Our goal in this study was to behaviorally characterize the property (or properties) that render negative quantifiers more complex in processing compared to their positive counterparts (e.g. the pair

Negative quantifiers like

Note that these are not mutually exclusive hypotheses. Theoretically, it could be the case that only negative polarity contributes to processing and not downward monotonicity. It could also be the opposite case, that only downward monotonicity contributes to processing and not negative polarity. Finally, both could play a role and have their own independent contributions to processing. In this study, we assess the amount of contribution of each source. The structure of the paper is as follows: First, we present linguistic data to convince that the comparison with adjectives is justified, as adjectives are similar to quantifiers in many respects but downward monotonicity. Then, we present a reaction time (RT) experiment to assess the contribution of these two sources. We finish with a discussion on each source.

Quantifiers denote a property of a set, which makes them second order predicates (

(1) | More than three students came: |

|{ |

(2) | Many students came: |

|{_{context} |

The numerosity argument d, whether explicit or implicit, is a degree on a linearly ordered scale. This scale is lower bounded by 0 (as the minimal intersection of two sets is the empty set, whose size is zero).

Gradable adjectives, by definition, also make reference to a scale with linearly ordered degrees. The gradable adjectives which will be discussed here are dimensional adjectives (cf.

(3) | John is 2 meters tall: |

(4) | John is tall: |

_{context} |

Dimensional adjectives are similar to quantifiers not only in their reference to a linear scale, but also in the lower bound of the scale. Both are lower bounded by zero—natural dimensions, as quantities, cannot get a negative value.

A second point of similarity is that of polarity. Both polar quantifiers and antonymous adjectives can be divided into pairs that constitute a positive and a negative. The negative counterpart reverses the ordering of the scale relative to the positive (i.e. if

(5) | a. | Many students came: |

|{_{context} |
||

b. | Few students came: | |

|{_{context} |

(6) | a. | John is tall: |

_{context} |
||

b. | John is short: | |

_{context} |

If the polarity effect in quantifiers stems from a “less than” computation (<), as given in the denotations in (5b) and (6b), a similar polarity effect in adjectives is also expected. Indeed, there are some hints that negative adjectives are more cognitively costly than positive adjectives (

A different perspective on the nature of negative polarity is through a morpho-syntactic analysis. Negative quantifiers are often analyzed as containing a hidden negation in their underlying structure (

To summarize, negative adjectives are similar to negative quantifiers. Both types of negation involve a “less than” computation on a linear scale, and possibly also a hidden negation. Therefore, it is natural to compare the polarity effects between these two types. If the polarity effect in quantifiers stems from the morpho-syntactic representation or from cognitive computations over degrees on scales, then we expect both quantifiers and adjectives to manifest a polarity effect, and this polarity effect should interact with truth value in the same way.

Negative quantifiers (

(7) | a. | If Q is a positive quantifier: |

b. | If Q is a negative quantifier: |

Downward monotonicity is a property of linguistic importance (

_{PL}_{PL}

First, let us be convinced that

(8) | a. | |

b. |

Although (8a) and (8b) seem synonymous, only (8a) can license negative polarity items (NPIs) like

(9) | a. | |

b. | * |

(10) | a. | To pass the exam, you should make |

b. | #To pass the exam, you should make |

The contrast in (10) suggests that

(11) | a. | |

|{_{context} |
||

b. | ||

c. | |{_{context} |

Prima facie,

To summarize, if the source of complexity in negative quantifiers is due to negative polarity, we expect to find a similar effect in adjectival expressions like

The goal of the experiment is to test whether the polarity effect in quantifiers stems from entailment pattern, the negative polarity, or both. We do that by measuring reaction times (RT) in a verification task, and comparing the polarity effect in quantifiers to the polarity effect in adjectives. If our cognitive system is indeed sensitive to downward monotonicity, we should expect a larger RT difference in

We introduce a verification paradigm with the factors Type (adjective/quantifier), Polarity (positive/negative) and Truth-value (true/false). To increase the number of items and add variability, we added another factor, Standard, which regarded the kind of comparison required from the participants: whether the degree is a proportion or not (cardinal/proportional). Having more than one downward entailing quantifier allows us to correlate performance—if downward monotonicity is a relevant property cognitively, then with enough between-subject variability in the size of the polarity effect, we should find a high correlation between the polarity effects of the quantifiers.

For a verification task, we used English sentences that described some ratio between blue and yellow circles using one of the degree operators presented in Table

The items of Type × Polarity for the cardinal level of the Standard factor. All sentences could be either true or false (Truth value factor).

Cardinal comparison | Polarity | ||
---|---|---|---|

Positive | Negative | ||

A large number | A small number | ||

Many | Few |

The items of Type × Polarity for the proportional level of the Standard factor. All sentences could be either true or false (Truth value factor).

Proportional comparison | Polarity | ||
---|---|---|---|

Positive | Negative | ||

A high proportion | A low proportion | ||

More than half | Less than half |

Each trial could be true or false. For sake of counter-balancing, sentences also varied in the referred Color (blue/yellow).

All sentences were recorded in English (female voice, native English speaker, Canadian accent), and later processed in Audacity® to equalize them in term of their average pitch, duration, average amplitude and speed.

Images either depicted more blue circles than yellow circles or vice versa. The number of blue circles was fixed to 16 and the number of yellow circles varied. The comparison task was kept easy by clustering circles of the same color close to each other and using ratios such that it is clear which color is the majority (4:16, 8:16, 32:16 and 64:16). Hence, for each truth value (yellow<blue and yellow>blue) there were 2 levels of Ratio, small (8:16, 32:16) and large (4:16, 64:16). Pictures were created in Mathematica™.

To conclude, each subject had to respond to a total of 320 trials (2 Type × 2 Polarity × 2 Standard × 2 Truth-value × 2 Color × 2 Ratio × 5 repetitions = 320 trials). Trials were evenly divided between 4 runs. A short break between runs was taken if needed. Each run was counterbalanced for Type (adjective/quantifier), Polarity (positive/negative), Standard (proportional/cardinal), Truth-value (true/false), Color (blue/yellow) and Ratio (small/large).

Experiment was run using Presentation™ (version 17.0). In each trial, subjects had to decide whether a sentence they heard on earphones correctly described a picture that later appeared on the screen. The procedure and timing of a single trial is summarized in Figure

Trial sequence.

Each trial started with a fixation cross on the screen (see Figure

35 students, aged 22 ± 3 (average ± standard deviation), native English speakers who were taking a summer course in Hebrew at the Hebrew University International School and participated in this experiment for payment, after signing informed consent approved by the Hebrew University Research Ethics Committee. 29 were right handed, 15 male and 20 female. One subject quit the experiment in the middle, so his data was not included in the analysis. Two other subjects were excluded from the analysis due to low accuracy rates (<70% correct responses).

Based on our theoretical considerations regarding the similarities between quantifiers and adjectives, these are our predictions for the RTs: (i) a main effect of Polarity; (ii) a Polarity × Truth-value interaction that is not significantly different between Types. If downward monotonicity also plays a role in processing, we expect also: (iii) a Type × Polarity interaction that stems from a larger effect for quantifiers versus adjectives; (iv) a relatively high correlation between subjects’ Polarity effects of quantifiers (i.e. between proportional quantifiers and cardinal quantifiers).

Analysis was carried out on 32 subjects, whose accuracy was relatively high (>70%, average = 83%). Misses and incorrect responses were removed (1.4% and 15.6% of data respectively). RTs were log-transformed.

As shown in Figure

Box plots, showing groups’ means (yellow diamonds and texts) and medians (thick horizontal black lines) within a box of 50% of the data. Whiskers show the extent of the data which is within 1.5 inter-quantile range (i.e. 1.5 of box’s height). Points beyond that are shown as black dots. Notice the small difference between the median and the average, strengthening the robustness of our data. N = 32.

Summary of the effects up to 2-way interactions (3-way and 4-way interactions were not significant); predictors were sum coded; average difference is calculated within subjects; p-values are obtained by using the Satterthwaite approximation of degrees of freedom (

111 ± 8 | 16.2 | <0.0001 | |

40 ± 5 | 10 | <0.0001 | |

25 ± 5 | 6.2 | <0.0001 | |

18 ± 4 | 4 | =0.0001 | |

–96 ± 14 | –8 | <0.0001 | |

36 ± 8 | 4.5 | <0.0001 | |

24 ± 7 | 2.8 | =0.005 | |

20 ± 7 | 2.9 | =0.004 | |

17 ± 8 | 2.1 | =0.04 | |

4 ± 10 | 0.4 | =0.7 |

In addition, a significant Polarity × Truth-value interaction was found (t = –8, p < 0.0001). This interaction was not found to be different between adjectives and quantifiers (non-significant Polarity × Truth-value × Type interaction: t = –0.6, p = 0.56), nor between cardinal comparison and proportional comparison (non-significant Polarity × Truth-value × Standard interaction: t = 1.9, p = 0.06). This is visualized in Figure

Same as Figure

The effects are summarized in Table

As shown in Figure

A second analysis we performed was to correlate the Polarity effects of all four classes of Standard × Type (cardinal/proportional, quantifier/adjective). Our prediction was to find the strongest correlation between the two quantifier Polarity effects (cardinal and proportional), as only they require processing downward monotonicity. First, for each subject in each Standard × Type condition, we calculated the Polarity effect by subtracting the averaged RT for the positive from the averaged RT for its negative counterpart, and dividing be the mean RT of that subject:

Correlation matrix (r-Pearson scores are given).

Of course, it might not be surprising that our predicted correlation turned out to be the highest among six, as there is an a-priori probability of 1/6 to get this result by chance. At least, this fact does not go against our hypothesis. However, as pointed out by an anonymous reviewer, the differences between the correlations are not major, casting doubt on whether being the highest in rank is even meaningful. We agree with this comment, but do want to point out that the difference between the highest correlation and the second highest correlation was relatively high (0.08, while the other differences, by order, are 0.02, 0.01, 0.11 and 0.03). However, without a larger sample we cannot draw a more decisive conclusion.

The current study examined the components of the polarity effect in quantifiers by comparing them to highly similar polar pairs—dimensional adjectives embedded in an indefinite construction. We used pairs of adjectives and pairs of quantifiers that have many properties in common—their reference to a scale with ordered degrees, and the classification into positive and negative antonyms—but differ in their logical properties. The negative quantifiers are downward monotone while the negative adjectival constructions are not. We found this difference to be of cognitive relevance, as a larger polarity effect was found in quantifiers. Furthermore, a correlation between the polarity effects in quantifiers was also found, not strong but yet consistent with our hypothesis that downward monotonicity has a distinct contribution to processing. We estimate that downward monotonicity contributes about 30% of the polarity effect in quantifiers, while the rest of the effect is explained by factors that are shared with negative adjectives, such as a hidden negation or a “less than” computation. We will now discuss these two sources and their cognitive underpinning.

In the beginning of this paper we presented negative polarity as triggering a “less than” computation. However, since every “<” is symmetrical with a “>”, it would be more accurate to define negative polarity as satisfying these two following properties (see also Footnote 2): (i) scale reversal; (ii) proximity to zero. Scale reversal means that any degree d that makes the positive proposition (

Scale reversal can be thought of as a consequence of an operator in the underlying structure, akin to the sentential negation

Denial is one of the main functions of sentential negation (

The zero point classifies antonyms into positives and negatives unequivocally, as only negative antonyms support stronger propositions the closer the degree argument is to zero. But for that, a zero point has to exist. The adjectives and quantifiers discussed in this paper have a scale with a natural zero point, hence polarity is defined. Positive polarity reflects the ordinary way we perceive the world, as the positive antonym points away from the zero point, toward a naturally observable direction (

Finally, we wish to point at a potential difficulty to the claim that negative polarity has an independent processing cost. As an anonymous reviewer kindly points out, Chemla et al. (

We now turn to discuss downward monotonicity as an independent source of the polarity effect. Prima facie, it is unclear why a logical property such as downward monotonicity should be determinant in language processing. One possibility is that part of understanding a sentence is to know what it entails and what it is consistent with (i.e. entailed by), and therefore entailment reversal is highly informative and has to be mentally represented. However, meaning is more than its semantics. For instance, although

A different view attributes the processing cost of downward monotone quantifiers to the verification strategies they invite (

A different verification algorithm was suggested by Bott et al. (

The question of which of the two properties is the one relevant for processing—empty-set or downward monotonicity—can be tested empirically. Bott and colleagues ran an experiment comparing two non-monotone quantifiers, only one of which was also an empty-set quantifier (

The upshot of this study is that the polarity effect in quantifiers is a more complex phenomenon than what was usually thought. It is not explained by only one factor, but rather there are at least two factors involved: negative polarity and downward monotonicity. We discussed the possible cognitive underpinnings of both sources, and suggested some interesting directions for future investigations in the realm of negation, adjectives and quantifiers.

To abstract away from complexities of adjectives’ internal structure that are irrelevant to this paper, we treat gradable adjectives as a relation between degrees on a scale rather than between intervals on a scale (cf.

To formalize, we can abstract over the degree element in a proposition. The relationship between a proposition with a negative adjective/quantifier (_{1}) ⊂ _{2}) ⇔ _{1}) ⊃ _{2}); (b) proximity to zero: _{1}) ⊂ _{2}) ⇔ _{1} < _{2}. Scale reversal means that if degree d_{1} makes _{2}), then it makes

The same tests can be applied to

Proof: assume Q(A)(B) is downward monotone:

∀

For all

∀

~∃

Contradiction. Hence, 1 is false: ~[

Although it is beyond the scope of this work, we would like to suggest two possibilities for future research on this topic: (i) the NP

An anonymous reviewer pointed out that Moxey & Sanford (

The sentences in the adjective condition were with the copula

We leave aside the issue of whether it is only a contextual effect (cf.

We thank an anonymous reviewer for raising this point.

This work was supported by ISF 2093/16 (YG), ISF 757/16 (YL), ISF 1366/14 (NB), and the Gatsby Charitable Foundation (YL).

The authors have no competing interests to declare.