \(t_{1}\) turn out +to be indefinite, as case memberships shift. + +\subsection{Applying dynamic supervaluationism} + +Having formalised the view, I will now apply it to the challenges of +HOV, starting with Williamson's D* argument. To attack the dynamic +approach, D* could be restated as the conjunction `DA at \(t_{1}\) \& DA +at \(t_{2}\) \& DA at \(t_{3}\) \& \ldots{} \& DA at \(t_{n}\)'. As +discussed in section 3, the D* argument establishes that, if D* is not +shown to be vague, then the cases where D* is true and the cases where +D* is false will both be ultimately definite. Hence, there will be no +borderline cases between D* categories, which provides a sharp boundary. +This contradicts the foundational supervaluationist claim that there are +no sharp boundaries. However, this argument loses its force under the +dynamic view. The dynamic framework allows us to easily account for the +vagueness of D*. Just as in the case of any D, we need to progress in +time to express D*'s vagueness. Thus, while D* may initially appear to +be non-vague, this is because we need to move to t + 1 to realize its +vagueness. + +Secondly, Keefe's view faced concerns about rigid hierarchies, but the +dynamic approach eliminates these. When two speakers disagree over a +case's definiteness, neither statement must be `prior'. They are simply +speaking from different p-sets that underwent different evolutions. +There is no rigid hierarchy of metalanguages since each discusses +categorizations in another metalanguage, and no pair can be clearly +ranked as `prior'. + +This lack of priority arises because it would be impossible to assign it +to any particular metalanguage. Surely, the metalanguage at \emph{t}+1 +must be a metalanguage of the metalanguage at \emph{t}, since it is able +to express facts about \emph{t}. Therefore, it is more `privileged' in +this sense. However, suppose that the p-sets evolve over time such that, +when moving from \emph{t}+1 to \emph{t}+2, we go back to the original +p-set from \emph{t}. Then, the \emph{t} and \emph{t}+2 metalanguages +gain their truth conditions from the same p-set. Therefore, in a sense, +the t metalanguage becomes `prior' to the \emph{t}+1 metalanguage. This +would undermine the strict, unidirectional Tarskian hierarchy. + +One could further argue that we could suppose a scenario in which two +identical people, A and B, undergo identical p-set evolutions. However, +A's evolution stops at \emph{t} and B's evolution stops at \emph{t}+1. +On the one hand, we might be tempted to assign priority to B's +statements, which would be counter-intuitive on the natural language +objection. However, there is no reason to suppose that A's evolution +should go the same way; she might consider a different part of the +Sorites spectrum. Therefore, although the metalanguages are in some +sense hierarchical, none has a clear priority in determining the truth +of one classification over another. Thus, the objections, such as the +ones made by Kripke, do not apply here. + +Thirdly, the dynamic view can help tackle the finite series paradox, +which was a critical blow to Keefe's account. I will explain how it +could achieve this through an example. Consider a 5-element Sorites with +objects \textbf{a}, \textbf{b}, \textbf{c}, \textbf{d}, and \textbf{e}. +Suppose that Alfred's initial categorizations are: + +\[D_{1}F = \{ a,b\}\] + +\[{\sim}D_{1}F\; \& \; {\sim}D_{1}{\sim}F = \{ c\}\] + +\[D_{1}{\sim}F = \{ d,e\}\] + +Alfred considers the pair \textbf{b} and \textbf{c} again. He realizes +that he cannot tell the difference, concluding that \textbf{b} is also +borderline. He adjusts his p-set accordingly, forming a new \(t_{2}\) +p-set. + +\[D_{2}F = \{ a\}\] + +\[{\sim}D_{2}F\ \&\ {\sim}D_{2}{\sim}F = \{ b,c\}\] + +\[D_{2}{\sim}F = \{ d,e\}\] + +% \includegraphics[width=4.50937in,height=2.12793in]{media/image3.emf} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} + +The \(t_{1}\) division, from the perspective of \(t_{2}\) becomes: + +\emph{\hfill\break +}\[{D_{2}D}_{1}F = \{ a\}\] + +\[{\sim}D_{2}D_{1}F\ \; \& \; {\sim}D_{2}{{\sim}D}_{1}F = \{ b\}\] + +\[D_{2}{\sim}D_{1}F\ \&\ D_{2}{\sim}D_{1}{\sim}F = \{ c\}\] + +% \includegraphics[width=4.79722in,height=2.19101in]{media/image4.emf} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} +Hence, in this part of the series, the vagueness of \(D_{1}\) is fully +accounted for since all \(D_{1}\) categories have borderline cases. + +Now suppose that at time \(t_{3}\), he looks at the pair $\mathbf{a}$ and +$\mathbf{b}$. Since he cannot tell the difference, he decides that b is +also a definite case, adjusting the p-set again. + +\[D_{3}F = \{ a,b\}\] + +\[{\sim}D_{3}F \; \& \; {\sim}D_{3}{\sim}F = \{ c\}\] + +\[D_{3}{\sim} F = \{ d,e\}\] + +% \includegraphics[width=4.46286in,height=2.2071in]{media/image5.emf} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} +Since $\mathbf{b}$ changed its category membership, from the perspective +of \(t_{3}\), $\mathbf{b}$ was not a definite borderline case at +\(t_{2}\). Thus, the \(t_{2}\) division, from the \(t_{3}\) perspective, +is: + +\[{D_{3}D}_{2}F = \{ a\}\] + +\[\sim D_{3}D_{2}F\ \&\ \sim D_{3}{\sim D}_{2}F = \{ b\}\] + +\[D_{3}\sim D_{2}F\ \&\ D_{3}\sim D_{2}\sim F = \{ c\}\] + +% \includegraphics[width=4.875in,height=2.1236in]{media/image6.emf} +\begin{center} + \begin{tikzpicture} + \node at (0, 0) {\texttt{[GRAPHICS FORTHCOMING]}}; + \end{tikzpicture} + \end{center} +Thus, vagueness of \(D_{2}\) is accounted for. + +In general, any bordering pair will exhibit change when reassessed. +Thus, any categorization at $t$ can prove indefinite at $t+1$. +In effect, you will never reach a point where there are more categories +than members in the series since the fluid categories will always be +filled. An object can fill different categories at different times. This +also does not mean that the \(t_{1}\) categories are definite at +\(t_{3}\), only that their vagueness cannot be expressed from the +\(t_{3}\) perspective. + +\section{Addressing possible objections} + +Dynamizing supervaluationism provides new methods to tackle the +paradoxes of HOV and other problems, for which standard +supervaluationism struggles to account. However, it also presents new +worries, which I will explore and sketch responses to in this section of +the essay. + +\subsection{Fixed time worry} + +The first possible objection to the view is that it breaks down when +time is fixed. This is because the account of HOV relies on shifty +p-sets, which in turn rely on the progress in time. More precisely, the +vagueness of some set of categories drawn in period $t$ can only be +expressed in period $t+1$. Thus, if we hold the time fixed, the +view breaks down: the categories drawn in period $t$ appear to be +sharply bounded, which contradicts the foundational claim that there are +no sharp boundaries. + +Although this might seem like a critical blow to the view, there are two +possible lines of response. First, we could simply reject the inference +from our inability to express the vagueness of some order when time is +fixed, to the claim that there are sharp boundaries. After all, the fact +that we cannot express it does not imply that it does not exist. This, +however, demands further explanation of why we cannot express it. One +response is that at a certain time, we are just using a well-defined but +arbitrary set of precisifications. However, this division is surely +wrong; it is made under one of many sets of equally good +precisifications. Thus, there is no reason to believe that the term was +made precise -- we just have not realized our mistake yet. + +A second and more powerful response is to deny the possibility of fixing +time in this sense. This could supplement the above argument. Suppose +that the critic of the view wants to prove to us that there are sharp +boundaries. However, in order to show that there are sharp boundaries, +they would have to find them in the series. Suppose that you manage to +find the extension-switching pair. Even if you do this, you will +realize, per tolerance, that you cannot tell the difference between the +two cases. In effect, you must conclude that one of the cases was +falsely classified when you made the division in the previous period. +Thus, your p-set changes. Therefore, the very considering of the sharp +distinction would automatically progress us to t+1, ensuring that there +was no sharp boundary. In conclusion, the fixed time objection is not a +significant worry to the dynamic view. + +\subsection{Collapse to contextualism worry} + +There is a second and more dangerous worry: one could argue that the +supervaluationist aspect of the dynamic view seems unimportant. By this, +I mean the use of supervaluationist semantics and classification of +vagueness through indecision between precisifications. It is only +directly applied to resolve FOV, and one could argue that the relativity +of classifications over time, which accounts for HOV, could be equally +applied to FOV. In effect, the supervaluationist method would disappear. +If this argument is accepted, and if we further assume that the +functioning of p-sets is sufficiently similar to that of contexts, then +the dynamic view risks collapsing into a contextualist one. This could +have some benefits, such as the preservation of bivalence (which +contextualists keep) and making the view more parsimonious by unifying +the approaches to vagueness at different orders. + +In what follows, I will defend the dynamic view from this objection. See +footnotes for background on contextualism\footnote{Contextualism rests + on the claim that vagueness is a species of context-sensitivity. This + roughly means that, in its application across different contextual + circumstances, a vague term maintains a constant \emph{character} but + shifts in \emph{content}. Therefore, vague terms function like + indexical terms. The relationship of vagueness and indexicality is a + contested matter for contextualists. Some hold that vague terms behave + \emph{like} indexicals, while others claim they \emph{are} indexicals. + However, this distinction is not directly relevant to the discussion, + and the objections raised here apply equally to both views. Consider + the word \emph{now}. It adheres to the same grammatical rules (i.e., + has the same \emph{character}) when uttered today and tomorrow. + However, when said today, it picks out a different time than it does + when used tomorrow (i.e., has different \emph{content}). Similarly, a + vague predicate like \emph{tall} is used in the same way when applied + to members of a group of pygmy peoples, as when applied to a group of + Dutch people. However, it would pick out radically different people. + In the first case, the extension of \emph{tall} likely includes some + of the world's shortest people; in the second, some of the tallest. + See Roy Sorensen, ``Vagueness,'' \emph{The Stanford Encyclopedia of + Philosophy} (Winter 2023 Edition), ed. Edward N. Zalta and Uri + Nodelman, + \url{https://plato.stanford.edu/archives/win2023/entries/vagueness/}.} +and their solution to the Sorites\footnote{Contextualists exploit this + idea of unstable extensions over contexts to solve the Sorites by + accusing it of equivocating different meanings of a vague term. + Similarly to the supervaluationists, the contextualists target the + inductive premise (2). The contextualist is committed to the claim of + weak tolerance (WT), which states that when two members of a bordering + pair are considered in the same context C, they will belong to the + same extension. However, WT permits that when one member is considered + in context C and the other in C', then they might belong to a + different extension. See Jonas Åkerman and Patrick Greenough, "Hold + the Context Fixed---Vagueness Still Remains," in \emph{Relative + Truth}, ed. Manuel García-Carpintero and Max Kölbel (Oxford University + Press, 2010), 275--76, + \href{https://doi.org/10.1093/acprof:oso/9780199570386.003.0016}. + + The WT explains why the inductive premise seems to hold. If we + consider any pair in the series, we will conclude that both members + belong to the same extension. But this is just because we are disposed + to view them in the same context C. The contextualist says that, in + fact, the context will gradually change across the series. This means + that even if we classify neighbouring terms the same at first, this + classification will not persist throughout the series. Thus, the + inductive premise of the sorites, such as `if n is short, then n+1 is + short', fails since the meaning of `short' is not the same for every + member \emph{n}. This is because, the shift of context C into C', + enables cases where `n is short' is true (in C) but `n+1 is short' is + false (in C'). See J. Åkerman, "Contextualist Theories of Vagueness," + \emph{Philosophy Compass} 7 (2012): 470--75, + \href{https://doi.org/10.1111/j.1747-9991.2012.00495.x}.}. +The first point that I address is the idea that supervaluation is +obsolete. On this view, its role at the first level could be replaced by +the context-reminiscent p-sets. The intuitive idea is that, since shifty +p-sets account for HOV, why not apply them to FOV and get rid of +additional semantic claims and concessions altogether? However, this +intuition is misguided, since the supervaluationist solution to FOV is +required to make the shifty p-set account of HOV work. This is because +the first-order divisions allow for the p-sets to shift in the first +place. At the first stage, we implicitly categorize objects into +positive, negative, and borderline cases. These categories are directly +determined by the p-set, which sets out the supervaluationist truth +conditions (i.e., DF iff true for all precisifications and so on). These +categorizations are provisional: they impose sharp boundaries where none +truly exist. This tension allows for future revisions of p-sets, and +thus for p-sets to shift. Hence, without supervaluation in the +beginning, the p-sets cannot shift. And if they cannot shift, they +cannot account for any order of vagueness. + +A stronger claim could be made that the p-sets are entirely purposeless +if we do not allow for supervaluation. To see the point, imagine that +you have some set of precisifications of tall $\{>170\text{cm}, +>180\text{cm}, >190\text{cm}\}$ and you use them to categorize a +group of people in the series. Without supervaluation, you end up with +six extensions, i.e., three positive and three negative extensions, one +per precisification. There are no borderline cases, since without +supervaluationist truth conditions --- where borderlines are true under +some precisifications and false under others --- such cases are not +defined. Since this is a key symptom of vagueness, as stressed in the +beginning, this result would require further explanation of why we think +there are borderlines at all. + +An enemy of the view could argue that this response misses the point --- +vagueness did not fail to arise due to the absence of supervaluation, +but rather because the p-sets did not shift. After all, on the dynamic +account, it is the shiftiness of p-sets that allows for HOV. To address +this, let us suppose, for the sake of the argument, that the p-set can +somehow shift without supervaluation. Imagine, for instance, that the +p-set expands by incorporating an additional precisification to the set. +You now have eight extensions, yet still no explanation for either +first-order or higher-order vagueness. Thus, even with shifty p-sets, +the dynamic view cannot function without supervaluation, showing it to +be an essential, not merely supportive, component of the account. + +Therefore, the case for the contextual collapse breaks down in the very +beginning. We simply cannot make the p-sets shifty without maintaining +the baseline supervaluationist aspects of the theory. If we cannot make +the p-sets shifty, they cannot resolve FOV, let alone HOV. Hence, +supervaluation is by no means obsolete. However, to strengthen the +defense, I will demonstrate that the next step needed for the +contextualist collapse fails. That is, I will show that p-sets and +contexts behave very differently. + +Although they might appear similar, the former crucially relies on the +characterization of vagueness as semantic indecision, while the latter +depend on context sensitivity. We might express this difference by +saying that the p-sets are inward-oriented, while contexts are more +outward-oriented. This is because the former shifts due to our +indecision among several equally good precisifications at the initial +stage. This indecision prompts us to make mistakes, which we +subsequently correct by revising the p-set into another equally +acceptable p-set. Thus, the changes directly follow our judgments. By +contrast, shifts in contexts seem to have an effect on our judgments - +contexts shift first, and judgments follow. Thus, the machinery appears +to be quite different. + +One could even argue that shifty p-sets rest on a firmer theoretical +ground -- their shiftiness is caused by our inconsistent judgments. On +the other hand, the contexts appear to shift arbitrarily. Thus, the +contextualist requires some external justification for this instability. +Additionally, the contextualist needs to show how contexts could become +shifty enough to prevent every instance of the Sorites. In other words, +enough shiftiness must be generated. I do not intend to digress further, +but the key takeaway is that despite their apparent similarities, p-sets +and contexts differ significantly. Thus, the threat of the `collapse' +does not seem to be so imminent. + +As a final point to strengthen my argument, I will provisionally assume +that the dynamic approach could collapse into contextualism. Even in +such a scenario, there remain independent reasons to prefer the former +view over the latter. One significant reason is that contextualism +undermines some of our most basic approaches to reasoning. Contextualism +requires extensions of vague terms to be unstable, which is precisely +what enables it to defeat the Sorites. However, these shifty contexts +become deeply problematic when applied outside of the paradoxical +setting. + +To see this, consider the following example. Saul and Jan are borderline +cases of tall. The former is 176.1cm, and the latter is 176cm. Suppose +you judge both of them to be tall. Now consider applying the following +instance of conjunction introduction: + +%Saul is tall Jan is tall + +%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \(\land I\) + +%Saul and Jan are tall + +However, if the extension of the vague predicate \emph{tall} is +unstable, we can easily imagine a situation in which both premises are +individually true, yet the conclusion turns out false. This would happen +if the context shifted midway through the argument. Thus, although +context sensitivity is useful for solving the Sorites, it is dangerous +when applied to everyday reasoning. Specifically, how can contexts +remain sufficiently stable to ensure our logic does not fail even in +such simple cases?\footnote{J. Åkerman, "Contextualist Theories of + Vagueness," \emph{Philosophy Compass} 7 (2012): 475--76, + \href{https://doi.org/10.1111/j.1747-9991.2012.00495.x}.} + +In contrast, dynamic supervaluationism does not provoke such worries. +Under supervaluationism, the rule of a conjunction introduction always +preserves validity. To illustrate, consider a p-set representing +precisifications for \emph{tall}: \{\textgreater170, \textgreater175, +\textgreater176\}. First two precisifications make both premises true +and the conclusion true as well. The third precisification makes one of +the premises true, the other false, and the conclusion false. This will +work for any possible precisification. Consequently, it applies to every +p-set.\footnote{This follows the exact same reasoning as that applied to + the failure of the inductive premise or the truth of the law of + excluded middle discussed in more detail at the beginning of the + essay.} + +One might argue that, similarly to a shifting context, the p-set could +shift over the course of an argument. For example, we might initially +classify both premises as true (e.g., using the set \{\textgreater170, +\textgreater175\}, but later we classify the conclusion as false (e.g., +shifting to the set \{\textgreater177, \textgreater180\}). However, this +objection reflects a misunderstanding of supervaluationist semantics, +since arguments must always be evaluated relative to a single p-set. If +we shifted the p-set to the second one, both premises would become false +along with the conclusion. Therefore, the validity of conjunction +introduction would remain intact. + +Why is this strategy not available to the contextualist? The +contextualist could simply deny that contexts can shift in such ways, +insisting instead that we always evaluate the premises and the +conclusion within a single context. However, this directly contradicts +the contextualist's equivocation strategy to the Sorites paradox. That +is, the strategy according to which bordering cases may differ in truth +value because their evaluation contexts differ. Hence the contextualists +need contexts to shift. In effect, they cannot deny that the above +scenario is possible. Instead, their strongest response would likely be +to argue that such cases rarely happen. + +I do not intend to argue that supervaluationism, or its dynamic version, +is superior to contextualism. Such a claim is clearly beyond the scope +of this essay and perhaps beyond the scope of any single essay. Rather, +my point is simply that there are independent reasons to prefer the +dynamic view over contextualism. Therefore, the claim that contextualism +explains everything that the dynamic view explains - but more simply, +and thus more parsimoniously - is clearly not accurate. + +Taking stock of these considerations, the collapse argument fails not +only at its initial stage but also on all subsequent fronts. Dynamic +supervaluationism is by no means contextualism in disguise; rather it is +its own theory, deeply grounded in Keefe's original supervaluationist +framework. + +\section{Conclusion} + +While Keefe's supervaluationism remains an attractive account of +vagueness, it ultimately struggles to account for higher-order +vagueness. Her adoption of a rigid, Tarskian infinite hierarchy may +block Williamson's D* argument, but at the cost of disconnecting the +theory from natural language. Even if, as I briefly explored, she could +respond to these problems, adopting an infinite metalanguage hierarchy +still leaves Keefe subject to a seemingly unresolvable finite series +paradox. I argued that Keefe's account could be dynamized by +incorporating ideas from Fu, thereby resolving the finite series paradox +and avoiding issues associated with a rigid hierarchy. Yet, the dynamic +model itself introduces new difficulties, notably the `fixed time' and +`collapse to contextualism' problems. To defend the view, I briefly +outlined potential replies to these issues, showing that they are not +fatal. Dynamizing supervaluationism may not resolve all problems, but it +is a promising development of the supervaluationist theory and would be +worth elaborating on and defending in future enquiries. + +\subsection*{Appendix} + +\subsubsection*{Why must Keefe deny the S4 and S5 principles?} + +\begin{enumerate} +\def\labelenumi{(\arabic{enumi})} +\item The S5 principle: If ${\sim}DF$ then $D{\sim}DF$. +\item The S4 principle: If $DF$ then $DDF$. +\end{enumerate} +Suppose that (1) and (2) hold and that we have the first-order +classification: +\begin{enumerate} + \def\labelenumi{(\roman{enumi})} +\item{$DF$ for definite positive cases.} +\item{${\sim}DF \; \& \; {\sim}D{\sim}F$ for borderline cases.} + \item{$D{\sim}F$ for negative cases.} + \end{enumerate} + +If (1) holds, it implies that at the second level, $DF$ and +$D{\sim}F$ transform into $DDF$ and $DD{\sim}F$ (see proofs a and +b). That is, the definite positive and definite negative case is +definitely definite positive and definitely definite negative, +subsequently. If (2) holds, it implies \(\sim DF\ \&\ \sim D\sim F\) +${\sim}DF \; \& \; {\sim}D{\sim}F$ transforms into $D{\sim}DF \; \& \; D{\sim}D{\sim}F$ (see proof c). That is, +the borderline case is definitely a borderline case. However, +second-order vagueness would require two more categories -- the +borderline between positive and borderline +(${\sim}DDF \; \& \; {\sim}D{\sim}DF$) and the borderline between borderline +and negative (${\sim}DD\sim F \; \& \; {\sim}D{\sim}D{\sim}F$). As a result, +sharp boundaries are drawn between the three categories since there are +no cases between them. + +% \includegraphics[width=3.125in,height=1in]{media/image7.png} + +%\includegraphics[width=2.87025in,height=0.94231in]{media/image8.png} +\bigskip +\noindent +\begin{minipage}[t]{0.48\textwidth} + \noindent \textbf{Proof a:} \\ + \begin{center} + \begin{tikzpicture} + \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}}; + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{minipage}[t]{0.48\textwidth} + \noindent \textbf{Proof a:} \\ + \begin{center} + \begin{tikzpicture} + \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}}; + \end{tikzpicture} + \end{center} + \end{minipage} +\bigskip + +\noindent \textbf{Proof c:} +\begin{center} + \begin{tikzpicture} + \node at (0,0) {\texttt{[TABLEAUX FORTHCOMING]}}; + \end{tikzpicture} + \end{center} + +%\includegraphics[width=6.26806in,height=1.61181in]{media/image9.png} + +\refsection + +\begin{hangparas}{\hangingindent}{1} +Åkerman, Jonas. 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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: -- cgit v1.2.3