Cumulative Comparison: Experimental Evidence for Degree Cumulation

Abstract

In this paper we address the question whether it makes sense to assume that the domain of degrees, as used in degree semantics, consists not just of atoms, but also of degree pluralities. A number of recent works have adopted that assumption, most explicitly (Fitzgibbons et al. Plural Superlatives and Distributivity, Proceedings of Semantics and Linguistic Theory, Vol. 18, 2008; Beck, The Art and Craft of Semantics: A Festschrift for Irene Heim, MITWPL, Vol. 70: 91–115, 2014; Dotlačil and Nouwen, Natural Language Semantics, 1–34, 2016). In this paper, we provide experimental evidence for degree pluralities by showing that comparatives may express cumulative relations between degrees.

1 Adjectives and Plurality

Adjectives can be distributive or collective. For instance, “tall” is related to the height of an individual and is, as such, intrinsically concerned with atoms only: there is no such thing as the height of John and Mary as a group. Other adjectives are different. For instance, the adjective “compatible” is inapplicable to atoms, since degrees of compatibility can only be assigned to groups of entities. This situation is familiar from the semantics of non-adjectival plurality. Predicates like “be a team” are collective in the same sense as “compatible” is, while predicates like “being wounded” are distributive in the same sense as “tall”.

A dominant way of thinking about this distinction is that collective predicates have no atomic individuals in their extension, while distributive predicate have nothing but atoms in their extension. One clear advantage of this is that it enables us to account for why collective predicates are never compatible with singular arguments, while distributive ones are compatible with plural arguments, as illustrated in (1) and (2).

To account for (2), all we need to assume is that, at least for plural cases like this, the extension of the predicate is closed under so-called sum formation: the operation that forms pluralities $a\sqcup b$ out of atoms a and b. The operator that enforces this kind of closure is often notated as $^{*}$ (after, Link 1983). What accounts for the contrast between (1) and (2) is the fact that $^{*}$ can create pluralities out of atoms, but it cannot create atoms out of pluralities. In other words, for a predicate with a non-empty extension, the extension of $^{*}P$ will always contain non-atomic entities, while it is not guaranteed to contain atomic ones.

Things are no different for adjectives. It would be natural to assume that “tall” expresses a relation between atomic entities and degrees and that “compatible” expresses a relation between non-atomic entities and degrees. Once the degree slot has been saturated, we will have a distributive predicate for the case of “tall” (as in “being tall” or “being two meters tall”) and a collective one in the case of “compatible”.¹

In other words, considerations of plurality appear to have to do with how predicates map to the domain of entities and in particular to the complexity of the members of their extension. It appears then that plurality does not play a role on the side of degrees. Comparison, for instance, is a purely atomic relation, even if the adjective involved is collective. Take (3) as a case in point.

What is at stake in (3) is whether the degree of compatibility of the sum of John and Mary exceeds the degree of compatibility of the sum of Peter and Sue. Clearly, even though the components of the comparative are plural, the degrees are not. The same point can be made for (4).

Here, the predicate “to be taller than Peter” is a distributive predicate, which can be closed under sum formation using the $^{*}$ operator. In this way, it can apply to plural arguments even though both the relation expressed by “tall” and the comparison relation expressed by the comparative only have atoms in their extension.

So far, then, we have seen that where adjectives interact with pluralities, there is no evidence for the need for plural degrees. What is plural in all the above cases is an e-type argument of an adjectival or a comparative relation. Recently, however, there have been a number of proposals that assume the existence of non-atomic degrees. Fitzgibbons et al. (2008), for instance, analyse sentences where a superlative predicate is combined with a plural subject (John and Paul are the tallest students) as involving a fully pluralised adjective, i.e. a relation between potentially plural entities and potentially plural degrees. Very recently, two other accounts of plural degrees were developed, both aiming to improve on existing analyses of comparatives: Beck (2014) and Dotlačil and Nouwen (2016). In the remainder of this paper, we will present experimental evidence for such proposals. Focusing on our own account, we will start by summarising the plural degree approach to comparatives.

2 Plural Degrees

At the heart of the plural degree approaches to comparatives lies a classical puzzle of the semantics of comparatives, namely how to account for the interpretation of comparatives that have quantified than clauses (von Stechow 1984; Larson 1988; Heim 2006). The best paraphrase for a sentence like (5) is one in which the quantifier takes scope outside of the than clause, as in (6).²

The same intuition holds for differentials: (8) is a very good analysis of (7), since it correctly predicts that (7) entails that every girl has the same height.

The issue is that than clauses are islands, cf. the ungrammatical status of the following example from Larson (1988):

This fact makes (6) and (8) useless as blue-prints for the semantic structure of comparatives and differentials, for they would require an island violation. Since Schwarzschild and Wilkinson (2002), semanticists have standardly observed this restriction and developed several accounts that derive the correct interpretation without the island violation.

In line with this tradition, in Dotlačil and Nouwen (2016) we also claim that the wide scope of the quantifier is an illusion. On our account, what rather happens is that the than clause denotes a plural degree, namely the sum of degrees containing (nothing but) the heights of every girl. The sentence is then interpreted distributively in the sense that for each atom in that sum, John’s height has to exceed that atom.

Before we say a little bit more about the framework that facilitates such an analysis, we zoom out a bit. If works like this or Beck (2014) or Fitzgibbons et al. (2008) are on the right track, then it suggests that the domain of degrees is no different from the domain of entities: both contain plural individuals and the relations we build on top of them are interpreted with respect to the same mechanisms, in particular, as we will see below, distributivity and cumulativity. In this paper, we follow this intuition. If degree plurality is like entity plurality, then we expect to see effects of plurality beyond the phenomena for which we designed the plural framework. That is, we should find evidence for plural interpretation beyond the simple comparatives in (5) and (7).

2.1 A Framework for Plural Degree Semantics

We take degrees to be discrete, atomic entities that are ordered by some ordering >. The plural degree semantics we developed in Dotlačil and Nouwen (2016) subscribes to the assumption that atomic degrees may combine to form sums. So, on top of the set of atomic degrees, there is also a set of non-atomic degrees, built from these atoms. If d and $d'$ correspond to two different heights, then $d\sqcup d'$ is the collection that contains nothing but these degrees. Since we take degrees to be discrete, $d\sqcup d'$ equals d only if $d=d'$.

Above, we discussed how distributive $\langle e,t\rangle$-type predicates (i.e. predicates which only have atomic entities in their extension) can take plural arguments by closing the extension under sum formation using the $^{*}$ operator. The interpretation of $^{*}$ for a predicate P is as follows:

To get a feel of what this definition does, let us briefly explain how, for instance, $\mathop {^{*}}{\{a,b\}}$ equals $\{a,b,a\sqcup b\}$. (10-a) states that $\{a,b\}\subseteq \mathop {^{*}}{\{a,b\}}$. The next condition states that $a\sqcup b \in \mathop {^{*}}{\{a,b\}}$. (10-c) adds that no other element is in $\mathop {^{*}}{\{a,b\}}$. In sum, $\{a,b\}\subseteq \mathop {^{*}}{\{a,b\}}$. For any atomic predicate P, the result is that $\mathop {^{*}}{P}$ is only true of a plurality if P is true of each of the atoms of that plurality.

In the literature, a parallel operator exists for $\langle e,\langle e,t\rangle \rangle$-type relations (Krifka 1989; Sternefeld 1998; Beck and Sauerland 2000), often written as $^{**}$. This is a generalisation of the $^{*}$ operator for sets of pairs of entities instead of just for sets of entities. For R a set of pairs:

For example, $^{**}\{\langle a_1,b_1\rangle ,\langle a_2,b_2\rangle \}$ equals $\{\langle a_1,b_1\rangle ,\langle a_2,b_2\rangle , \langle a_1\sqcup a_2,b_1\sqcup b_2\rangle \}$. The effect on an originally atomic relation R is that two pluralities A and B stand in the $^{**}R$ relation if for each atom x in A there is at least one atom y in B such xRy and for each atom y in B there is at least one atom x in A such that xRy. This means that interpretative effect of $^{**}$ is a cumulative reading (Scha 1981). For instance, when (12) is interpreted as (13), it yields the truth-conditions in (14). This makes the sentence true in a situation in which one boy carried two of the boxes and the other one carried the remaining boxes. In such a situation, the distributive reading is false.

The operations $^{*}$ and $^{**}$ suffice to account for distributive, collective and cumulative readings of (12). Applying the $^{*}$ operator to the predicate [ carried [ the four boxes ] ] allows it to take a plural subject. The resulting reading is compatible with both a distributive and a collective understanding of the sentence (depending on whether or not the extension of the predicate already contained plurality—i.e. boys jointly carrying boxes—or not). On the collective reading, neither of the boys carried the four boxes by themselves, they only did so collectively. Note that this is different from the cumulative reading, which does not entail that any collective carrying took place.³

The resulting framework is a minimal plural semantics for (predicates over) the domain of entities. For the case of degrees, we now assume that: (i) the domain of degrees contains both atoms and sums, just like the domain of entities; (ii) predicates and relations that involve degrees can be interpreted using the $^{*}$ and $^{**}$ operators; (iii) the degree comparison relation > is a relation between atomic degrees.

2.2 Quantified Than-Clauses as Degree Pluralities

This framework can now be used to solve the puzzle of quantified than clauses. The idea is that than clauses denote potentially plural degrees, using the interpretation scheme in (15). (See Dotlačil and Nouwen 2016 for details of an underlying compositional semantics, and Beck 2014 for an alternative.)

For a DP like Mary, this scheme is going to return the smallest plurality that contains the height of Mary, which is simply the atomic degree Mary’s height. For a QP like every girl, this scheme is going to return the smallest plurality that contains the height of every girl: girl$_1$’s height$\sqcup \ldots \sqcup$girl$_n$’s height.⁴

If the than clause denotes a non-atomic degree, it is in principle incompatible with the comparative semantics, since, as we said above, only atomic degrees are ordered and, so, degree comparison is comparison of atoms only. This means that in order to interpret a comparative with a than clause containing a quantificational element, we need to pluralise the comparison relation. For the case of John is taller than every girls is, we get:

The relation $^{**}\!\!>$ is true of pluralities A and B if and only if each atom in A exceeds some atom in B and each atom in B is exceeded by some atom in A. If A itself is atomic, this simply boils down to this atom exceeding each atom in B, and so, for (16) to be true, John’s height has to exceed all the atoms in the plurality of girl heights, which entails him being taller than the tallest girl. In other words, what in (5)–(8) seemed like a distributive quantifier taking wide scope is really the distribution over atoms in a plurality stemming from the need to pluralise an atomic relation that got given a non-atomic argument.

2.3 A Predicted Effect: Cumulative Comparison

We are assuming that if degrees can be plural then all the interpretation mechanisms we observe for the domain of entities should in principle also be available for degrees. We see no reason to assume a watered-down version of plural semantics for degrees, for instance where degree pluralities exist but relation cumulativity over degree relations does not. The account we sketched for quantified than clauses already suggests that this assumption is on the right track. This kind of view, however, also predicts that we should be able to observe further effects of plurality. In particular, the availability of $^{**}$ accounts for cumulative readings for sentences like (12) and, so, we would expect to see true cumulative readings for $^{**}\!\!>$. The interpretation (16) of John is taller than every girl is is not evidence for that, since that interpretation is equivalent to the distributive reading we get by pluralising a derived predicate $\lambda d.John's height>d$ and applying it to the plurality denoted by the than clause.

In order to find true cumulative readings, we need two plural arguments. The literature contains at least one influential example of where we might find such a reading.

One possible interpretation of (18) is one in which there were groups of ships and in each group the frigates in that group were faster than the carriers in that group. On that reading, the subject distributivity reading is false, since there may be carriers that were faster than one or more frigates, as long as they were not in the same group.

In order to account for this reading, it is natural to resort to $^{**}$. But for cases like (18), one need not assume that such an operator functions in the domain of degrees. Indeed, Scha and Stallard (1988), Schwarzschild (1996) and Matushansky and Ruys (2006) all analyse (18) as a cumulative relation between entities. That is, since (18) is a phrasal comparative, we can analyse it as a relation between entities (here, the frigates and the carriers) and so we can cumulate that relation using $^{**}$.

This means that examples like (18) are not evidence for a cumulative interpretation of the degree relation >, but one could think that its clausal counterpart (19) is.

Clearly, (19) shares with (18) the same cumulative-like reading. However, in order to analyse (19) as a relation between entities, we would need to move out the subject of the than clause.⁵ This is because clausal comparatives cannot be understood as relations between entities, given that one of the ‘comparees’ is a clause. To turn it into a relation over entities, we would somehow need to abstract over the subject in that clause, something we assume not to be a viable option, given that it would constitute an island violation. This suggests, then, that perhaps (19) does not involve a cumulative relation between entities, but one between degrees. Still, as we explain in Dotlačil and Nouwen (2016) in more detail, (19) is still not definitive proof that cumulative comparison exists. This is because we could arrive at exactly the same truth-conditions using distributivity and dependency. As Winter (2000) shows, cumulative readings are often indistinguishable from distributive readings. For (19), that reading would be along the lines of (20).

The idea is that the definite the carriers is interpreted as being dependent on the frigates. All one needs to assume is that distributivity can bind definites, something we need anyway to account for examples like (21), which has one reading in which each boy thinks that he is the tallest, instead of attributing the contradictory thought to him that all the boys are the tallest.

This means that if we want to show that cumulative comparison exists we need to use examples with two features: (i) we have to avoid phrasal comparatives, like (18), and use clausal comparatives instead, since phrasal comparatives may be understood as cumulative relations between entities, not degrees; (ii) we need to exclude the option of cumulative-like truth-conditions arising through dependent interpretation. We can accomplish the latter by resorting to distributive quantifiers. Consider for instance a minimal variation on (21): (22).

Whereas (21) has a reading in which they depends on the distributive quantification over boys, (22) lacks such a reading. The reason is that if them in (22) is interpreted dependently, it will refer to single boys and this renders the distributive quantification by each inappropriate, cf:

Using this for our quest to find cumulative comparison, we arrive at examples like (24): This is an example of a clausal comparative, where there is no option of the subject of the comparative clause to depend on distribution over the matrix subject.

Intuitions are admittedly murky, here, and there are several complications: not least of all the fact that the distributive reading tends to be more readily available than the cumulative one, even already for the much simpler (19). For this reason, we turn to an experimental setting, in which we probe the truth-conditions participants assign to sentences of the shape in (24).

3 The Experiment

We tested interpretations of comparatives in a simple verification task. The goal was to find to what extent cumulative readings of clausal comparatives are accepted and how the level of acceptance compares to other readings one might associate with clausal comparatives. The experiment was run in Dutch.

3.1 Experimental Setup

In the experiment, participants were first given a cover story which told of a fictional study that compared people’s ability to write (by hand) and type (on a keyboard) in a wide array of different circumstances. For each trial, this study recorded the writing and typing speeds of the participants in the cover story. Each stimulus of our experiment consisted of a fictional graph from the fictional study, depicting the typing and writing speed of three participants for a single trial. Figure 1 shows an example of such a graph. (The original stimuli were in Dutch and contained colours instead of shading.) Here, the speeds of three (fictional) participants (p.1, p.2 and p.3) are displayed. Shaded bars indicated the speed of their handwriting, non-shaded bars the speed of their typing in the trial.

Graphs like these were displayed with sentences that were supposed to provide a true statement about the trial in question. Participants in our experiment had to decide whether the statement was indeed correct.

Fig. 1

An example plot used in the experimental stimuli

There were two types of test items appearing with graphs. In the test items, the than clause included a distributive universal quantifier (glossed as dist⁶), (25-a), or a plural definite anaphor, (25-b).

The test items had a verb in the than clause and consequently, they had to be treated as clausal comparatives.

Each barplot graphically summarized six data points representing the typing and writing speed of the three participants, as illustrated in Fig. 1. For ease of exposition, we will represent the graphs used in stimuli by enlisting the typing speed/writing speed pairs of the three participants. For instance, the shorthand for Fig. 1 is $\langle 4-8,6-9,7-10\rangle$.

It depended on the available readings whether a sentence was compatible with its accompanying barplot or not. We focused on two readings, the distributive and the cumulative reading. These are represented by the propositions in (26) and (27), respectively.

There were 5 tested scenarios for the experimental items. They are summarized in the following table:

Name	Example	Distr. reading	Cumul. reading
Dist	$\langle 8-5,10-6,12-3\rangle$	True	True
Cumul1	$\langle 6-5,10-7,12-3\rangle$	False on 1 account	True
Cumul2	$\langle 8-6,6-5,5-4\rangle$	False on 2 accounts	True
Noreading1	$\langle 4-8,9-5,7-6\rangle$	False	False on 1 account
Noreading2	$\langle 7-8,9-5,2-3\rangle$	False	False on 2 accounts

To illustrate the idea behind this setup let us go through the examples. First of all, the example given for dist clearly verifies (26), since $8>5$, $8>6$, $8>3$, $10>5$, $10>6$, $10>3$, $12>5$, $12>6$ and $12>3$. Since in our setup, the distributive reading entails the cumulative one, (27) is true too. In the case of cumul1, the distributive reading is false. This is because participant 2 wrote faster than participant 1 typed: $7>6$. The cumulative reading is still true though, since $6>5$, $10>7$ and $12>3$. That is, the fact that for each participant it was the case that the typing speed exceeded the writing speed satisfies the requirements for the cumulative reading as stated in (27). This requirement also holds in the case of cumul2, but here the distributive reading is false on two accounts. Firstly, the typing speed of participant 2 does not exceed the writing speed of participant 1 and the typing speed of participant 3 does not exceed the writing speed of either participant 1 or 2.

In the cases of noreading1 and noreading2 both the cumulative and the distributive readings are false. We distinguish two cases here. In noreading1, two participants satisfied the cumulative relation imposed by the comparative, while one participant violated it. In the example in the table above, the problematic participant is participant 1 since he typed slower than he wrote (4 vs. 8). In noreading2, two participants violated the cumulative relation imposed by the comparative (in the example above, these are participant 1⁷ and participant 3). For this reason, the cumulative reading is false in noreading1 on one account (participant 1) and false in noreading2 on two accounts (participants 1 and 3).

3.2 Predictions

The theory of Dotlačil and Nouwen (2016) predicts the following. For the test sentence without the distributive quantifier, i.e. the PlDef item, both the distributive and the cumulative reading should be available. The former should be the default interpretation, derivable via pluralisation of the matrix predicate. The latter is available in two distinct ways: (i) via the cumulative operator, $^{**}$; (ii) via the distributive operator in tandem with a dependent interpretation of the pronoun. For the test sentence with the distributive quantifier, the Universal item, it should also be the case that both readings are available. However, now the option of arriving at the cumulative interpretation via a dependent analysis of the pronoun is excluded. If for some reason inserting $^{*}$ is preferred over inserting $^{**}$, then we would furthermore predict higher rates of acceptance for dist than for cumul1 and cumul2.

Theories that do not have the option of interpreting the comparison relation cumulatively would potentially make the same prediction as Dotlačil and Nouwen (2016) for the PlDef item, since in the absence of cumulativity, cumul1 and cumul2 are still compatible with a distributivity plus dependency reading. However, for Universal items this reading is unavailable, so here one would predict low acceptability for all conditions, except for dist. The only way we could imagine higher scores is if participants somehow allow exceptions on the distributive quantification. In this case, you’d expect a slippery scale from universal acceptance for the case of dist, lower acceptance for cumul1 and then continuously lower acceptance for cumul2, noreading1 and noreading2.⁸ The differing predictions are summarized in Table 1.

Table 1 Predictions in terms of proportion of responses in which participants respond that the sentence correctly describes the graph for the Universal item

3.3 Methodology

3.3.1 Participants

44 native Dutch speakers participated in the experiment. 38 of them were students from the University of Groningen who either volunteered or received a course credit for their participation. 6 participants were volunteers from Utrecht University.

3.3.2 Materials and Procedure

The experiment consisted of graph-sentence pairs, as described in Sect. 3.1. Participants had to decide whether the sentence was true or false given the situation captured in the graph. Two sentence types were tested (PlDef vs. Universal) in five scenarios (dist, cumul1, cumul2, noreading1 and noreading2). Two items per scenario were created (10 items in total). Two lists were created out of the items, so that in each list only one sentence type was present for each item. Every participant was assigned to one of the lists.

Apart from 10 experimental items, the experiment consisted of 2 practice items and 24 fillers. The fillers were unambiguously true/false (e.g., for Fig. 1 one filler might be the true sentence Participant 3 typed slower than he wrote). Fillers and experimental items were randomly ordered and each stimulus appeared on a separate screen (with no backtracking possible).

The whole experiment was run in Ibex and hosted on Ibex Farm (see http://spellout.net/ibexfarm/).

3.4 Results

Just one participant made more than 3 mistakes in the 24 fillers. Except for this one individual, we kept all the participants for the analysis.

Figure 2 shows the results. The percentages indicate proportionally how many participants responded that the sentence correctly describes the graph.

Fig. 2

Experimental results

For the analysis, we focus on the Universal factor since this is the part at which theories make different predictions. We consider logistic regression with one factor—Reading. We consider two different models. In the first one, Reading consists of two levels, distributive reading (consisting only of dist) and no reading (consisting of all the other cases, i.e., cumul1, cumul2, noreading1, noreading2). This is the model that is appropriate for theories that assume no $^{**}$. In the second model, cumul1 and cumul2 are treated as a separate factor from noreading1/2, that is, Reading consists of three levels. This is the model appropriate for Dotlačil and Nouwen (2016). Somewhat unsurprisingly (given the graphical summary in Fig. 2), we see that using the three-level factor of Reading improves the model fit compared to the two-level factor ($\chi ^2(1)=96, p<0.001$).

As we noted in Sect. 3.2, the theory lacking $^{**}$ for comparatives could predict higher scores in cumul1 than, say, noreading1 if it somehow allowed exceptions on the distributive quantification. But in that case there should be a slippery scale from universal to noreading2. This is not the case when we look at Fig. 2. Here’s one way to quantify this claim. If the acceptability decreased with the number of exceptions, we might expect that responses in readings cumul1, cumul2, noreading1 and noreading2 would form some form of linear function. Therefore, fitting it using logistic regression with one independent variable (the number of exceptions to make the dist reading true) would be appropriate. On the other hand, if Dotlačil and Nouwen (2016) are right, such a linear fit simplifies the picture. Instead, we can consider a general additive model, in which the model itself is left to find the best smooth function over the number of exceptions (using the mgcv package, see Wood 2006). This logistic model has one dependent variable, the number of exceptions to make the dist reading (with 5 knots), and Response is the dependent variable. It turns out, perhaps unsurprisingly, that the logistic general regression model fits our data significantly worse than the logistic general additive model ($\chi ^2(2.8)=38, p<0.001$). One potential worry is that we might be overfitting the model in the former case. Importantly, though, the logistic general additive model is not significantly better than the simple model we considered above: logistic regression with one variable, Reading, which has three levels (distributive reading, cumulative reading and no reading) ($\chi ^2(1.8)=2.5, p>0.1$). In conclusion we can say that in our search for the right model to fit the data with the distributive quantifier, the model that assumes that there is a distributive and cumulative reading (and nothing else) is the best. This supports Dotlačil and Nouwen (2016).

Finally, we note that it is clear from Fig. 2 that cumul1/2 readings are more acceptable in PlDef than in Universal. This is compatible with our account under the assumption that $^{*}$ is preferred over $^{**}$ and this preference is further corroborated by the higher acceptability of dist readings in Universal.

3.5 Discussion: Collective Readings?

Readers familiar with Scontras et al. (2012) might recall other cases in which comparatives do not seem to be interpreted distributively. For instance, one may judge (28) to be true of a depiction of blue and red dots, even if there is one red dot that is smaller than every blue dot, as long as the average size of red dots exceeds that of blue dots.

The experiments of Scontras et al. (2012) suggest that plural comparatives like (28) are indeed sometimes interpreted collectively. This means that subjects tend to interpret such sentences in terms of a comparison between an aggregate degree of size for the red dots and an aggregate degree of size for the blue ones.

It is not immediately clear whether the observations in Scontras et al. (2012) are relevant to our present study. First of all, the sentences in their experiments were always phrasal comparatives. Our theory in Dotlačil and Nouwen (2016) is a theory of clausal comparatives and so our current experiment deliberately only contains clausal comparatives as stimuli. Second, the sentences used by Scontras et al. have definite plurals in the than clause. It is not clear whether the collective reading is available once this definite is replaced by a distributive quantifier, as in the crucial stimuli in our experiment.

Ignoring these questions, could our results be understood as cases of collective comparison? We do not think so. First, while dot sizes might be easily imaginable as aggregate, supporting the collective interpretation for comparatives, our setup stressed individuals’ writing/typing achievements. The focus on individual performances makes it unlikely that participants would consider collective comparisons in our experiment. Second, in our items, all conditions, including the noreading ones, were created in such a way that the collective reading would be true. For a sentence like Participants typed faster than they wrote the total typing speed was always faster than the total writing speed, in any of the test conditions. Consequently, the average typing speed exceeded the average writing speed for such a sentence.⁹ Consequently, if the average-based collective reading is an option, we would expect that no condition would be rejected. But this is not the case: both noreading conditions were almost universally rejected.

Fig. 3

Responses compared to differences between average speeds

In some cases, however, the difference between the average speeds is hard to gauge. It could be that the collective reading only results in “correct” responses when the difference between the averages is clear enough. That is, the likelihood of accepting the sentence increases when the difference between the average increases.

To test this, we looked at the average-differences in the test item and how these influenced responses, see Fig. 3. If the average difference played a role, we would expect that the proportion of accepting responses increases with the difference. This is clearly not the case. In particular, the difference of 2 is almost fully rejected even though lower differences between averages are almost fully accepted. To test this further, we considered a logistic regression model in which the difference in averages is a linear predictor. The model is significantly worse than the model we considered above as the one supporting Dotlačil and Nouwen (2016) (i.e., the one which has a factor with three levels, distributive reading, cumulative reading, no reading): $\chi ^2(1)=175, p<0.001$. Thus, categorizing our data into three reading types clearly has much more predictive power than considering the difference in averages.

4 Conclusion

We discussed the semantics of comparatives and a new analysis that employs plural degrees (Fitzgibbons et al 2008; Beck 2014; Dotlačil and Nouwen 2016). Focusing on our own account, we argued that comparisons of plural degrees predicts a hitherto undiscussed reading, cumulative comparison. Controlling for several factors, we presented an experiment in which the relevant reading clearly surfaces. We take this as supporting evidence for the plural degree analysis of the comparative.

Appendix: Experimental Items

The following table contains all the experimental scenarios. As explained in Sect. 3.1, the scenarios are given by means of barplots. We represent these barplots here using the shorthand introduced before: each pair in the triple represents a participant, where the left number is their typing speed and the right number their handwriting speed. By orientation we mean whether the sentences that accompany the barplot compare typing to handwriting (left) or handwriting to typing (right), see below. The scenario types are those introduced in Sect. 3.1.

Scenario	Orientation	Scenario type
$\langle 6-5,10-5,12-3\rangle$	left	dist
$\langle 3-5,4-10,2-9\rangle$	right	dist
$\langle 6-5,10-7,12-3\rangle$	left	cumul1
$\langle 3-5,7-10,2-9\rangle$	right	cumul1
$\langle 8-6,6-5,5-4\rangle$	left	cumul2
$\langle 4-5,6-7,7-10\rangle$	right	cumul2
$\langle 4-8,9-5,7-6\rangle$	left	noreading1
$\langle 4-8,6-9,7-6\rangle$	right	noreading1
$\langle 7-8,9-5,2-3\rangle$	left	noreading2
$\langle 10-8,4-5,3-3\rangle$	right	noreading2

The stimuli themselves were as in (29). Here, (29-a) and (29-b) go with left-oriented bar plots and (29-c) and (29-d) go with right-oriented barplots. The stimuli in (29-a) and (29-c) are of the PluDef type and those in (29-b) and (29-d) are of the Universal type.

There were 36 fillers were, half of them presented in scenarios that made them false, half of them in scenarios that made them true true. The sentences that made up the fillers were of the following types: