Grounding a Pragmatic Theory of Vagueness on Experimental Data: Semi-orders and Weber’s Law

Abstract

One of the traditional pragmatic approaches to vagueness suggests that there needs to be a significant gap between individuals or objects that can be described using a vague adjective like tall and those that cannot. In contrast, intuitively, an explicit comparative like taller does not require fulfillment of the gap requirement. Our starting point for this paper is the consideration that people cannot make precise measures under time pressure and their ability to discriminate approximate heights (or other values) obeys Weber’s law. We formulate and experimentally test three hypotheses relating to the difference between positive and comparative forms of the vague adjectives, gap requirement, and Weber’s law. In two experiments, participants judged appropriateness of usage of positive and comparative forms of vague adjectives in a sentence-picture verification task. Consequently, we review formal analysis of vagueness using weak orders and semi-orders and suggest adjustments based on the experimental results and properties of Weber’s law.

Thanks to the Dutch NWO gravitation project Language in Interaction, the ESSENCE Marie Curie Initial Training Network, and the project ‘Logicas no-transitivas. Una nueva aproximacion a las paradojas’ for financially supporting this project. The authors also thank Nicole Standen-Mills for professional proof-reading.

Appendices

Appendix A. Experimental Procedure and Participants

Procedure The experiment was built using a JavaScript library for online chronometric experiments—JsPsych—and run in participants’ web browsers. This library allows for collection of participants’ responses as well as fairly accurate reaction time data (Leeuw 2015; Leeuw and Motz 2016). The first screen that participants saw displayed general information about the experiment. On the second screen, they had to give consent to participate, and agree with storage of the data obtained. They then filled in a short questionnaire regarding their background, which served as a check that indeed all eligibility requirements were met. This was followed by detailed instructions for the task. Participants then did three practice trials, after which the experiment itself started. There were two blocks—one with sentences with positive vague adjectives and one with sentences with comparative vague adjectives. Each of the blocks contained 56 trials. Between the blocks, participants could take a break.

Trials within each block were presented in random order. Each trial started with a display of the sentence in the middle of the screen. When the participants finished reading the sentence, they pressed the space bar to see the picture itself. They then had to press either P if they agreed that the sentence could be used to describe this picture, or Q if they did not agree. The participants had 2300 ms to give a response, and if no response was given, the trial ended automatically and it was recorded as a missing response. The next trial then started after a random inter-stimulus interval between 700 and 1200 ms.

The instructions that participants received were the following:

In this experiment, you will see pairs of sentences and pictures. Your task is to indicate whether you think the sentence is a good description of the given picture. In other words, do you think people would use this description for the given picture? You will need to think about the exact meanings of words in order to make correct judgments.

Every trial will start with a sentence [...] Please pay attention to the meanings of sentences and give correct answers, but also try to do it quickly [...].

Participants Eligible participants had to meet the following criteria: native speaker of English; age 18–35 years old; born and currently living in the USA; right-handed. Completing the experiment should have taken approximately 15 min and the participants were paid 1.80 £.

Experiment 1: Twenty-five participants completed the task. One participant was excluded due to reading the instructions for under 10 s. The remaining 24 participants were included in the analysis. Nine were male, 15 female; their mean age was 24.96 (range 18–33). They took 11:29 min on average to complete the task (min. 08:33 and max. 17:11).

Experiment 2: Forty participants completed the task. One participants reported being color-blind and was excluded. Six further participants were excluded for reading the instructions for under 10 s. Two participants were excluded for pressing a single button as a response throughout the experiment. Thus, 9 participants in total were excluded and the analyses were performed on the remaining 31 participants. The mean age of these participants was 26.26 (range 20–34). Thirteen were female, 17 male, and 1 of other gender. Participants took 11:19 on average to complete the task (min. 07:11, max. 17:32).

Appendix B. Proof of Fact 3

To prove Fact 3, we have to show that the resulting order obeys the conditions for being a semi-order. That is, it has to be (i) irreflexive (IR), (ii) satisfy the interval order condition (IO), and (iii) be semi-transitive (STr).

(IR) Irreflexivity follows immediately if $\epsilon (x) \ge 0$.

(IO). Suppose xPy and vPw. To prove xPw or vPy. Suppose $\lnot x P w$. It follows that (i) $g(x) > g(y) + \epsilon (y)$, and (ii) $g(v) > g(w) + \epsilon (w)$, and (iii) $g(x) \not > g(w) + \epsilon (w)$.

Now either (a) $\epsilon (v) = \epsilon (x)$, (b) $\epsilon (v) < \epsilon (x)$, or (c) $\epsilon (v) > \epsilon (x)$.

• (a) But then by the constraint it follows that $g(v) = g(x)$, and with (ii) and (iii) we have a contradiction.

• (b) But then by the constraint it follows that $g(x) > g(v)$ and thus with (ii) $g(x) > g(w) + \epsilon (w)$, which is in contradiction with (iii).

• (c) But then by the constraint it follows that $g(v) > g(x)$ and thus with (i) $g(v) > g(y) + \epsilon (y)$. But then vPy.

Similarly, we can prove that if $\lnot vP y$ it follows that xPw, which is enough to prove what we wanted.

(STr). Suppose $x > y$ and $y > z$. To prove for any v: xPv or vPz. So suppose xPy and yPz and $\lnot x Pv$. That is, suppose (i) $g(x) > g(y) + \epsilon (y)$ (ii) $g(y) > g(z) + \epsilon (z)$, and (iii) $g(x) \not > g(v) + \epsilon (v)$. Now either (a) $\epsilon (v) = \epsilon (x)$, (c) $\epsilon (v) > \epsilon (x)$ or (c) $\epsilon (v) < \epsilon (x)$.

• (a) By the constraint it follows that $g(x) = g(v)$. Notice that from (i) $g(x) > g(y) + \epsilon (y)$ and (ii) $g(y) > g(z) + \epsilon (z)$ it immediately follows that $g(x) > g(z) + \epsilon (z)$ (transitivity). Thus vPz.

• (b) By the constraint it follows that $g(v) > g(x)$. Notice that from (i) and (ii) it follows that $g(x) > g(z) + \epsilon (z)$, as in (a). Thus now also $g(v) > g(z) + \epsilon (z)$. But this means that vPz.

• (c) By the constraint it follows that $g(x) > g(v)$, and thus $g(v) + \epsilon (v) \ge g(x) > g(v)$. Now there are three possibilities: either (c1) $\epsilon (v) = \epsilon (y)$, (c2) $\epsilon (v) < \epsilon (y)$, or (c3) $\epsilon (v) > \epsilon (y)$.

  • (c1) By the constraint it follows that $g(y) = g(v)$. But this is impossible, because we have assumed that xPy and $\lnot x P v$.

  • (c2) By the constraint it follows that $g(y) > g(v)$. Thus, $g(y) + \epsilon (y) > g(v) + \epsilon (v)$. But this is impossible, because we have assumed that xPy and $\lnot x P v$.

  • (c3) So, (c3) has to be the case. But this means with the constraint that $g(v) > g(y)$. Because $g(y) > g(z) + \epsilon (z)$, it follows that $g(v) > g(z) + \epsilon (z)$. Thus vPz.

Thus, if $\lnot x Pv$, then vPz.

Similarly, we can also prove that if $\lnot v Pz$, then xPv. Now we have proved semi-transitivity.