2. The sorites and measurement theory

The idea underlying many sorites arguments is that vague predicates are naturally understood to apply to the elements of structures A describable as either concrete or perceptual continua—e.g., points on concrete lines or transitions between colors or sounds. These are taken to admit to so-called indiscriminable differences—i.e., a change in the relevant magnitude which cannot be directly perceived or measured—and are thus regarded as unable to affect whether a point in A satisfies the vague predicate. But the accumulation of sufficiently many such differences can lead to a discriminable difference resulting in a change in the truth value.

The properties of the structures in question are typically illustrated via examples rather than specified axiomatically—e.g., by taking the vague predicate $\Phi _0(x)$ to be short and A to contain a sequence of objects varying in height or $\Phi _0(x)$ to be orange and A to be a color spectrum. But of course the possibility of formulating a sorites-like argument for a given predicate will depend on the principles which the relevant structures are assumed to satisfy. This is paradigmatically true of the continuous sorites whose formulation is based on the fundamental assumption that

Here $\Phi (x)$ is assumed to be the image of $\Phi _0(x)$ in the real numbers under the envisioned mapping—e.g., of the real numbers measuring points on a ruler corresponding to the heights of short individuals or the wavelengths of orange positions in the spectrum. We thus see from the outset that the formulation of the forms of the sorites which will be relevant here are reliant on representation in a mathematical domain. But once this is acknowledged, we can also inquire into what must be assumed about the extension of $\Phi _0(x)$ in A to ensure the existence of a function $\phi : A \rightarrow \mathbb {R}$ with properties allowing a given variant of the sorites to go through.

Reflection convinces us that the sorts of predicates which figure in soritical arguments typically come along with an associated order $\prec $ on A—e.g., shorter (or strictly less tall) or oranger (or strictly less red). Such relations typically figure in so-called penumbral connections (see Fine [Reference Fine17]) involving $\Phi _0(x)$ —e.g., that if a is short and a is taller than b, then b is also short or symbolically

1. (2) $\forall a , b \in A[(\Phi _0(a) \wedge b \prec a) \rightarrow \Phi _0(b)].$

It is this sort of property which is presumed to be preserved by the aforementioned kind of mapping $\phi $ of A into the real numbers—e.g.,

1. (3) $\forall a , b \in A[a \prec b \rightarrow \phi (a) < \phi (b)].$

It thus becomes apparent that measurement theory in the tradition of Krantz et al. [Reference Krantz, Luce, Suppes and Tversky21] provides a useful—and by our lights necessary—framework for taking into account the relationship between non-mathematical continua A and the real numbers. The importance of this subject to theorizing about vagueness is now widely acknowledged.Footnote ² To articulate the relation the continuous sorites bears to its better-known discrete counterparts we will review some of its basic conventions.

Measurement theory seeks to provide an account of conditions under which a given non-mathematical structure $\mathcal {A}$ can be represented by some mathematical structure $\mathcal {M}$ such that abstract reasoning about the latter can be used to facilitate drawing concrete conclusions about the former. Here “structure” should be understood in the logician’s sense of model. $\mathcal {A}$ is thus specified as a tuple $\langle A, R_1, \ldots ,R_n, f_1, \ldots , f_m \rangle $ consisting of a domain A relations $R_i$ and functions $f_i$ —e.g., as described by $\prec $ above or the concatenation operation $\circ $ considered below. These are assumed to satisfy a set of axioms $\Gamma $ which specify the properties of the non-mathematical domain—e.g., that $\prec $ is a linear order or that $\circ $ is associative.

One of the central aims of measurement theory is to provide an account of when it is possible to establish the existence of a structure preserving mapping, called a homomorphism, from an arbitrary structure $\mathcal {A}$ satisfying $\Gamma $ to a prescribed sort of mathematical structure $\mathcal {M} = \langle \mathbb {M}, S_1, \ldots ,S_n, g_1, \ldots , g_m \rangle $ . In the prototypical cases we will consider, $\mathbb {M}$ will be the real numbers $\mathbb {R}$ or a subfield thereof and $S_i$ and $f_i$ will thus respectively be relations on $\mathbb {R}^{j_i}$ and functions of type $\mathbb {R}^{k_i} \rightarrow \mathbb {R}$ . A result establishing that there exists a mapping preserving the relevant structure is known as a representation theorem and takes the following form:

1. (4) If the model $\mathcal {A}$ satisfies the axioms $\Gamma $ , there exists $\phi :A \rightarrow \mathbb {R}$ such that for all $\vec {a} \in A^{j_i}$ , if $R_i(\vec {a})$ , then $S_i(\phi (\vec {a}))$ and for all $\vec {a} \in B^{k_i}$ , $\phi (f(\vec {a})) = g(\phi (\vec {a}))$ .

Theorizing about vagueness often implicitly assumes the availability of a representation theorem for the predicates in question. This is paradigmatically so for what we will refer to as vague measure predicates—i.e., adjectives such as short, or brief, lightweight and physical or psychological magnitudes which “admit to degrees”.Footnote ³ This is illustrated by the following prototypical soritical argument:Footnote ⁴

Several hallmarks of measurement theory are apparent in this and many similar formulations. This includes a choice of an empirical unit c—i.e., a millimeter—which is assumed to be less than a so-called just-noticeable difference d for length measurement. It also includes the specification of a domain $A = \{a_0, \ldots , a_{1000}\}$ of items whose heights are measurable relative to c and which are assumed to be listed in order from shortest to tallest with $a_{i+1}$ being exactly one millimeter taller than $a_i$ . The existence of a measurement function $\phi : A \rightarrow \mathbb {R}^{+}$ is then assumed which preserves the shorter relation $\prec $ in the manner required by (3) and is such that $\phi (c) = 0.001$ (e.g., one millimeter $= \frac {1}{1000}$ meters) and $\phi (c) < \phi (d)$ .

The choice of a specific $c \prec d$ , the decision to measure c relative to a particular system of units, and the choice of a particular unit within that system to represent the length of c are all arbitrary. But if we wish to formalize the prior passage, then the selection of some specific magnitude and corresponding numerical value for its length is required. For it is only once such a value has been designated that we are able to formalize a tolerance principle like (5) in the following manner:

1. (6) $\forall a ,b \in A[(|\phi (a) - \phi (b)| \leq \phi (c) \wedge \Phi _0(a)) \rightarrow \Phi _0(b)].$

To state such a principle also requires that we employ a symbol for the measurement function in our object language description of the sortical rasoning. But this is prototypical of how we speak about various magnitudes in practice—e.g., when we say that someone is “1.2 meters tall”. The statement of (6) also illustrates how the envisioned reasoning is dependent on our ability to refer to “tiny variations” to which a vague measure predicate like short is tolerant. What is assumed here is thus that the domain from which the elements comprising the envisioned sorites sequence are selected contains a sufficiently small object to play the role of a unit.

This in turn illustrates how the additive properties of the relevant mathematical scale also play a role in the envisioned reasoning. For it is only after such an object has been selected as a unit c shorter than d that we are able to assert that the objects $a_0,\ldots ,a_{1000}$ stand in the appropriate relation—i.e., that they are such that the height of object $a_{i+1}$ is the same as that of a compound object resulting from laying $a_i$ end-to-end with c.Footnote ⁵ Suppose we introduce ‘ $\circ $ ’ to denote this concatenation operation and ‘ $a \sim b$ ’ to denote the empirical same length relation defined as

1. (7) $a \sim b$ if and only if not $a \prec b$ and not $b \prec a.$

We must thus have that $a_{i+1} \sim a_{i} \circ c$ for all $0 \leq i < 1000$ . This is required to ensure that $a_i$ and $a_{i+1}$ stand in the indiscriminability relation and also that $a_{1000}$ has been reached via transitions “adding up to” a discriminable difference.Footnote ⁶

For the soritical derivation to have its intended force, we must thus ensure that $\phi $ not only preserves the structure of the shorter relation $\prec $ in the sense of (3) but also that of the empirical concatenation operation $\circ $ . In other words, we additionally require

1. (8) $\forall a , b \in A[\phi (a \circ b) = \phi (a) + \phi (b)].$

This illustrates how even familiar forms of the sorites presuppose mathematical representation.Footnote ⁷ But here the existence of an appropriate measurement function is ensured by a seminal result of measurement theory known as Hölder’s Theorem.Footnote ⁸

That $\mathcal {A}$ is an ordered semigroup means that $\prec $ is a linear order on A, $\circ $ is associative, and that $x \preceq y$ implies $z \circ x \preceq z \circ y$ and $x \circ z \preceq y \circ z$ . Positivity requires that $x \prec x \circ y$ (and thus there is no identity element). Regularity requires that if $x \prec y$ , then there is z such that $x \circ z \preceq y$ . The Archimedean property requires that

1. (9) For all $a \in A$ with $a \prec b$ , there is $n \in \mathbb {N}$ such that $b \preceq a \circ \ldots \circ a$ (n times).

These conditions are paradigmatically satisfied in cases of extensive measurement which are often in the background of sorites arguments.Footnote ⁹

Such an argument is conventionally formulated as a series of conditionals:

1. (10) $\Phi _0(\texttt {a}_0), \Phi _0(\texttt {a}_0) \rightarrow \Phi _0(\texttt {a}_1), \Phi _0(\texttt {a}_1), \ldots , \Phi _0(\texttt {a}_{999}), \Phi _0(\texttt {a}_{999}) \rightarrow \Phi _0(\texttt {a}_{1000}), \Phi _0(\texttt {a}_{1000}).$

Herein the conclusion $\Phi _0(\texttt {a}_{1000})$ is intended to express that the 1000th individual in the envisioned series is short. But to derive this conclusion we must confirm that each of the conditionals $\Phi _0(\texttt {a}_i) \rightarrow \Phi _0(\texttt {a}_{i+1})$ are true. This in turn requires not the only the truth of the tolerance principle (6) but also that each of the empirical statements $|\phi (\texttt {a}_i) - \phi (\texttt {a}_{i+1})| \leq 0.001$ for $0 \leq i < 1000$ be assumed as premises of the argument. But the latter statement employs a symbol for the measurement function $\phi :A \rightarrow \mathbb {R}^+$ satisfying (3) and (8) which in turn is only guaranteed to exist in virtue of Hölder’s theorem. We thus reach the intermediate conclusion—to which we will return in §6—that not only Theorem 1 but also the mathematical principles required to prove it must be counted amongst the premises of the conditional sorites.

3. From the discrete to the continuous sorites

The foregoing is illustrative of how formulations of traditional forms of the sorites paradox are often implicitly reliant on mathematical representation. This is true to an even greater extent in the case of the continuous and covering sorites. Inspiration for the former derives from the observations that had we elected to formulate (6) using a smaller choice of the unit c—e.g., a micrometer (i.e., $10^{-6}$ meters), a nanometer (i.e., $10^{-9}$ meters), etc.—our confidence in the corresponding version of this form of tolerance principle would presumably increase. This comes into clearest focus when we examine versions of the sorites involving not just discrete sequences of objects drawn from non-mathematical continua but the continua themselves.

These are exemplified by predicates describing temporally or spatially extended regions which are said to have “fuzzy boundaries”—e.g., a period of time asserted to contain the moments of mid-morning or a region asserted to contain the points on Mount Everest. A related class of cases concerns what are often explicitly referred to as perceptual continua—e.g., a crescendo from quiet to loud tones or a spectrum containing wavelengths varying from a paradigm shade of orange to one of red.

In such cases the empirical or perceptual domain A may be assumed to contain a subinterval $C = \{c \in A : a \preceq c \preceq b\}$ consisting of the points “between” the endpoints $a,b \in A$ in the relevant sense—e.g., the points on a line between the summit of Mount Everest and a sufficiently distant point or between shades in an orange-red spectrum. In such cases we typically think that the predicate in question holds of the left endpoint—i.e., $\Phi _0(a)$ —and fails to hold of the right endpoint—i.e., $\neg \Phi _0(b)$ . But such a description does not provide us with a sequence of objects from C by which we can discretely step through the sort of reasoning described above.

To formulate a sorites argument in such cases, it thus seems that we have little choice but to assume the existence of an appropriate measurement function $\phi :A \rightarrow \mathbb {R}$ for which $\phi (a) = u$ and $\phi (b) = v$ . In such cases it is also typically assumed that the image of C under $\phi $ is a closed interval $[u,v] := \{x \in \mathbb {R} : u \leq x \leq v\}$ in the real numbers.Footnote ¹⁰ But although we may also feel that $\Phi _0(x)$ is insensitive to small variations, we will often be unaware of a relevant magnitude in $d \in A$ to serve as a just noticeable difference or of its numerical image.Footnote ¹¹ The question thus arises how we can formulate a tolerance principle which can be regarded as part of the meaning $\Phi _0(x)$ in a manner which still gives rise to a deductively mediated contradiction.

One way of avoiding the artificiality of choosing a specific c is to maintain that what we in fact accept is that if $\Phi _0(x)$ is vague and holds of $a \in A$ , then $\Phi _0(x)$ also holds of points $b \in A$ which are “vanishingly close” to a. Or we may abandon the metrical connotations of “close” by asserting that if $\Phi _0(x)$ is vague and b is within a “neighborhood” of a point a satisfying $\Phi _0(x)$ , then b also satisfies $\Phi _0(x)$ .

The terms neighborhood and cognates like region or vicinity evidently express topological concepts. But it is still not obvious how to formulate a tolerance principle which takes advantage of any of these notions. To do so, Weber & Colyvan rely on the assumption that the real numbers satisfy Dedekind completeness—i.e.,

1. DC: Suppose $X \subseteq \mathbb {R}$ , $X \neq \emptyset $ and X has an upper bound—i.e., there is $y \in \mathbb {R}$ such that $\forall x(x \in U \rightarrow x \leq y)$ . Then there exists a $u \in \mathbb {R}$ which is the supremum (or least upper bound) of X—i.e., $\forall x(x \in U \rightarrow x \leq u)$ and if v an is upper bound of X, then $u \leq v$ . In this case we write $u = \sup (X)$ .

In general a bounded set need not contain its supremum. But if $u = \sup (X)$ then u is as close as possible to points in X in the metrical sense—i.e., $\forall \varepsilon> 0 \exists x \in X(|\sup (X)-x| < \varepsilon )$ . And any neighborhood of u in the Euclidean topology on $\mathbb {R}$ —i.e., $(s,t) := \{x \in \mathbb {R} : s < x < t\}$ with $u \in (s,t)$ —will also contains points in X.

In order to formulate a tolerance principle which exploits this property of the real numbers, Weber & Colyvan presuppose that the field A of $\Phi _0(x)$ has been mapped in the manner we have been discussing into a closed subinterval of $\mathbb {R}$ which they take to be $[0,1]$ . Their proposed tolerance principle is then formulated for a mathematical predicate $\Phi (x)$ which is understood as the image of $\Phi _0(x)$ in $\mathbb {R}$ under a suitable measurement function $\phi (x)$ —i.e.,

1. (11) $\Phi (x) := \exists a \in A(\Phi _0(a) \wedge \phi (a) = x)$ .

Writing $\mathrm {BA}(X)$ to abbreviate that a set X is bounded above, we can now state Weber & Colyvan’s proposed tolerance principle as follows:Footnote ¹²

(LCC_sup(Ψ)) $$\begin{align} \forall X \subseteq \mathbb{R}(X \neq \emptyset \ \& \ \text{BA}(X) \ \& \ \forall x(x \in X \rightarrow \Psi(x)) \rightarrow \Psi(\sup(X))). \end{align} $$

Weber & Colyvan refer to this as the Leibniz Continuity Condition [LCC] and suggest that it holds for $\Phi (x)$ obtained via (11) from a vague predicate $\Phi _0(x)$ . We will have more to say about its motivation in §4. But for the moment we will focus on reconstructing its role in mediating a sorites-like contradiction.

Suppose $\mathcal {A}$ satisfies the hypotheses of Theorem 1 and also the following hold:

1. (12)

   1. i) $a,b \in A$ and $a \prec b.$

   2. ii) There exists $\phi :A \rightarrow \mathbb {R}^+$ satisfying the conclusion of Theorem 1 such that $\phi (a) = 0$ and $\phi (b) = 1$ and also that $\phi $ is onto $[0,1]$ —i.e., $\forall x \in [0,1]\exists a \in A (\phi (a) = x).$

   3. iii) $\Phi _0(a)$ and thus $\Phi (0).$

   4. iv) $\neg \Phi _0(b)$ and thus $\neg \Phi (1).$

Inspection of the proof of Hölder’s theorem ensures that $\phi (x)$ can also be constructed to satisfy the first part of (12ii).Footnote ¹³ Part i), iii), and iv) are traditional soritical premises. The goal is then to use $\textrm {LCC}_{\sup }(\Phi )$ to show that if $\Phi (x)$ holds of $0$ then it holds of $1$ . But unlike (6), $\textrm {LCC}_{\sup }(\Phi )$ does not allow us to extend the set of points satisfying $\Phi (x)$ rightward by any fixed increment. Thus despite the similarity in form between these principles, there is no apparent way of harnessing this principle to mimic the reasoning (10) of the conditional sorites.

There are indeed some subtleties in selecting principles sufficient to derive a contradiction from the premises (12) in conjunction with $\textrm {LCC}_{\sup }(\Psi )$ . One complication is that the argument presented by Weber & Colyvan requires not just $\textrm {LCC}_{\sup }(\Psi )$ but also an analogous principle about lower bounds:Footnote ¹⁴

(LCC_inf(Ψ)) $$\begin{align} \forall X \subseteq \mathbb{R}(X \neq \emptyset \ \& \ \mathrm{BB}(X) \ \& \ \forall x(x \in X \rightarrow \Psi(x)) \rightarrow \Psi(\inf(X))). \end{align} $$

Here $\mathrm {BB}(X)$ abbreviates that X has a lower bound—i.e., $\exists y(\forall z \in X \rightarrow y \leq z)$ —and $\inf (X)$ denotes its infimum (i.e., greatest lower bound).

Adopting this assumption their argument can now be set out as follows:Footnote ¹⁵

1. (13)

   1. i) Assume $\textrm {LCC}_{\sup }(\Phi )$ and $\textrm {LCC}_{\inf }(\Phi )$ for a vague measure predicate $\Phi $ .

   2. ii) Assume for a contradiction that $\neg \Phi (1)$ .

   3. iii) Define the following subsets of real numbers

      $U = \{x \in [0,1] : \Phi (x)\}$ and $V = \{x \in [0,1] : \neg \Phi (x)\}.$

   4. iv) We obtain the following from i), property (2) of $\prec $ , and (12i-iii):

      1. a) $U \neq \emptyset $ and U is bounded above.

      2. b) $V \neq \emptyset $ and V is bounded below.

      3. c) $U \cup V = [0,1]$ and $U \cap V = \emptyset $ .

   5. v) By iv a), DC entails that $\sup (U) := u$ exists. Thus $\Phi (u)$ by the definition of U and $\textrm {LCC}_{\sup }$ applied to $\Phi (x).$

   6. vi) By iv b), DC entails that $\inf (V) := v$ also exists. Thus $\neg \Phi (v)$ by the definition of V and $\textrm {LCC}_{\inf }$ applied to $\neg \Phi (x).$

   7. vii) By the trichotomy property of $<$ on $\mathbb {R}$ one of the following must hold: $u = v$ ; $u < v$ ; $v < u$ . We now reason by cases as follows:

      1. a) If $u = v$ , then $\Phi (u)$ and $\neg \Phi (u)$ by v) and vi). Contradiction.

      2. b) If $u < v$ , then by the density of $<$ on $\mathbb {R}$ , there exists w such that $u < w < v$ . Since $w < v$ , $w \in U$ and thus $\Phi (w)$ . But since $u < w$ , $w \in V$ and thus $\neg \Phi (w)$ . Contradiction.

      3. c) If $v < u$ , a contradiction in reached analogously with b).

   8. viii) Discharging the reductio assumption ii), we conclude $\Phi (1)$ .

The foregoing constitutes a demonstration of $\Phi (1)$ from the premises (12i-iii), the two forms of the LCC and Dedekind Completeness. This then yields a contradiction with the other soritical assumption $\neg \Phi (1)$ (premise 12iv). It is this overarching reasoning which we henceforth refer to as the continuous sorites argument.Footnote ¹⁶

4. The Leibniz Continuity Condition and Open Induction

We have presented the continuous sorites argument as showing that $\Phi (1)$ follows from $\Phi (0)$ and the two forms of the LCC. But a similar argument may be used to show that these assumptions entail that all real numbers $x \in \mathbb {R}$ fall under $\Phi (x)$ . Indeed in analogy to mathematical induction on $\mathbb {N}$ , we have the following.

As we will discuss further in §5, the foregoing style of argument was repeatedly anticipated in the development of real analysis from at least the late 19th century.Footnote ¹⁷ One of the latter chapters was the isolation of a principle which Lorenzen [Reference Lorenzen24] called analytical induction. This is simply the assertion that if $\Psi (x)$ satisfies the hypotheses of Proposition 2 then it also satisfies its conclusion.Footnote ¹⁸ Writing $\mathrm {Open}(\Psi )$ to abbreviate (14), $\mathrm {Down}(\Psi )$ for (15), and $\mathrm {Prog}(\Psi )$ (or Progressive) for (16), this can be formulated schematically as follows:

(OI₀(Ψ)) $$\begin{align} \exists x \Psi(x) \ \wedge \text{Open}(\Psi) \wedge \text{Down}(\Psi) \wedge \text{Prog}(\Psi) \rightarrow \forall x \in \mathbb{R}\hspace{.4ex} \Psi(x). \end{align} $$

The proof just given for this principle is similar in many respects to the continuous sorites argument. Better conformity to the reasoning of (13) can be obtained by applying a version of OI $_0(\Psi )$ restricted to the unit interval $[0,1]$ to which Coquand [Reference Coquand8] gave the name Open Induction. Using the abbreviation

(Prog^{[0, 1]}(Ψ)) $$\begin{align} \forall x \in [0,1]\big((\forall y \in [0,1])(y < x \rightarrow \Psi(y)) \rightarrow \Psi(x)\big) \end{align} $$

this takes the formFootnote ¹⁹

(OI₁(Ψ)) $$\begin{align} (\mathrm{Open}(\Psi) \ \wedge \ \mathrm{Prog}^{[0,1]}(\Psi)) \rightarrow (\forall x \in [0,1]) \Psi(x). \end{align} $$

In these principles the assumption that $\Psi (x)$ is progressive plays a role similar to LCC $_{\sup }(\Psi )$ . Thus even better conformity to the continuous sorites argument can be had by considering a restricted version of the former

(LCC_sup ^{[0, 1]}(Ψ)) $$\begin{align} \forall X \subseteq [0,1](X \neq \emptyset \ \wedge \forall x(x \in X \rightarrow \Psi(x))) \rightarrow \Psi(\sup(X))) \end{align} $$

to formulate another version of Open Induction as follows:

(OI₂(Ψ)) $$\begin{align} (\mathrm{Open}(\Psi) \ \wedge \ \Psi(0) \ \wedge \textrm{LCC}^{[0,1]}_{\sup}(\Psi)) \rightarrow (\forall x \in [0,1]) \Psi(x). \end{align} $$

A disadvantage of this formulation is that it quantifies over subsets of the reals and as such is a third-order statement. But Weber & Colyvan also discuss a form of the LCC in which the variable X is understood to vary not over subsets of $\mathbb {R}$ but rather over Cauchy sequences. We can also thus consider

1. LCC^{[0, 1]} _seq(Ψ) : For any $x\in [0,1]$ , if for any sequence $(x_{n})_{n\in {\mathbb {N}}}$ in $[0, x]$ with $\lim _{n\rightarrow \infty }x_{n}=x$ , if we have $\Psi (x_{n})$ for all $n\in {\mathbb {N}}$ , then $\Psi (x)$ .

and the corresponding version of Open Induction

(OI₃(Ψ):) $$\begin{align} (\mathrm{Open}(\Psi) \ \wedge \ \Psi(0) \ \wedge \ \textrm{LCC}^{[0,1]}_{\mathrm{seq}}(\Psi)) \rightarrow (\forall x \in [0,1]) \Psi(x). \end{align} $$

We may now formulate a derivation similar to the continuous sorites argument which consolidates the reasoning of steps (13.iii-xiii) into a single application of $\mathrm {OI}_{i}$ for $j \in \{1,2,3\}$ . Retaining the basic assumptions (12), we have:

1. (13′)

   1. i) Suppose $\mathrm {Open}(\Phi )$ , $\Phi (0)$ , and $\mathrm {Prog}^{[0,1]}(\Phi )$ (for $j = 1$ ), $\textrm {LCC}^{[0,1]}_{\sup }(\Phi )$ (for $j = 2$ ) or $\textrm {LCC}^{[0,1]}_{\mathrm {seq}}(\Psi )$ (for $j = 3$ ).

   2. ii) Suppose $\mathrm {OI}_j(\Phi )$ .

   3. iii) By i) and ii), $\forall x \in [0,1]\Phi (x)$ and thus, in particular, $\Phi (1)$ .

As with (13), this yields a contradiction with the further assumption $\neg \Phi (1)$ .

In comparing the derivations (13) and (13′), it should be kept in mind that $\mathrm {OI}_{1}(\Psi ), \mathrm {OI}_2(\Psi )$ and $\mathrm {OI}_3(\Psi )$ are all theorem schemas of classical real analysis. Just as mathematical induction is taken as a valid schema for deriving universal statements about natural numbers, these principles may all be regarded as valid schema for deriving statements about the unit interval.

The validity of mathematical induction is often regarded as constitutive of the structure $\mathbb {N}$ . But it can also be regarded as a consequence of some but not all definitions of the natural numbers. Similarly, the proof of Open Induction is related to the assumption that the real numbers satisfy Dedekind Completeness which, in turn, plays a central role in the traditional “construction of the reals” in terms of Dedekind cuts. But as we will discuss in §6, there are other definitions of $\mathbb {R}$ which lack this property. We will also see that in axiomatic formulations of analysis there is a sense in which OI $_0$ -OI $_3$ are all equivalent to Dedekind Completeness.

Presuming that such principles are retained, we can now also see that the assumptions which the continuous sorites brings into conflict are the openness of the extension of $\Phi (x)$ and that $\Phi (x)$ satisfies either Progressiveness or one of the latter forms of the LCC.Footnote ²⁰ As we will discuss further in §5, the role of $\mathrm {Open}(\Phi )$ in the proof of Open Induction (in any of its forms) may itself seem reminiscent of an intuitively plausible kind of tolerance principle extending the extension of $\Phi (x)$ from a given $u \in [0,1]$ to an open set $(u - \varepsilon , u + \varepsilon )$ surrounding it. But in the continuous setting this on its own is not sufficient to infer $\Phi (1)$ from $\Phi (0)$ as a bounded open sets need not contain its least upper bounds. It is such a transference of $\Phi (x)$ from the points in X to $\sup (X)$ which the various forms of the LCC are intended to effect. So we must finally attend to their motivation.

The expression “Leibniz Continuity Condition” is taken from Priest [Reference Priest28] who in turn attributes a principle similar to $\textrm {LCC}_{\sup }$ to Leibniz. Leibniz referred to the principle in question as his Law of Continuity which he first published in 1687. One of his more succinct formulations was as follows:

Leibniz intended this principle to express part of his conviction that physical transitions had to be continuous in a manner analogous to geometrical limiting processes. It is in virtue of this interpretation that Leibniz’s principle was later sloganized as “what is true up to the limit is true at the limit” (see, e.g., [Reference Cajori6]).

It is tempting to read the sequential formulation $\textrm {LCC}^{[0,1]}_{\mathrm {seq}}(\Psi )$ of the LCC into such historical sources. But such principles are not in general true for arbitrary predicates of real numbers.Footnote ²¹ While Weber & Colyvan [Reference Weber and Colyvan35] flag their awareness of this, they still assert that such a principle is constitutive of how we should understand vague predicates. And indeed debate about the Law of Continuity continued to play a role in the development of the calculus during the 18th century. But far from confirming the principle, the discovery that modeling more complex kinds of physical dynamical systems required the recognition of broader classes of discontinuous functions was taken as a decisive refutation.Footnote ²² It thus seems that the ideas behind the LCC stand in bad company alongside an outmoded understanding of both and mathematical and concrete continua.

5. The Covering Sorites and the Heine–Borel Theorem

The forms of the sorites we have considered thus far rely on tolerance principles which push the extension of the image $\Phi (x)$ of a vague predicate $\Phi _0(x)$ rightward from $0$ so that it ultimately surpasses some designated endpoint. The variant we will consider in this section is grounded in the prima facie weaker assumption that each point a in an empirical or perceptual interval is surrounded by some region $U^{a} = \{b \in A : a \prec b \circ c_a \ \& \ b \prec a \circ c_a\}$ of points which are sufficiently “nearby” a such that all points closer than the “tiny variation” $c_a$ agree with a on $\Phi _0(x)$ .

If this is true of every $a \in A$ , it follows that there must be some means of partitioning such a continuum into a not-necessarily disjoint family $\mathcal {D} = \{U_i : i \in I\}$ of such regions with the following properties:

1. (17)

   1. i) Every $a \in A$ is contained in some region $U_i$ .

   2. ii) The regions are sufficiently “small” such that all $b,c \in U_i$ , $\Phi _0(b) \leftrightarrow \Phi _0(c)$ .

The intuition motivating the existence of such a family $\mathcal {D}$ is similar to that for $\mathrm {Open}(\Phi )$ . Its is supported by the relationship between vagueness, margins of error arising from finite sensitivity, and the use of open sets to model them.Footnote ²³ The plausibility is bolstered further by observing that a family witnessing (17i,ii) might consist of infinitely or even uncountably many regions of sufficiently “fine grain”.

The tolerance principle on which the covering sorites is based results from using the framework of §2 to formalize the properties (17i,ii). In this case regions U in A go over to open sets $O = \{z \in \mathbb {R}^+ : \phi (c) - \varepsilon _a < z < \phi (a) + \varepsilon _a\}$ in $\mathbb {R}$ for a suitable $\varepsilon _a = \phi (c_a)$ . We assume that $\phi :A \rightarrow \mathbb {R}^+$ is a bijection between A and $[0,1]$ satisfying the properties of Theorem 1. In this case the conditions (17) go over to the claim that there exists a family of intervals $\mathcal {C} = \{O_i : i \in I\}$ satisfying

1. (18)

   1. i) $\forall x \in [0,1] \exists i(x \in O_i)$

   2. ii) $\forall i \in I \forall x,y \in O_i(\Psi (x) \leftrightarrow \Psi (y)).$

We will write $\exists \mathcal {C}\mathrm {Cov}(\Psi , \mathcal {C})$ to abbreviate that a cover satisfying (18i,ii) exists.

The covering sorites seeks to derive a contradiction from the same soritical premises (12) as the continuous sorites by relying on the compactness of the Euclidean topology on $[0,1]$ rather than Dedekind Completeness or one its equivalents. The former is reported by what is now called the Heine–Borel theorem.

We will return to the proof of Theorem 3 in a moment. But to see how it can be used to formulate a soritical derivation, we record the following consequence.

While this is immediate from Theorem 3, it is still useful to isolate Proposition 4 as a corollary as it bears a similar relation to the Heine–Borel theorem as does Proposition 2 (and thus Open Induction) to Dedekind Completeness. In particular, we may use it to formulate a schema encapsulating how the compactness of the unit interval may be used to uniformly mediate a general inference:

(CL(Ψ)) $$ \begin{align} (\exists \mathcal{C}\mathrm{Cov}(\Psi,\mathcal{C} ) \ \wedge \ \Psi(0)) \rightarrow \forall x \in [0,1] \Psi(x). \end{align} $$

We will call this principle the Creeping lemma.Footnote ²⁴

The covering sorites for a vague predicate $\Phi (x)$ is now as follows:Footnote ²⁵

1. (19)

   1. i) Suppose $\exists \mathcal {C}\mathrm {Cov}(\mathcal {C},\Phi )$ and $\Phi (0)$ .

   2. ii) Suppose $\mathrm {CL}(\Phi )$ .

   3. iii) By i) and ii), $\forall x \in [0,1]\Phi (x)$ and thus, in particular, $\Phi (1)$ .

Again as in the case of derivation (13), this reasoning yields a contradiction with the further soritical assumption $\neg \Phi (1)$ . But the Creeping lemma is again a theorem schema of classical analysis. As with the continuous sorites and the LCC, whether one regards the above as a genuine paradox will thus likely covary with the stock one places in $\exists \mathcal {C}\mathrm {Cov}(\mathcal {C},\Phi )$ for a given predicate. But there is still a connection between the continuous sorites and the manner in which the Heine–Borel theorem was originally obtained.

The Heine–Borel theorem can be credited directly to Borel [Reference Borel5] in the case of countable coverings. Both his original proof as well as several others which followed exemplify two salient features.Footnote ²⁶ First, assuming that $\mathcal {C}$ is a covering of $[0,1]$ , they employ sorites-like arguments to show that it is possible to derive a contradiction from the assumption that there is a point in $x \in [0,1]$ which cannot be reached via a finite sequence of overlapping intervals from $\mathcal {C}$ the first of which contains $0$ . Second, in order to do so, they employ principles equivalent to Dedekind Completeness.

Both points are illustrated by Lebesgue’s Lebesgue [Reference Lebesgue22] proof of Theorem 3.

Proof. Let $\mathcal {C} = \{O_i : i \in I\}$ be an open covering of $[0,1]$ and suppose for reductio that there does not exist a finite sub-cover of $\mathcal {C}$ . Say a point is reached if it satisfies

$$ \begin{align*} \Psi(x) = \textit{the interval} \ [0,x]\ \textit{can be covered by a finite number of intervals from} \ \mathcal{C}. \end{align*} $$

Our reductio assumption implies that the set $W = \{x \in [0,1] : \neg \Psi (x)\} \neq \emptyset $ . But since $\mathcal {C}$ covers $[0,1]$ , it follows that $0 \not \in W$ since $0$ is reached as witnessed by some $O_{i_0} = (x_{i_0},y_{i_0})$ . As $W \neq \emptyset $ and is bounded below, $\inf (W) = w$ exists by DC. But then $w \in O_k = (x_k,y_k)$ for some $k \in I$ . By the density of the reals there also exist $u,v$ such that $x_k < u < w < v < y_k$ . By the definition of W, u is reached. There thus exists a finite sub-cover $\mathcal {C}^-_0 \subseteq \mathcal {C}$ of $[0,u]$ . But then $\mathcal {C}_0 = \mathcal {C}^-_0 \cup \{(x_k,y_k)\}$ is a finite covering of $[0,v]$ , contradicting the definition of w as $\inf (W)$ .

The method of this proof is similar to that of Open Induction which is in turn similar to the continuous sorites argument itself. The development of the topology of reals in the 19th century can thus be understood as anticipating the covering sorites in much the same way that reflection on limits in the 18th century anticipated the continuous sorites. But since the proof of Theorem 3 just given also relies on Dedekind Completeness, one might conclude that the covering and continuous sorites share a common mathematical core. As will see, however, argument (19) requires strictly weaker assumptions than does (13). It thus it seems reasonable to regard the continuous and covering sorites as genuinely distinct forms of the paradox.

6. Reverse Mathematics of the sorites

In this section we use the methods of reverse mathematics to show that the central assumption underlying the continuous sorites is equivalent to a non-constructive principle known as Arithmetical Comprehension while that underlying the covering sorites is equivalent to a strictly weaker (but still non-constructive) principle known as Weak König’s lemma. This will inform our discussion in §7.

6.2. Classificatory theorems

The mathematical results presented in this section illustrate two points on which we will build in §7. First, the three sorites arguments considered above —i.e., discrete (§2), and covering (§5), continuous (§3)—rest on increasing non-constructive principles. Second, this provides a principled reason for regarding the three arguments as distinct forms of the paradox.

In regard to the first point, the formalization of mathematics in $\mathsf {RCA}_0$ has traditionally been regarded as compatible with the development of constructive analysis [Reference Simpson30, sec. I.8]. For instance within $\mathsf {RCA}_0$ it is possible to formalize proofs describing effective approximations of real numbers such as the non-enumerability of the reals. Another example is the proof of original statement of Hölder’s theorem—i.e., every order Archimedean group is isomorphic to a subgroup of $\langle \mathbb {R},<,+ \rangle $ .

Solomon [Reference Solomon31] showed that this statement is provable in $\mathsf {RCA}_0$ . By adapting his construction to accommodate the weaker axioms for $\mathcal {A}$ we can obtain the following.

Proposition 5. Using the coding of the reals described above, the $L_2$ -formalization of Theorem 1 of §2 is provable in $\mathsf {RCA}_0$ .

This result is significant for two reasons. First, it exemplifies a presupposition of measurement theory that representation theorems ought to be accompanied by constructive procedures for associating magnitudes with real numbers. Indeed the proof of Hölder’s theorem given by Krantz [Reference Krantz, Dantzig and Cwinorr20] describes a method for computing the measurement function $\phi : A \to \mathbb {R}$ via a process of successive bisecting approximations so that the value of $\phi (a)$ is given as a sequence of rationals $q_0, q_1, \ldots $ with an effective modulus of convergence.Footnote ²⁹ Second, it illustrates that while the formalization of discrete sorites arguments are often reliant on such representations, they do not require additional non-constructive principles beyond $\mathsf {RCA}_0$ .

Turning next to the continuous sorties, we begin by observing that it is possible to formulate a version of Open Induction as a single statement rather than a schema by quantifying over open sets directly rather than formulas which define them:

1. OI₄: For any open $O \subseteq [0,1]$ , we have $\forall y \in [0,1]((\forall z\in [0,1])(z<y\rightarrow z\in O)\rightarrow y\in O) \rightarrow (\forall x\in [0,1])(x\in O)).$

The relation of this principle to the continuous sorites is clarified by the following.Footnote ³⁰

Theorem 6. The following statements are provably equivalent over $\mathsf {RCA}_0:$

1. (a) The schema of Arithmetical Comprehension $\mathrm {ACA}$ .

2. (b) A bounded monotone sequence $(x_{n})_{n\in {\mathbb {N}}}$ of reals in $[0,1]$ converges to a limit.

3. (c) If $(x_n)_{n \in \mathbb {N}}$ is a Cauchy sequence of reals, then $\lim _{n \rightarrow \infty } x_n$ exists.

4. (d) If $(x_{n})_{n\in {\mathbb {N}}}$ is a bounded sequence of reals, then $\sup (\{x_n : n \in \mathbb {N}\})$ exits.

5. (e) If C is a (code for) a closed subset of $[0,1]$ , the infimum $\inf (C)$ exists.

6. (f) If O is a (code for) an open subset of $[0,1]$ , the supremum $\sup (O)$ exists.

7. (g) Open Induction in the form $\mathrm {OI}_{1}(\Psi )$ for $\Sigma _{1}^{0}$ -formulas $\Psi (x)$ .

8. (h) Open Induction in the form $\mathrm {OI}_{3}(\Psi )$ for $\Sigma _{1}^{0}$ -formulas $\Psi (x)$ .

9. (i) Open Induction in the form $\mathrm {OI}_{4}(\Psi )$ .

Proof. The equivalence between the first six items is well-known [Reference Simpson30, IV.2.11, III.2.2], while the equivalence between $\mathrm {OI}_{1}(\Psi )$ and $\mathrm {OI}_{1}(\Psi )$ is immediate as $\Sigma _{1}^{0}$ -formulas can be effectively converted into codes for open sets, and vice versa, [Reference Simpson30, II.5.7]. For the implication (a) $\Rightarrow $ (i), assume ${\mathsf {ACA}}_{0}$ and let O be a code for an open set as in item (i)—i.e., we have for all $y\in [0,1]$ that

1. (21) $(\forall z\in [0,1])(z<y\rightarrow z\in O)\rightarrow y\in O.$

Note that fixing $y=0$ in (21), we (trivially) obtain $0\in O$ . Suppose $x_{0}\not \in O$ for some $x_{0}\in [0,1]$ —i.e., $C:= [0,1]\setminus O$ is closed and non-empty. Now consider $\inf (C)$ which satisfies $0<\inf (C)<1$ by assumption and in light of $0\in O$ and the assumption that O is open. Now consider the instance of (21) for $y=\inf (C)$ , i.e.,

$$\begin{align*}(\forall z\in [0,1])(z<\inf(C) \rightarrow z\in O)\rightarrow \inf(C)\in O. \end{align*}$$

Since the antecedent holds, so does the consequent. However, $\inf (C)\in O$ implies that $\{(\inf (C)-r,\inf (C)+r) \subset O$ for some $r>0$ . As this contradicts the definition of $\inf (C)$ , $(\forall x\in [0,1])(x\in O)$ must hold. One derives item (h) in the same way.

For (i) $\Rightarrow $ (a), assume item (i) and let $(x_n)_{n \in {\mathbb {N}}}$ be an increasing sequence of reals in $[0,1]$ . Consider the (code for an) open set defined as $O=\bigcup _{n\in {\mathbb {N}}} [0, x_n)$ . If $1\not \in O$ , the contraposition of $\mathrm {OI}_{4}$ yields that there is $y\in [0,1]$ such that (21) is false, i.e.,

$$\begin{align*}(\forall z\in [0,1])(z<y\rightarrow z\in O)\wedge y\not\in O. \end{align*}$$

Since y is the limit of $(x_n)_{n \in N}$ , the Monotone Convergence theorem as in item (b) follows; by the above, we obtain ${\mathsf {ACA}}_{0}$ . For the derivation of the latter via (g), define $\Phi (x)\equiv (\exists n)(x<x_{n} )$ and use the prior proof. The proof based on (h) is similar.

This illustrates that the various forms of Open Induction already have strong consequences about set existence for relatively simple formula.Footnote ³¹ Turning now to the covering sorites, we obtain a similar classificatory theorem:

Theorem 7. The following statements are provably equivalent over $\mathsf {RCA}_0$ :

1. (a) Weak König’s lemma $\mathrm {WKL}$ .

2. (b) The Heine–Borel theorem for countable coverings.

3. (c) The Creeping lemma $\mathrm {CL}(\Psi )$ for $\Sigma _{1}^{0}$ -formulas $\Psi (x)$ .

Proof. The equivalence between the first two items is well-known [Reference Simpson30, IV.1]. Now assume the Creeping lemma as in item (c) and consider a countable covering $(O_{i})_{i\in {\mathbb {N}}}$ of $[0,1]$ . Define $\Psi (x)\equiv (\exists j)( [0, x] \subset \bigcup _{i<j} O_{i} )$ , which can be shown to be $\Sigma _{1}^{0}$ relative to coding of open sets described above. We also have $\Psi (0)$ and $\exists \mathcal {C}\mathrm {Cov}(\Psi ,\mathcal {C} )$ . Using item $\mathrm {CL}(\Psi )$ , we obtain $\Psi (1)$ , i.e., $(O_{i})_{i\in {\mathbb {N}}}$ has a finite sub-covering of $[0,1]$ . Thus, item (b) follows as required.

To prove the Creeping lemma as in item (c), assume item (b) and let $(O_{i})_{i\in {\mathbb {N}}}$ be a covering of $[0,1]$ . Define $\Psi (x)$ be as above and note that this covering again witnesses $\exists \mathcal {C}\mathrm {Cov}(\Psi ,\mathcal {C})$ . Using item (b), there is $j\in {\mathbb {N}}$ such that $[0,1]\subset \cup _{i<j}O_{i}$ . Since we assume $\Psi (0)$ , we have $\Psi (z)$ for all $z \in O_{i_{0}}$ where $i_0$ is such that $0\in O_{i_{0}}$ . Since $[0,1]\subset \cup _{i<j}O_{i}$ , there is $i_{1}<j$ with $O_{i_{0}}\cap O_{i_{1}}\ne \emptyset $ , implying $\Psi (z)$ for all $z \in O_{i_{0}}\cup O_{i_{1}}$ . Continuing in this fashion, one concludes $\Psi (z)$ for all $z \in \cup _{i<j}O_{i}$ after finitely many steps. This reasoning can be formalized using $\Sigma _{1}^{0}$ -induction.

WKL does not imply ACA over $\mathsf {RCA}_0$ [Reference Simpson30, VIII.2.12]. The equivalence of WKL and CL thus illustrates that the continuous sorites requires strictly stronger mathematical principles than the covering sorites. This substantiates our claim that they are distinct argument forms.Footnote ³² At the same time, the original proofs of Heine–Borel theorem (such as the one given in §5) all made use of principles implying ACA. The separation of the two arguments thus also emerges as a distinctive contribution of reverse mathematics to the study of the sorites.Footnote ³³ But the fact that CL entails WKL shows that the reasoning of the covering sorites still depends on non-constructive principles. For while $\mathsf {WKL}_0$ is unlike $\mathsf {ACA}_0$ in that it does not prove the existence of specific non-computable sets, it still implies some such sets must exist [Reference Simpson30, XIII]. We will now see how such computability-theoretic facts may be used to construct models in which the premises of both forms of the sorites hold but their putatively paradoxical conclusions fail.

7. Resolutions

The continuous sorites and covering sorites rest on two sorts of major premises:

1. (P1) The schemas $\mathrm {Open}(\Psi )$ , $\mathrm {Prog}^{[0,1]}(\Psi )$ , $\textrm {LCC}^{[0,1]}_{\sup }(\Psi )$ , $\textrm {LCC}^{[0,1]}_{\mathrm {seq}}(\Psi )$ , and $\exists \mathcal {C}\mathrm {Cov}(\Psi )$ which we will continue to refer to collectively as tolerance principles.

2. (P2) The schemes $\mathrm {OI}_1(\Psi )$ , $\mathrm {OI}_2(\Psi )$ , $\mathrm {OI}_3(\Psi )$ , and $\mathrm {CL}(\Psi )$ which Theorems of 6 and 7 provide further justification for calling mathematical principles.

The latter class are all conditional in form while the former serve as their antecedents in the arguments (13′) and (7). To block the reasoning it hence suffices to reject either (P1) or (P2). We will refer to these as Type-1 and Type-2 responses.

To see how the prior analysis bears on these options consider:

In light of Theorems 6 and 7, we may take $\mathcal {M}$ to be the model . The formulas in question can then be obtained from traditional constructions known as recursive counterexamples—i.e., computable objects satisfying the antecedent of a classical theorem for which there does not exist a computable witness to its consequent. For since $\mathsf {RCA}_0$ proves the existence of objects satisfying the following definitions, it follows that the Monotone Convergence Theorem (item b) in Theorem 6) and the Heine–Borel theorem are both false when interpreted in the computable reals—i.e., the sets satisfying the $L_2$ -definition ‘x is a real number’ in the model .

The relevant features of such objects (as reprised in the Appendix) are as follows:

1. (R1) From the perspective of the classical continuum $\mathbb {R}$ —e.g., as characterized by Dedekind Completeness— contains “gaps” corresponding to non-computable real numbers. If $(s_{n})_{n\in {\mathbb {N}}}$ is a Specker sequence and $s = \lim _{n\in \mathbb {N}} s_n = \sup \{s_{n} : n\in {\mathbb {N}}\}$ then illustrates such a gap.

2. (R2) If $\mathcal {S} = \{(s_i,t_i) : i \in \mathbb {N}\}$ is a singular covering, then for each there exists an i such that $x \in (s_i,t_i)$ . But since $\sum _{i \in \mathbb {N}} |s_i - t_i| < 1$ , any attempt to cover all of must “leave a gap” in $[0,1]$ . As such, we cannot pass from $0$ to $1$ via any finite set of overlapping intervals $\mathcal {S}_0 = \{(s_{i_j},t_{i_j}) : j < k\}$ .

These properties are already suggestive of how the computable real numbers may be useful for providing a semantics for vague predicates.Footnote ³⁴ But it is not initially clear whether they motivate a Type-1 or a Type-2 response to our current paradoxes.

Consider first a Specker sequence $(s_n)_{n \in \mathbb {N}}$ with $s = \lim _{n \rightarrow \infty } s_n$ and define

1. (22) $\Sigma (x) := 0 \leq x \ \wedge \ \exists n(x < s_n).$

The set $S = \{x \in \mathbb {R}: \Sigma (x)\}$ will thus be an interval $[0,s)$ . It may be confirmed that the premises of the continuous sorites $\Sigma (0)$ , $\mathrm {Open}(\Sigma )$ , $\mathrm {Prog}^{[0,1]}(\Sigma )$ , $\textrm {LCC}^{[0,1]}_{\sup }(\Sigma )$ , and $\textrm {LCC}^{[0,1]}_{\mathrm {seq}}(\Sigma )$ all hold in .Footnote ³⁵ But since $s = \sup (S) < 1$ , it also follows that $\neg \Sigma (1)$ . It also follows from the construction given in the Appendix that there is a total computable function $f(n)$ such that $f(n) =$ (the code of) $s_n$ . $\Sigma (x)$ thus exemplifies a $\Sigma ^0_1$ -predicate for which each of the principles $\mathrm {OI}_1,\mathrm {OI}_3$ and $\mathrm {OI}_4$ fail in .

A singular covering can be similarly employed to show how the premises of the covering sorites—other than CL itself—can hold in without entailing a contradiction. For let $\mathcal {S} = \{(s_i,t_i) : i \in \mathbb {N}\}$ be a such a covering and define

1. (23) $\Pi (x) := \exists n([0,x] \subseteq \bigcup _{i < n} (s_i,t_i)).$

It is easy to see that that $\Pi (0)$ and that $\forall i \forall x,y \in (s_i,t_i)(\Pi (x) \leftrightarrow \Pi (y))$ hold in .Footnote ³⁶ $\mathcal {S}$ thus witnesses . But since $\mathcal {S}$ does not admit a finite sub-cover in , . Thus $\Pi (x)$ is an example of a $\Sigma ^0_1$ -predicate for which $\mathrm {CL}$ fails in . Similarly if we let $s' = \sup \{x \in \mathbb {R}: \Pi (x)\}$ then we can see that —i.e., the supremum of the set of points reachable from $0$ via a finite sequence of overlapping intervals in $\mathcal {S}$ —must itself be a non-computable real.

What one makes of these constructions may depend on one’s views about both the proper response to traditional discrete forms of the sorites and also the proper foundations for analysis and its applications. For instance if one maintains the classical view that the mathematical continuum is appropriately characterized by properties such as Dedekind Completeness, then one will presumably accept a mathematical theory at least strong as $\mathsf {ACA}_0$ . In this case, one will accept the mathematical principles (P2) and thus presumably be predisposed to reject one of the principles under (P1) in the manner of a Type-1 response to the sorites.

But even in this case, properties (R1) and (R2) of may still play a role. For recall that the rejection of tolerance principles has traditionally been regarded as an inevitable—although potentially undesirable—consequence of theories of vagueness known as supervaluationism (e.g., after Fine [Reference Fine17]) and epistemicism (e.g., after Williamson [Reference Williamson36]). In particular, both supervaluationists and epistemicists hold that principles like (6) will have false instances.

Supervaluationists, for instance, hold that a statement $\Theta $ containing a vague predicate $\Phi _0(x)$ with field A should be counted as true only if it is in fact supertrue. If we let $P^+ \subset A$ be the definite extension of $\Phi _0(x)$ , this is to say that $\Theta $ is satisfied in all interpretations in which $\Phi _0(x)$ corresponds to an admissible precisification $P \supseteq P^+$ —i.e., a means of exhaustively deciding indefinite (or borderline) cases so that $P \cup \overline {P} = A$ and also $P \cap P^- = \emptyset $ where $P^-$ is the definite anti-extension $P^-$ of $\Phi _0(x)$ . But of course each such precisification P will yield an instance $b \in A$ making the antecedent of principles like (6) true but its consequent false.

But now observe that the predicate $\Sigma (x)$ will determine an interval $[0,s)$ whose boundary occurs at a non-computable real . On the other hand, it seems reasonable to maintain that the definite extension $S^+$ of $\Sigma (x)$ must be fixed by a sequence of observations which can be carried out so as to determine an interval of the form $[0,t]$ or $[0,t)$ where $t < s$ is a computable real number—e.g., as resulting from a measurement procedure with a computable modulus of convergence similar to the envisioned construction in the proof of Theorem 1 described above.Footnote ³⁷

Putting aside the question of whether non-computable real numbers exist, there is still the further question of whether $S \supseteq S^+$ must be acknowledged as an admissible precisification of the definite extension of $\Sigma (x)$ . For suppose there is a rationale for excluding such sets from the class of precisification which must be considered in a supervaluationalist semantics. It will then follow that even though the relevant instances of the tolerance principles falling under (P1) are false in the full model $\mathcal {N}$ , they will be supertrue when we restrict attention to precisifications corresponding to intervals whose endpoints exist in the model . This conclusion might in turn be embraced by supervaluationists who must otherwise provide an account of why the intuitions underlying tolerance principles are misleading.Footnote ³⁸

It also seems possible for epistemicists to similarly avail themselves of the features of properties (R1) and (R2) of . For recall that such theorists are also compelled to reject tolerance principles because they wish to adhere to classical (in particular non-supervaluational) semantics. This in turn requires positing a sharp boundary between the extension of a vague and its anti-extension. Epistemicists then attempt to mitigate the necessity of abandoning tolerance intuitions by offering an auxiliary account of why we cannot know where such a boundary is located.

In the context of a discrete sorites like (10), the boundary between the extension of $\Phi _0(x)$ and its anti-extension must coincide with a fixed object $a_i$ in a sequence $a_0, \ldots ,a_{n}$ . Taking into account that the measurement procedures which we have suggested in §2 are required to formalize the relevant argument, it seems likely the corresponding numerical boundary $\phi (a_i)$ will be given by a rational number $q_i \in \mathbb {Q}$ .Footnote ³⁹ But since such a value is finitely specifiable, it is then difficult to substantiate the epistemicist’s claim that its value cannot be known even in principle.

But of course the situation is again different in the case of the continuous and covering sorites wherein the relevant boundaries can in principle be determined by arbitrary real numbers. The constructions under consideration call attention to the fact that such a value may be non-computable. But when we examine the methods by which such values might be determined, it seems there is indeed even an in practice sense in which we cannot come to learn their exact values. This in turn suggests that epistemicists may also find it useful to employ recursive counterexamples to illustrate the sense in which they deem such boundaries to be unknowable.Footnote ⁴⁰

Turning now to Type-2 responses, readers from logic will be aware that such constructions were originally presented within foundational programs which seek to deny various features of the classical continuum. In this context Specker sequences and singular covers serve as weak counterexamples to continuity properties of $\mathbb {R}$ or to its compactness—if we accept such principles, then these constructions show that we must also accept non-constructive principles. Within the historical development of intuitionism, the availability of independent arguments against such principles were presupposed.Footnote ⁴¹ Practicing intuitionists have shown little interest in the interpretation of non-mathematical language. But readers from philosophy will be aware that there is a related tradition of attempting to reply to sorites arguments by rejecting—or at least failing to assert—that vague predicates are bivalent.

A precedent can be traced to Bernays [Reference Bernays3] who observed a contradiction could be avoided by abstaining from applying the law of the excluded middle to the evidently soritical predicate feasible natural number—i.e., one up to which we can count in practice. This example was famously repurposed by Dummett [Reference Dummett13] as part of his critique of strict finitism.Footnote ⁴² Building on Dummett’s broader view that a theory of meaning for natural language ought to be based on intuitionistic logic, Wright [Reference Wright37], reiterated the proposal that semantics for vague predicates should be based on an intuitionistic interpretation of the logical connectives. This proposal was recently put into a more precise form by Bobzien & Rumfitt [Reference Bobzien and Rumfitt4].

These considerations are relevant here in light of our prior characterization of as containing “gaps” corresponding to non-computable real numbers. This naturally invokes comparisons not only to failures of excluded middle but also to approaches which posit truth value gaps for vague predicates. Indeed the basic analogy on which the aforementioned theorists build is the claim that bivalence can only be asserted for decidable predicates. For on the one hand, mathematical examples motivating the non-assertability of $\forall x(\Phi (x) \lor \neg \Phi (x))$ relative to the proof (or BHK) interpretation of the logical connectives make use of cases in which $\forall x \Phi (x)$ is undecided (or even undecidable).Footnote ⁴³ On the other, philosophical accounts often characterize vague predicates $\Phi _0(x)$ as ones for which there exist borderline cases $a \in A$ such that the truth value of $\Phi _0(a)$ is difficult (or even impossible) to decide.

It would thus seem that theorists inclined toward constructivist approaches to vagueness may find Type-2 responses congenial. For if $(s_n)_{n \in \mathbb {N}}$ is a Specker sequence with limit s, then the $L_2$ -formalization of $\Sigma (s)$ is formally undecidable in $\mathsf {RCA}_0$ .Footnote ⁴⁴ If we then apply the proof interpretation to $\Sigma (s) \lor \neg \Sigma (s)$ , this provides a basis for failing to assert $\forall x(\Sigma (x) \lor \neg \Sigma (x))$ .Footnote ⁴⁵ But at the same time, this also underscores that $\Sigma (s)$ and $\neg \Sigma (s)$ are both consistent with $\mathsf {RCA}_0$ . Once this is acknowledged, it seems that it should also be conceded that models like (in addition to the full model $\mathcal {N}$ ) can serve as felicitous interpretations of vague predicates. Of course such models fail to satisfy the principles falling under (P2). On its own, however, this does not recommend intuitionistic logic as a medium for reasoning within them.Footnote ⁴⁶