Tolerance and higher-order vagueness

Abstract

The idea of higher-order vagueness is usually associated with conceptions of vagueness that focus on the existence of borderline cases. What sense can be made of it within a conception of vagueness that focuses on tolerance instead? A proposal is offered here. It involves understanding ‘definitely’ not as a sentence operator but as a predicate modifier, and more precisely as an intensifier, that is, an operator that shifts the predicate extension along a scale. This idea is combined with the author’s earlier approach to the semantics of vague expressions, which builds on the idea of a central gap associated with a predicate. The central gap approach is generalized to handle arbitrarily many iterations of ‘definitely’.

Appendices

Appendix

Proof of fact 1

For any finite domain E, any $F \in A_{S}$, and any measure function ${\mathscr {H}}_{F}$, there are ${\mathscr {G}}_{F}$, $\delta _{F}$ and $\epsilon _{F}$ such that (${\mathscr {H}}$-req) is satisfied.

Proof

Note first that if the domain is empty, (${\mathscr {H}}$-req) is trivially satisfied. For nonempty domain, we proceed by induction on the length of T sequences. The idea is to show that we can construct ${\mathscr {G}}_{F}$, $\delta _{F}$ and $\epsilon _{F}$, step by step, so as to make sure that (${\mathscr {H}}$-req) is met. Given ${\mathscr {H}}_{F}$ and E, we have a finite set V of real numbers such that $V=\{{\mathscr {H}}_{F}(a): a \in E\}$.

For the base case, with $l(\xi )=1$, we need to find a pair of real numbers i, j, with $i > j$, such that for any $r \in V$, $r>i$ or $r<j$. This can clearly be done: pick any elements of V that are adjacent in the natural order, and pick i and j between these two, with $i>j$. Then set ${\mathscr {G}}_{F}(\langle + \rangle )=i$ and ${\mathscr {G}}_{F}(\langle - \rangle )=j$.

For the inductive step, assume that (${\mathscr {H}}$-req) is satisfied for sequences of length k. There are $2^{k+1}$ sequences $\xi ^{\smallfrown }\langle * \rangle$, where $l(\xi )=k$. Again, let $\overline{*}$ be $-$ if $*$ is $+$, and vice versa. For each such $\xi$, we need extend $\xi ^{\smallfrown }\langle * \rangle$ into $\xi ^{\smallfrown }\langle *,+ \rangle$ and $\xi ^{\smallfrown }\langle *,- \rangle$ such that, where $d=|{\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle * \rangle )-{\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle \overline{*} \rangle )|$,

in accordance with clauses iiia) - iiic) of Definition 2. To do this, we need to find two numbers $\alpha$ and $\beta$ such that for any of the $2^{k+1}$ sequences $\xi ^{\smallfrown }\langle * \rangle$:

Since $\beta -\alpha$ can be arbitrarily small, this can again be done: Pick $\beta$ such that for no $r \in V$, ${\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle + \rangle )+\beta = r$, nor ${\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle - \rangle )-\beta = r$. $\beta$ must in addition be chosen such that

for $d'$ defined below, in order that clause iib) of Definition 2 is satisfied (we return this below).

Then pick some $\alpha < \beta$. If some $r' \in V$ satisfies either (i) or (ii), for some sequence $\xi$ of length k, then replace $\alpha$ by $\alpha '$ such that $\alpha <\alpha '<b$ and $\alpha '$ does not satisfy (##) with respect to $\xi$, $\beta$, and $r'$. This is possible, since $\beta -\alpha '$ can be arbitrarily small. Repeat the procedure until you have a number $\alpha ^{*}$ such that no $r \in V$ satisfies (##) with respect to $\alpha ^{*}$ and $\beta$. Since V is finite and there are $2^{k+1}$ sequences to consider, this process will come to an end.

Then extend the definition of $\epsilon$ by setting $\epsilon (d)=\beta$. Preliminarily extend the definition of $\delta$ by setting $\delta (d)=\beta -\alpha ^{*}$. We must check that clauses i) and ii) of Definition 2. are met. Let $\xi =\xi '^{\smallfrown }\langle + \rangle$. Let $d'=|{\mathscr {G}}_{F}(\xi '^{\smallfrown }\langle + \rangle )-{\mathscr {G}}_{F}(\xi '^{\smallfrown }\langle - \rangle )|$. For clause ib) of 2 it must hold that $\delta (d)\le d$. If not, choose $\alpha '$ such that $\alpha ^{*}<\alpha '<\beta$, that meets this condition. It will also satisfy (##), since it is smaller than $\beta$. Let $\alpha '$ be the final choice. That is, we finally extend the definition of $\delta$ by setting $\delta (d)=\beta -\alpha '$.

We must also check that clause iib) is met. This holds if $\epsilon (d)\le \epsilon (d')$, since by the ind. hyp. $d \le d'$ (because by the ind. hyp. $d = \delta (d')$). Since $\beta$ was chosen to satisfy $(\dagger )$, and $\epsilon (d)=\beta$, this condition is met.

Finally, extend ${\mathscr {G}}_{F}$ by setting

We can verify that (i)–(iii) of (#) are satisfied. In case (iii), with $*$ as +, we have:

The other cases are similar.

Finally, we verify that (${\mathscr {H}}$-req) is satisfied for ${\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle +,+ \rangle )$ and ${\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle +,- \rangle )$. Assume the contrary. Let $r_{b} \in V$ be such that

$$\begin{aligned} {\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle +,- \rangle )\le r_{b} \le {\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle +,+ \rangle ) \end{aligned}$$

By (###), this means that

$$\begin{aligned} G_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )+ \alpha ' \le r_{b} \le G_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )+ \beta , \end{aligned}$$

contradicting (##) (with $\alpha '$ instead of $\alpha$). The case for $\langle -,+ \rangle$ and $\langle -,- \rangle$ is similar. This completes the induction.$\square$

Proof of fact 2

For any higher-order gap model M for SL and any predicate $F \in P_{S}$, for any $a \in E$, if $a \in [\![F ]\!]$, then for any $b \in E$, if $|{\mathscr {H}}_{F}(a)-{\mathscr {H}}_{F}(b) |\le {\mathscr {T}}(F)$, then $b \in [\![F ]\!]$.

Proof

By straightforward induction over degrees of predicates (number of occurrences of D), from (${\mathscr {T}}$) and Definition 2, we can show that

Further, it is immediate that

Let’s say that atomic predicates are positive; and if F is positive [negative], then $\lnot F$ is negative [positive]; and if F is positive [negative], then DF is positive [negative]. Let the degree of a predicate F be the number of occurrences of D in F. We show that the extensions of predicates follow the T-assignment in the following sense:

Showing this is is sufficient, for the assumption that M is a higher-order gap model entails by definition that for any $\xi \in T$, and any $a \in E$, ${\mathscr {H}}_{F^{\circ }}(a)>G_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )$ or ${\mathscr {H}}_{F}(a)<G_{F}(\xi ^{\smallfrown }\langle - \rangle )$. Then the Fact follows by (*) and (**): (***) together with the assumption that M is gap model (i.e. satisfies ${\mathscr {H}}-rec$) delivers the antecedent of (**). (**) itself then delivers the consequent of (**). Using (*), we substitute the left-hand side of (*) for the right-hand side in the consequent of (**). We then have the Fact.

We prove (***) by induction over predicate degree. It holds for atomic F immediately by clause (ii) of (SLS). For the induction step, assume that F is a positive predicate of degree $k+1$. Then either F is $DF'$, where $F'$ is a positive predicate, or F is $\lnot DF'$, where $F'$ is a negative predicate, of degree k. Assume the former. The other case is similar. By the induction hypothesis, there is a sequence $\xi ^{\smallfrown }\langle + \rangle$ such that $[\![F' ]\!]=\{a \in E: {\mathscr {H}}_{F^{\circ }}(a)>{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )\}$, where $l(\xi )=k$.

By clause (iv) of (SLS), and the definitions (${\mathscr {T}}$) and (${\mathscr {S}}$), $a \in [\![DF' ]\!]$ iff for any $b \in E$, $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(b) |\le \epsilon ({\mathscr {T}}(F'))$ it holds that $b \in [\![F' ]\!]$. By the induction hypothesis, if $b \in [\![F' ]\!]$, then ${\mathscr {H}}_{F^{\circ }}(b)>{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )$. Hence, if $a \in [\![DF' ]\!]$, then ${\mathscr {H}}_{F^{\circ }}(a)>{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )+\epsilon ({\mathscr {T}}(F'))$. By (*), ${\mathscr {T}}(F')={\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )-{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle - \rangle )$. And by clause iiib) of Definition 2, ${\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle +,+ \rangle )={\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )+\epsilon ({\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )-{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle - \rangle ))$. Therefore, if $a \in [\![DF' ]\!]$, then ${\mathscr {H}}_{F^{\circ }}(a)>{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle +,+ \rangle )$, as desired.

Then consider a negative predicate of degree $k+1$. Assume that $F'$ is positive, and consider $\lnot DF'$. By (iii) of (SLS), $a \in [\![\lnot DF' ]\!]$ just in case $a \in E-[\![DF' ]\!]$. By the result for the positive case above, an object $b \in [\![DF' ]\!]$ iff for some sequence $\xi$ of length k, ${\mathscr {H}}_{F^{\circ }}(b)>{\mathscr {G}}_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle )$. Since M is a higher-order gap model, there is no object $c \in E$ such that $G_{F^{\circ }}(\xi ^{\smallfrown }\langle + \rangle ) \ge {\mathscr {H}}_{F^{\circ }}(c) \ge G_{F^{\circ }}(\xi ^{\smallfrown }\langle - \rangle )$. Therefore, $a \in [\![\lnot DF' ]\!]$ iff ${\mathscr {H}}_{F}(a)<{\mathscr {G}}_{F}(\xi ^{\smallfrown }\langle - \rangle )$, as desired. The other two cases (DF negative and $\lnot DF$ positive) are similar. This completes the proof of (***).$\square$

Proof of fact 3

(SLS) validates (a) T, (b) RN, (c) B, and (d) modus ponens.

Proof

a)    The predicate counterpart of T is

The validity is immediate from clause (iv) of (SLS), by clause ii) of Definition 2 and the definition (${\mathscr {S}}$) of the safety function ${\mathscr {S}}$. Since ${\mathscr {S}}(DF)\ge 0$, and hence ${\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(a) \le {\mathscr {S}}(DF)$, $a \in [\![F ]\!]$ if $a \in [\![DF ]\!]$.

b) The predicate counterpart of RN is

It is valid. If $\models F_{M}$, $[\![F ]\!]_{M}=E_{M}$. Then any $a \in E_{M}$ satisfies the condition for $[\![DF ]\!]$, since it trivially holds for any $a \in E_{M}$ that $\forall b \in E_{M} \ (|{\mathscr {H}}_{F}(a)-{\mathscr {H}}_{F}(b) |\le {\mathscr {S}}(DF)) \longrightarrow b \in [\![F ]\!]$.

c)    The predicate counterpart of B is

(B$_{\text {pred}}$) is valid iff for any model M, $[\![F ]\!]_{M} \subseteq [\![D \lnot D \lnot F ]\!]_{M}$. To show that this holds, assume $a \notin [\![D \lnot D \lnot F ]\!]$. By clause (iv) of (SLS), and clause (b) of (${\mathscr {S}}$), it then holds that

Let $b'$ be such a b. By (iii) of (SLS) we also have

Let $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(b')| = j$, where $0 \le j \le {\mathscr {S}}(DDF)$. This means that either ${\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(b')=j$, or ${\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(b')=-j$.

Again, by clause (iv) of (SLS), it holds that

Since ${\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(c)={\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(b')+{\mathscr {H}}_{F^{\circ }}(b')-{\mathscr {H}}_{F^{\circ }}(c)$, it holds either that $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(c)|=|{\mathscr {H}}_{F^{\circ }}(b')-{\mathscr {H}}_{F^{\circ }}(c)|+j$, or else that $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(c)|=|{\mathscr {H}}_{F^{\circ }}(b')-{\mathscr {H}}_{F^{\circ }}(c)|-j$

It therefore follows from (§) and (§§) that

Since ${\mathscr {S}}(G)\ge 0$, for any G, it holds that $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(a)| \le j + {\mathscr {S}}(DF)$. Further, by the definitions of ${\mathscr {S}}$ and ${\mathscr {T}}$, and by clauses ib) and iib) of Definition 2, it holds that ${\mathscr {S}}(DDF) \le {\mathscr {S}}(DF)$, and hence that $j \le {\mathscr {S}}(DF)$. Therefore, it also holds that $|{\mathscr {H}}_{F^{\circ }}(a)-{\mathscr {H}}_{F^{\circ }}(a)| \le {\mathscr {S}}(DF) -j$. Either way, it then follows from (§§§) that $a \notin [\![F ]\!]$. Hence, if $a \notin [\![D \lnot D \lnot F ]\!]$, then $a \notin [\![F ]\!]$, which establishes c).

d) The tolerance logic counterpart of modus ponens is valid:

Immediate from (STL): if, by (ii) and (i), $[\![F ]\!]_{M} \subseteq [\![G ]\!]_{M}$ and $[\![F ]\!]_{M}=E_{M}$, then $[\![G ]\!]_{M}=E_{M}$.$\square$

Proof of fact 4

a):    

b):    

Proof

Proof of a). By means of a simple countermodel. Consider a set of men and the predicate ‘tall’ T. The central gap has size 1 cm and the upper edge at 180 cm. The $\delta$ and $\epsilon$ function are such that $\delta (x)=0.9x$ and $\epsilon (x)=10x$.

Then the upper boundary for being $\lnot T$ is at 179 cm. The upper boundary for being $D \lnot T$ is $179-10\times 1 = 169$ cm. The lower boundary for being $\lnot D \lnot T$ is $169 + 0,9\times 1 = 169.9$ cm. The lower boundary for being $D \lnot D \lnot T$ is $169.9 + 10\times 0.9= 178.9$ cm. The lower boundary for being $DD \lnot D \lnot T$ is then $178.9 + 10\times 0.9\times 0.9=178.9+8.1=187$ cm. Repeating this, we find that the lower boundary for being $D^{6}\lnot D \lnot T$ is greater than the lower boundary for being $D^{4}T$, but that the result of prefixing both with D yields the reverse result: the lower boundary of being $D^{7} \lnot D \lnot T$ is slightly lower than the lower boundary of being $D^{5} T$. Hence, we have $D^{6} \lnot D \lnot T \, \models _{M} D^{4}T$, but .

Proof of b). By means of counterexample. Let F be $\lnot D \lnot D G$. We can show that $F \models G$, by a reasoning that parallels the proof that (SLS) validates B. Here, I shall simplify the argument. Assume without loss of generality that G is a positive predicate (elements of its extension have higher measure than elements of its anti-extension). Let $\gamma$ be ${\mathscr {T}}(G)$, and let $\beta$ be the upper boundary of the G gap. We must now show that $[\![F ]\!] \subseteq [\![G ]\!]$, which means if an object a is in $[\![F ]\!]$, then its measure is not below the G gap. That is: ${\mathscr {H}}_{G^{\circ }}(a)\ge \beta -\gamma$.

The upper boundary $F^{+}$ of the gap for F is, by Definition 2, calculated as:

Since no element is in the G gap, any object above the lower boundary of the G gap in in G. So what we need is that $F^{+}>\beta - \gamma$. That is

Simplifying, this holds iff

Since, by Definition 2, a) it holds that $\gamma \ge \delta (\gamma )$, b) it holds that $\epsilon (\gamma )\ge \epsilon (\delta (\gamma ))$, and c) it holds that $\delta (\delta (\gamma ))>0$, ($\circ \circ \circ$) is true. Hence, $[\![F ]\!] \subseteq [\![G ]\!]$.

Nevertheless, by suitable choices of $\epsilon$ and $\delta$, we can have a model, along the lines of the proof of part a) of the Fact, where $[\![DF ]\!] \not \subseteq [\![DG ]\!]$ $\square$