When “All the Five Circles” are Four: New Exercises in Domain Restriction

Abstract

The domain of a quantifier is determined by a variety of factors, which broadly speaking fall into two types. On the one hand, the context of utterance plays a role: if the focus of attention is on a particular collection of kangaroos, for example, then “Q kangaroos” is likely to range over the individuals in that set. On the other hand, the utterance itself will help to establish the quantificational domain, inter alia through presuppositions triggered within the sentence. In this paper, we concentrate our attention on constructions like the following, in which “the square to which ... ” is the critical presupposition trigger:

• (i) Q circles ...

• (ii) Q of these circles ...

• (iii) Q of these five circles ...

  ... have the same colour as the square to which they are connected.

Many theories predict that all instances of these schemata will give rise to the presupposition that every circle is connected to a square. We present an analysis which predicts that these sentences should generally be accepted in a context in which not all the circles are connected to a square, with one exception only: if a quantified sentence is of type (iii) and Q is non-intersective, then the sentence should be more likely to be rejected. Furthermore, we predict that manipulating the context so as to make the connected circles more or less salient should have an effect on statements with non-intersective quantifiers only. These predictions were tested in a series of experiments.

Appendix: Further Details on Experimental Design and Data

1.1 Experiment 1

1.1.1 Participants

We posted surveys for 88 participants on Amazon’s Mechanical Turk (mean age: 39; range: 17–77; 47 females). Only workers with an IP address from the United States were eligible for participation. These workers were asked to specify their native language, but payment was not contingent on their answer to this question. Two participants were excluded from the analysis because they were not native speakers of English.

1.1.2 Materials and Procedure

A trial consisted of a sentence and a picture. Participants were instructed to indicate if the sentence was true or false as a description of the corresponding picture. Participants could also choose the answer “Don’t know”. The full instructions went as follows:

On each of the following pages of this survey, you will see a picture and a sentence. In each case, we ask you to decide whether the sentence gives a correct description of the picture or not. If it does, check “True”. If it doesn’t, check “False”. If you feel you cannot decide whether the sentence is true or false, check “Don’t know”.

The survey consisted of 15 items: 5 targets and 10 fillers. The sentences in both conditions were of the form “Q {circle has/circles have} ...”. The form of the quantifier Q was varied between participants: plain (24 participants), D-partitive (24 participants), or DN-partitive (40 participants). The corresponding pictures consisted of four squares and five circles on a 3$\times$3 grid. The shapes were coloured red, green, or blue.

The target sentences were of the following form:

• Q {circle has/circles have} the same color as the square to which {it is/they are} connected.

In the corresponding pictures, four of the circles were each connected to one square, while the remaining circle was unconnected. The colours were distributed so that the sentence was false on a universal construal of the presupposition and true otherwise. For each of the quantifiers, five pictures were constructed, varying the distribution of the colours, the connections between circles and squares, and the position of the unconnected circle. Sample target pictures for each of the quantifiers are given in Fig. 6.

Fig. 6

Example pictures used in Experiment 1

Filler sentences were structurally similar to the target sentences. Three examples with plain quantifiers are:

• (1) a. Every circle is connected to a circle of a different color.

  b. More than two squares are connected to less than two circles.

  c. Less than three squares are connected to a red circle.

The corresponding pictures for the filler items unequivocally verified or falsified these sentences.

Five lists were created, varying the order of the items and the correct responses to the filler items. The first two items were always fillers, and target items were always separated by at least one filler item.

1.1.3 Results

Filler items were answered correctly 79 % of the time. One response was missing. The answer “Don’t know” was extremely rare ($<$1 %). Since it is unclear how to interpret these responses, they were discarded from the analysis. The percentages of “True” responses are given in Table 3.

Table 3 Percentages of participants in Experiment 1 who indicated that the target sentence was true in a situation falsifying the universal inference

We compared the proportions of positive responses by means of pairwise Z-tests. None of the proportions of positive responses were significantly different (all $p$’s $>$ $.05$), except for the proportion of positive responses for “each of these n”, which differed significantly from the proportions of positive responses for all other quantifiers (all $p$’s $<$ $.001$). Furthermore, there was a highly significant correlation between the number of incorrect answers to filler items and the number of “False” responses to target items ($r = .51,\, t(84) = 5.37,\, p < .001$). So the more errors participants made in the filler items, the more likely they were to give “False” responses to target items. This correlation suggests that some of the “False” responses to target items may be attributed to mistakes.

1.2 Experiment 2

1.2.1 Participants

We posted surveys for 50 participants on Amazon’s Mechanical Turk (mean age: 30; range: 18–59; 17 females). Only workers with an IP address from the United States were eligible for participation. These workers were asked to specify their native language, but payment was not contingent on their answer to this question. All participants turned out to be native speakers of English.

1.2.2 Materials and Procedure

A trial consisted of a sentence and a picture. Participants were instructed to indicate if the sentence was true or false as a description of the corresponding picture. The instructions went as follows:

In the following survey, we will show you pairs of sentences and pictures. In each case, we ask you to decide whether or not the sentence gives a correct description of the picture. If you feel that the sentence is true, check “True”. If not, check “False”.

We are interested in your spontaneous judgments, so please don’t think too long about your answers.

The survey consisted of five items: 1 target and 4 fillers. The target sentence was “All of the squares are red”. The corresponding picture alternated between picture A (25 participants) and picture B (25 participants) from Fig. 3. Filler items were either ambiguous, involved some kind of visual illusion, or required analytic thinking. An example of the last category is shown in Fig. 7. One list was created. The target item was the fourth item in the list.

Fig. 7

Example of a filler item used in Experiment 2

1.2.3 Results

Participants were divided about the ambiguous and illusory filler items. All participants gave the correct answer to the filler item in Fig. 7. Responses to the target item depended on the picture. For picture A, 52 % of the participants judged that the sentence was true; for picture B, 16 % did. This difference was statistically significant ($Z = 2.69,\, p < .01$).

1.3 Experiment 3

1.3.1 Participants

We posted surveys for 84 participants on Amazon’s Mechanical Turk (mean age: 34; range: 20–80; 44 females). Only workers with an IP address from the United States were eligible for participation. These workers were asked to specify their native language, but payment was not contingent on the answer to this question. Three participants were excluded from the analysis because they were not native speakers of English.

1.3.2 Materials and Procedure

A trial consisted of a sentence and a picture. Participants were instructed to indicate if the sentence was true or false as a description of the corresponding picture. Participants could also choose the answer “Don’t know”. The instructions were the same as for Experiment 1.

We constructed three types of pictures: one with seven circles and seven squares on a 2$\times$7 grid and one with five circles and five squares on a 3$\times$4 grid. There were two kinds of 3$\times$4 pictures, depending on whether two (= 3$\times$4|2) or three (= 3$\times$4|3) colours were used. In the 3$\times$4 pictures, there was one unconnected circle. We varied whether this circle had the same colour as the corresponding unconnected square, but this didn’t have an effect. In the 2$\times$7 pictures, five circles were unconnected. Some but not all of these had the same colour as the corresponding square.

Examples of the three types of pictures are shown in Figs. 4 and 5. For each picture type, we constructed five pictures, varying the selection and distribution of the colours, and the position of the unconnected circle or circles. There were three groups of participants, one for each type of picture: 20 participants saw the 3$\times$4|3 pictures, 24 participants saw the 3$\times$4|2 pictures, and 40 participants saw the 2$\times$7 pictures.

The target sentences were the same as in Experiment 1, but we only tested DN-partitive quantifiers. So all target sentences were of the following form, with n being the total number of circles in the picture:

• Q of these n circles {has/have} the same color as the square to which {it is/they are} connected.

For the 3$\times$4|3 pictures, we tested all five DN-partitive quantifiers listed in Table 1, along with ten fillers. Since the pattern of results was similar to that of Experiment 1, we only tested the quantifiers “each of these n” and “none of these n” for the other two picture types, reducing the number of fillers to eight. In the surveys with 3$\times$4|2 and 2$\times$7 pictures, we also included two control items to gauge if participants correctly parsed the target sentences and pictures. These were of the following form, and were paired with pictures that made the sentences unambiguously true or false:

• Q of these n circles which is connected to a square has {the same/a different} color {as/than} the square to which it is connected.

We created four lists of 2$\times$7 items, five lists of 3$\times$4|3 items, and eight lists of 3$\times$4|2 items, varying the order of the items and the correct responses to the filler items. The first two items were always fillers, and target items were separated by at least one filler item.

1.3.3 Results

Filler items were answered correctly 79 % of the time. Control items were answered correctly 92 % of the time. The answer “Don’t know” was extremely rare ($<$4 %). Since it is unclear how to interpret these responses, they were discarded from the analysis. The results for the quantifiers “each of these n” and “none of these n” are shown in Table 2.

The distribution of answers for “none of these n” was similar to the results for that quantifier in Experiment 1. The distribution of answers for “each of these n” differed depending on the type of picture. The proportion of positive responses for the 3$\times$4|3 pictures (35 %) was significantly higher than for the 3$\times$3 pictures tested in Experiment 1 (13 %, $Z = -1.99, p = .047$), and significantly lower than for the 3$\times$4|2 pictures (68 %, $Z = 2.05, p = .040$) and the 2$\times$7 pictures (65 %, $Z = 2.07, p = .039$). The difference between the proportions of positive responses for the last two pictures was not statistically significant ($Z < 1$).