The Domains of Monotonicity Processing

Abstract

This paper reports an investigation into the nature of Negative Polarity Item (NPI) licensing conditions from a processing perspective. We found that the processing cost of Downward Entailingness (a k a the Monotonicity Effect) is determined by the number of monotonicity reversals of NPI domains, rather than by the number of Downward-Entailing (DE) operators. This conclusion is not based on the standard judgment paradigm, but rather, on the measurement of continuous variables (error rates, Reaction Times (RTs)) in a verification task, in which the truth value of a sentence is determined against a scenario. We conducted two experiments with sentences that contained one or two DE operators, which featured in different syntactic configurations. We explored how RT is affected by the manipulation of both the number of DE operators, and the syntactic environments in which they reside. We ran these experiments in Hebrew and in English, with different participant populations and different testing methods. Despite the linguistic subtlety of the theoretical issues, our results were remarkably sharp, leading to two firm conclusions: (i) that processing time is determined not by the number of DE operators, but rather, by the monotonicity of the minimal constituent in which they reside; (ii) that DE-ness is not a property of operators, but of environments. We show how our results bear directly on the current debate about the nature of monotonicity, which we describe below. Finally, we provide quantitative tests of alternative, non-semantic explanations, and show how our results do not support them.

1. INTRODUCTION

This paper presents the results of two experiments that provide new insight into the debate on the licensing conditions of Negative Polarity Items (NPIs). Specifically, we approached this issue from the perspective of processing, and asked whether monotonicity has measurable processing consequences. To this end, we manipulated the number of Downward-Entailing (DE) operators as well as the syntactic environments in which they reside, and measured Reaction Time (RT) in a verification task.

NPIs such as any and ever in English have interested linguists since the 1960s (Klima, 1964; Baker, 1970; Fauconnier, 1975; Ladusaw, 1980; Gajewski, 2011; Xiang et al., 2009, 2013, 2016; passim). These items must be licensed by some sort of negativity—a negative marker, such as no or not, or a DE operator, such as less and few.¹ Hence, the unlicensed NPIs in the sentences in (1a,c,e) result in ungrammaticality, but sentences (1b,d,f) are grammatical, as their NPIs are licensed by a negation (1b) or a DE quantifier (1d,f).

• (1)

  • ^*The kids ate any cookies at the birthday party.

  • The kids did not eat any cookies at the birthday party.

  • ^*More than 3 kids ate any cookies at the birthday party.

  • Less than 3 kids ate any cookies at the birthday party.

  • ^*Many kids ate any cookies at the birthday party.

  • Few kids ate any cookies at the birthday party.

Whereas it is easy to see that an NPI requires a licensor, to formulate the precise licensing conditions is more difficult. Everyone agrees that a DE operator is needed to license an NPI, yet the question is how the licensor and licensee are related. Homer (2021) divides theories of NPI licensing into two groups: an Operator-Based Approach (OpBA), by which all an NPI needs is to be in the scope of a DE operator, as stated in (2), and a stricter Environment-Based Approach (EnvBA), by which the environment needs to be DE, as in (3).

• (2) Operator-Based Approach (OpBA): An NPI is licensed only if it is in the scope of a DE expression. (Fauconnier, 1975; Ladusaw, 1980).

• (3) Environment-Based Approach (EnvBA): An NPI α is licensed in sentence S only if there is a constituent A of S containing α such that A is DE w.r.t the position of α. (Gajewski, 2005).

At issue is the domain of an NPI (Monotonicity Domain henceforth). The OpBA claims that it is the scope of a DE operator. Namely, whether or not it is licensed depends solely on its structural relation to that operator. Nothing else is relevant. This approach is contrasted with the EnvBA, which argues that the Monotonicity Domain that licenses an NPI is determined by the DE-ness of the syntactic environment in which the NPI resides. This type of DE-ness, moreover, may not be necessarily computed in whole sentences, but rather, in constituents that make up the syntactic environment necessary for licensing. As our experimental inquiry seeks to adjudicate between the OpBA and the EnvBA empirically, we focus on specific syntactic considerations, carefully selected out of the dense literature on the licensing conditions. We begin with empirical arguments that rely on plausibility and acceptability judgments in the context of two DE operators. Here, a sentence is needed, in which an NPI is in the scope of a DE operator, but at the same time, the relevant environment is non-DE. If the sentence is acceptable, the DE-ness of that environment may not matter, as the OpBA would have it; yet if the sentence is unacceptable, then the DE-ness of the environment does matter, as the EnvBA suggests.

Homer (2021) provides evidence from French in favor of the EnvBA. He first presents evidence that quoi que ce soit (= anything) is a (weak) NPI, licensed by a negative adjective (4) or a negation (5):

• (4) Il est impossible que Jean ait fait quoi que ce soit pour aider la Mafia.it is impossible that Jean have.subj done what that this be.subj to help the Mafia.‘It is impossible that Jean did anything to help the Mafia.’

• (5) Il n’ est pas possible que Jean ait fait quoi que ce soit pour aider laIt ne is neg possible that Jean have.subj done what that this be.subj to help theMafia.Mafia.‘It is not possible that Jean did anything to help the Mafia.’

The acceptability of (4)–(5) is expected by the OpBA, because both contain a DE operator that can serve as licensor. If so, then the presence of both licensors must certainly license the NPI. Yet Homer points out that in “flip-flop” sentences, an NPI is anti-licensed:

• (6) ^*Il n’ est pas impossible que Jean ait fait quoi que ce soit pour aiderIt ne is neg impossible that Jean have.subj done what that this be.subj to helpla Mafia.the Mafia.‘It is not impossible that Jean did anything to help the Mafia.’ (Homer, 2021)

Homer shows how this unacceptability follows from the EnvBA. The intuition is that one cannot find in this sentence a DE environment that contains quoi que ce soit and only one of pas and impossible (which are both in the same TP). These two operators (each individually licensing the NPI) are coupled in this configuration, and thus NPI licensing fails. Homer proposes that this failure is because licensing here may only be done by the node that dominates both of them, which is Upward-Entailing (UE), as the combination of the two DE operators n’...pas and impossible leads to UE-ness. Therefore, quoi que ce soit resides in a UE environment, and unacceptability follows.

The EnvBA succeeds because it requires that licensing is dependent on a DE environment, not a DE operator. Homer calls this environment “the domain of an NPI”:

• (7) Domain of an NPI: A constituent γ which contains the NPI π is a domain of π if and only if the acceptability of π can be evaluated in γ.²(Homer, 2021)

(8)–(11) are the configurations at issue, their actual acceptability, and the predictions of each approach in keeping with its definition of Monotonicity Domain. Below is a presentation in which γ marks the Monotonicity Domain of an NPI according to the EnvBA (⁠ denotes a UE constituent and ↓ denotes a DE one):

But what is γ? Homer proposes that a Polarity Phrase (PolP) is the smallest, but not the only, domain of quoi que ce soit. Since (6) is ungrammatical, n’...pas and impossible must be in the same domain there; on the other hand, in the acceptable (12a), the configuration contains a domain with the NPI and a single licensor. He further points out that in (12b), both the conditional-if and impossible are DE, and quoi que ce soit is nonetheless licensed.

• (12) a. Il est impossible que Jean n’ ait pas fait quoi que ce soit pourIt is impossible that Jean ne have.subj neg done what that this be.subj toaider la Mafia.help the Mafia.‘It is impossible that Jean didn’t do anything to help the Mafia.’b. S’il est impossible que Jean ait fait quoi que ce soit pour aider la mafia, je lui présenterai mes excuses.‘If it is impossible that Jean did anything to help the Mafia, I will apologize to him.’

He concludes that these two DE operators cannot be in the same minimal domain, and therefore, there must be a domain below CP, but above VP, where the NPI quoi que ce soit is licensed. This domain is PolP.³ Homer’s LFs of (4)–(6) and (12a-b) are shown below. They feature the relevant domains (i.e., PolP and TP), and annotated for their monotonicity with respect to quoi que ce soit:

• (4)’ …[_TP T [_PolP  impossible [_CP que Jean T [_PolP [quoi que ce soit]₁ faire t₁]]]].

• (5)’ …[_TP T [_PolP  pas possible [_CP que Jean T [_PolP [quoi que ce soit]₁ faire t₁]]]].

• (6)’ ^*…[_CP [_TP T [_PolP  pas impossible [_CP que Jean T [_PolP [quoi que ce soit]₁ faire t₁]]]].

• (12a)’ …[_TP T [_PolP  impossible [_CP que Jean T [_PolP  pas [quoi que ce soit]₁ faire t₁]]]].

• (12b)’ …[_TP [_CP  Si T [_PolP  impossible [_CP que Jean T [_PolP [quoi que ce soit]₁ faire t₁]]]] T].

This perspective on a Monotonicity Domain enables Homer to derive a more refined version of the Environment-based licensing condition for NPI:

• (13) Environment-based Licensing Condition for NPIs (refined): An NPI α is licensed in sentence S only if α has a DE domain in S.

Homer’s data support the EnvBA. An operator-based approach cannot explain why (6) is unacceptable, as the NPI is in the scope of a DE operator. Note that Homer’s discussion, while pertinent to our specific experimental inquiry, addresses few among the many issues in the analysis of NPIs. A proper analysis must involve semantic and pragmatic assumptions, which, together with syntactic ones, conspire to account for the richly complex array of monotonicity-related phenomena as revealed through the distribution of NPIs (cf. Bar-Lev & Fox, 2020; Chierchia, 2013; Crnič, 2019; Guerzoni & Sharvit, 2007; von Fintel, 1999; Gajewski & Hsieh, 2014, to mention just a few). The phenomena on which Homer and the present paper focus mostly call for a syntactically-constrained analysis of monotonicity, by pitting the OpBA and the EnvBA against one another.

2. THE PROCESSING COST OF DOWNWARD-ENTAILINGNESS

Here, we investigate the nature of DE-ness from a processing perspective. The empirical basis for hypotheses regarding NPI licensing is typically grammaticality/acceptability judgments, whereas we seek to predict, and then precisely measure, the processing cost of monotonicity. Below, we construct a mapping between representation and processing, from which we can then derive explicit predictions about measurable processing costs that the DE-ness of a string incurs, and the manner in which they are dictated by each of the grammatical approaches to monotonicity. In doing so, we will expand the range of empirical evidence that bears on the definition of NPI licensing conditions. As extant evidence is presently limited to patterns of NPI licensing, we now show that results from experiments with continuous variables, mainly Reaction Times (RTs), provide a sharp picture that bears on this issue directly.

Our starting point is the well-known Monotonicity Effect in processing: main clauses that contain a DE operator (but no filler-gap dependencies) take longer to process than their UE counterparts (e.g., Just & Carpenter, 1971; Clark & Chase, 1972; Tian et al., 2010; Deschamps et al., 2015; Tian & Breheny, 2016; Agmon et al., 2019, 2022; Dudschig et al., 2019; Schlotterbeck et al., 2020; Wang et al., 2021). Many, if not most, of these experiments control for psychological, morphophonological, and syntactic confounds, which strongly suggests that the monotonicity effect is not due to frequency, length or generic complexity (cf. Deschamps et al., 2015; Grodzinsky et al., 2021; Agmon et al., 2022 for recent discussions). Yet robust as this result may be, it is not easy to interpret. A thought that most naturally comes to mind is that the parser never expects DE-ness. Its encounter with DE-ness therefore forces a costly monotonicity reversal:

• (14) Monotonicity Processing Hypothesis (MPH):a. The parser never expects DE-ness (as defined by the theory of NPI licensing)b. Each monotonicity reversal incurs a processing cost.

The MPH derives the Monotonicity Effect. The idea, to be clear, is that the theory of NPI licensing dictates the processing of monotonicity, for which the parser has a default UE expectation (14a); a monotonicity reversal incurs a measurable cost (14b). The question here is how the monotonicity reversal is counted? MPH is expected to pit the OpBA and EnvBA against one another in terms of their different definitions of the notion “Monotonicity Domain”. Thus (14), coupled with each of these accounts, has clear cost predictions. Below, we report experiments that sought direct experimental support for it. We hoped, that is, to obtain evidence to adjudicate between the two theories—the OpBA and EnvBA. Coupling the MPH with the former (i.e., MPH + OpBA) predicts that the cost is determined by the number of DE operators in the incoming sentence; by contrast, MPH + EnvBA predicts that the processor is taxed by the number of Monotonicity Reversals of Domains (MRDs) of an NPI.

We therefore investigated the conditions under which DE-ness incurs a processing cost. We measured continuous variables—mainly Reaction Time (RT)—and explored their relation to the number of DE operators in the incoming sentence, as well as to the domain in which these operators are contained. When the OpBA is mapped onto RT (as an index of processing cost), it predicts RT to grow with the number of DE operators, as each monotonicity reversal is said to be costly; the EnvBA also predicts RT growth with the number of monotonicity reversals, but only if these reversals occur in distinct NPI domains. The table below details these predictions as the sign of ΔRT between members of minimal pairs of sentences, in which DE-ness and domain size are varied systematically (γ and ß denote the domains of an NPI):

As noted above, the contrast in (15) has been tested extensively in main clauses (e.g., few/many of the dots are red, Just & Carpenter, 1971), resulting in a positive ΔRT, denoted by “+”, illustrating to the Monotonicity Effect. These sentences feature a single DE/UE operator contrast, and likewise one DE/UE contrast within the domain γ. Thus, both approaches to NPI licensing (coupled with the MPH) predict it.

Equipped with the MPH and a positive ΔRT₁₅, we sought to distinguish between the two approaches to NPI licensing by measuring the effects of processing two operators in a single domain γ, as depicted in (16). Homer’s flip-flop effect (6) suggested that while (16a) is DE, (16b) is UE. As the table shows, the EnvBA+MPH predicts an RT flip-flop as well—reversal in the sign of ΔRT₁₆ relative to ΔRT₁₅. Thus, negative ΔRT₁₆ is expected (denoted “-” above), as RT_16a should be greater than RT_16b. A ΔRT sign reversal between (15) and (16) has actually been found by Grodzinsky et al. (2021) and Tan et al. (2023). The EnvBA would appear to be vindicated, as the OpBa predicts the opposite, namely that the processing of two DE operators would be more taxing than one, regardless of where they stand relative to γ. Yet the sign reversal found for (the negative) ΔRT₁₆ relative to (the positive) ΔRT₁₅ stops short of distinguishing between the two approaches to NPI licensing: the sentences in (16) contain more operators (and more words) than (15). A reversal in the sign of ΔRT may be due to that. Further controls are missing. The remedy comes in the form of (17), that has the very same words as (16), yet the two operators are in different domains—one in γ, and the other, in ß. If the (15)–(16) sign reversal is due to added operators, then it should persist in (17); this is what OpBA would predict. Yet if the negative ΔRT₁₆ is due to flip-flop in γ, it should reverse to positive in (17), because the two DE operators are in separate domains. Overall, then, OpBA and EnvBA diverge in prediction, as the table indicates. Thus (15)–(17) provide the full array of sentence types, and feature as stimuli in our experiment.

3. EXPERIMENTAL METHODS

3.1. A speeded sentence-picture verification task that measures processing cost

In the current study, we attempt to adjudicate between the OpBA and the EnvBA by providing quantitative processing evidence from two experiments: one is a lab experiment with native Hebrew speakers, and the other is an online experiment with native English speakers. We used the common speeded sentence-picture verification task (SSPVT) paradigm. In this task, participants first hear a sentence, and then see an image that appears immediately thereafter.⁴ The SSPVT requires them to indicate, as fast as they can, whether the image makes the sentence true or false.

We used two DE operators, not and less, to construct two different structures of double negation: the intra-domain structure and the cross-domain structure. In the intra-domain structure, there is no domain that contains only one DE operator. Namely, both of the eligible domains contain either zero or two DE operators (e.g., Not  less than half of the circles are blue). In the cross-domain structure, there is one domain which contains only one DE operator (e.g., Less than half of the circles are not blue). We measure RTs for these, as well as for their more baseline counterparts, in the SSPVT. To forecast, in both experiments, participants responded faster to the conditions with two intra-domain DE operators than the conditions with two DE operators across-domains. This is predicted by MPH + EnvBA but not the OpBA.⁵

3.2. Experimental materials in Hebrew

When the Hebrew is used, and the NPI ‘ey-pa’am (=ever) replaces quoi que ce soit, Hebrew mimics French (cf. Appendix 1 for Hebrew replications of Homer’s effects). We test the flip-flop effect in Hebrew with monoclausal sentences, instead of the embedded clausal structures that Homer provided. We use two DE operators, less and not, to build two double negative structures, (18d) and (18f). These contrast with sentences including zero or one negation, (18a, b, c, e). Note that no NPIs are present, because the experiment is about the processing of multiple negations and domains. Therefore, all sentences are acceptable:

• (18)

  • [_TP [Yoter mi-xezi me-ha-igulim][T [_PolP hem kxulim]]]more than-half of-the-circles are blue‘More than half of the circles are blue’ (0 DE operator; 0 MRD)

  • [_TP [Paxot mi-xezi me-ha-igulim][T [_PolP hem kxulim]]] less than-half of-the-circles are blue‘Less than half of the circles are blue’ (1 DE operator;1 MRD)

  • [_TP [Lo yoter mi-xezi me-ha-igulim][T [_PolP hem kxulim]]]Not more than-half of-the-circles are blue‘Not more than half of the circles are blue’ (1 DE operator; 1 MRD)

  • [_TP [Lo paxot mi-xezi me-ha-igulim][T [_PolP hem kxulim]]].Not less than-half of-the-circles are blue‘Not less than half of the circles are blue’ (intra-domain structure: 2 DE operators; 0 MRD)

  • [_TP [Yoter mi-xezi me-ha-igulim][T [_PolP hem lo kxulim]]] more than-half of-the-circles are not blue‘More than half of the circles are not blue’ (1 DE operators; 1 MRD)

  • [_TP [Paxot mi-xezi me-ha-igulim][T [_PolP hem lo kxulim]]] less than-half of-the-circles are not blue‘Less than half of the circles are not blue’ (cross-domain structure: 2 DE operators; 2 MRDs)

To derive the predictions from the MPH + EnvBA, we compare how many MRDs occurs in each sentence in the relevant minimal pairs in (18). We have three sets of predictions: (i) RT_{not less} < RT_{not more} because not less includes zero MRDs (both PolP and TP are UE), and not more includes one (PolP is UE but TP is DE); (ii) RT_less…not > RT_more…not because MRDs occurs twice in less…not (the minimal domain PolP reverses from UE to DE, and then TP becomes UE again) but only once in more…not (an MRD only occurs in PolP); (iii) RT_less…not > RT_{not less}, as less…not has more MRDs. Note that the prediction (i) has been tested previously in a study which adopted a similar paradigm as the current study (RT_{not less} (926.9 ms) < RT_{not more} (987.4 ms) and RT_more (804 ms) < RT_less (888.2 ms), Tan et al. (2023)). By contrast, the prediction of the MPH + OpBA is easy to derive, since for this hypothesis, the processing cost increments are determined by the number of DE operators.⁶

Before we move on to our experiments, we seek to settle the question of whether the distinct predictions of the OpBA and EnvBA may be confounded with a difference in linear adjacency. That is, whether two DE operators are perceived as a single UE operator by virtue of their immediate proximity rather than being within the same domain. The flip-flop effect in (6) and (10) is consistent with this perspective: the NPI quoi que ce soit is anti-licensed when the two DE operators are adjacent (6) and (10), but licensed when they are separated (12a-b). To distinguish between domain and linear adjacency, we need an example in which two DE operators are adjacent, but no flip-flop effect occurs (or vice versa). Homer shows this for French (19a), and (19b) demonstrates the same effect for Hebrew: (19b) contains two DE operators, ‘im ‘if’ and le-xol-ha-yoter xamiša ‘at most five people’, and an NPI ey pa’am ‘ever’.⁷ (19a) and (19b) are both grammatical while the two DE operators are adjacent to each other, just like the case in (6). Thus, linear adjacency cannot account for the flip-flop effect of (6).

• (19) a. Si au plus cinq personnes ont fait quoi que ce soit pour aider la Mafia,If at most five people have done what that this be.subj to help the mafia,nous sommes sauvés. we are saved. ‘If at most five people did anything to help the Mafia, we are good.’

• b. ‘im le-xol-ha-yoter xamiša ‘anašim siy’u‘ey pa’am la-mafia, ‘anaxnu beseder if at-most five people assisted ever the Mafia, we good. ‘If at most five people ever assisted the Mafia, we are good.’

4. EXPERIMENT I

Experiment I was a lab study, implemented in Hebrew. We used two DE operators in Hebrew, lo ‘not’ and paxot ‘less’, along with yoter ‘more’, to juxtapose between the OpBA and the EnvBA. We asked the participants to verify the aforementioned sentences against pictures in the paradigm of speeded sentence picture verification task. According to the MPH + EnvBA, we would expect to observe RT_{not less} < RT_{not more} but RT_{less not} > RT_{more not} based on the number of MRDs. By contrast, if the MPH + OpBA is correct, we would observe RT_{not less} > RT_{not more} and RT_{less not} > RT_{more not} instead, since both 2*DE structures contain one more DE operator than their more counterparts.

4.1. Materials

We used four Sentence Types along with the pair of polar quantifiers, more and less, to build up our sentential stimuli. As shown in Table 1, there are eight conditions in total. Sentence Type plain was used as a baseline for comparison with other conditions and as a sanity check to make sure the participants are doing the task dutifully. The Sentence Type intra-domain not and cross-domain not, as discussed in section 2, were the ones which include two DE operators in different configurations. Finally, the Sentence Type that-clause was added to counterbalance the number of stimuli with not so we would have an equal number of stimuli with and without not. The sentences were recorded in Hebrew by a male native speaker, and processed in Audacity to minimize their pitch, amplitude and length variability. However, since the sentences in that-clause contain much more words than the other groups, we did not match their length with others (4490 msec vs. 3350 msec), while within group uniformity was controlled.⁸

Experimental design

As for the test images, we used a set of images in which the blue and yellow circles are arranged in a 5 x 5 array. The ratios between the blue and yellow circles are 5:20, 10:15, 15:10 and 20:5. To add variety, for each ratio, there were two types of arrangement of location of the circles. The circles of the same color clustered together to keep the verification task simple.

In order to counterbalance all the conditions, the experiment featured 128 trials (4 Sentence Type x 2 Quantifier Type x 2 Referred Color x 4 Ratio x 2 Arrangement = 128 trials). The Truth Value (true/false) was thus counterbalanced accordingly. Each combination of the trial appeared twice in the experiment. To eliminate the potential learning effect over time, we randomized the order of the trials for each subject.

4.2. Procedure

Before beginning an experimental session, the experimenter explained the task to the participant. They were told that they would hear a sentence describing the relationship between circles in two colors, after which, an image with circles in the two colors would appear. Participants were asked to determine whether or not the sentence they heard matched the image, and respond as fast and as accurately as possible. Each experiment included two blocks, each containing 16 practice trials and two runs with 64 trials each. Participants thus responded to 32 practice trials and 256 experimental trials in total. To reduce the difficulty of the task, each block contains only two groups of Sentence Types. Considering the occurrence of not, two types of grouping are possible: one is [plain + intra-domain not] vs. [that-clause + cross-domain not]; the other is [plain + cross-domain not] vs. [that-clause + intra-domain not]. As in the first type of grouping, the combination plain and intra-domain was the same combination as the stimuli in a previous study (Tan et al., 2023), we decided to assign most of the participants to the second type of grouping and only five participants to the first type of grouping. All the relevant sentences and a few examples of images were shown to participants before the beginning of each block. To minimize fatigue, a 3-minute break was enforced between every two runs.

In every trial, participants first saw a fixation cross on the screen. After 800 ms, they heard a sentence from the headset while the fixation cross remained on the screen. Each sentence was either 3350 ms long or 4490 ms long, depending on condition. Right after the end of the sentence, participants saw an image in the middle of the screen. They were allowed to respond between 300 ms—5000 ms after the onset of the image (Fig. 1).⁹ Once participants responded, the trial terminated and the next trial would start. If a participant did not respond within 5000 ms, the response would be recorded as a “miss” and the next trial would start. Participants responded by pressing the right arrow key (TRUE) or the left arrow key (FALSE) on the keyboard. RTs were measured from image onset to key press.

4.3. Participants

23 students (12 male and 11 female), aged 26 ± 2.9 (mean ± SD), native Hebrew speakers. All signed an informed consent form approved by the Hebrew University Research Ethics Committee. They received either payment or course credit for participation.

4.4. Results and analyses

The data of 22 participants were included (our exclusion criterion was error-based—average accuracy in at least one run < 75%. It led to the removal of only one participant’s data). The group’s mean accuracy after this screening = 95.12%. In the error domain, within each Sentence Type, the less conditions showed lower average accuracy than their more counterparts. We show the data of intra-domain not and cross-domain not in Fig. 2, where the mean accuracy of each condition is marked in dark red in each boxplot (the rest of the data: accuracy_more = 99.23%; accuracy_less = 96.92).¹⁰ Among all the conditions, the lowest accuracy and widest spread was in less...not (88.9%).

When analyzing the RT data, only correct responses were taken into account (since it is unclear what cognitive processes are involved in erroneous responses). The results of the plain and intra-domain not conditions converge on findings from previous studies (Deschamps et al., 2015; Tan et al., 2023): |$\overline{RT}$|_more (1106.8 ms) < |$\overline{RT}$|  _less (1389.0 ms) and |$\overline{RT}$|_{not more} (1594.8 ms) > |$\overline{RT}$|_{not less} (1510.7 ms). As for the pair of conditions named cross-domain not, |$\overline{RT}$|_more..not (1639.9 ms) < |$\overline{RT}$|_less...not (1885.4 ms). The details of the data of the intra-domain not and the cross-domain not conditions are in Fig. 3, where means are marked as a dark red diamond. Reviewing the RT data, we notice that: (i) less…not was the most costly condition in terms of RT; (ii) within every Sentence Type pair, the less condition took longer to verify than its more counterpart (except the intra-domain not condition); (iii) even though both less… not and not less contain two DE operators, the former is much more difficult to process than the latter (longer RT, lower accuracy, and greater variance in accuracy in the former than the latter).

To further test the difference between not less and less…not, we fitted a linear mixed effects model to the data of the intra-domain not and cross-domain not with the logarithmic transformation of RT as the dependent variable (using R and lme4, Bates et al., 2015). Quantifier Type, Sentence Type and Truth Value and their interactions were used as the fixed effect factors. As random effects, we had an intercept for participants as well as by-participant slope for the effect of Sentence Type.¹¹ P-values were derived via the lmerTest package in R (Kuznetsova et al., 2017). Visual inspection of residual plots revealed no obvious deviations from homoscedasticity or normality. The statistics are shown in Table 2. Among the statistical results, there are two effects relevant to our research questions: first, there was a main effect of Sentence Type (t = 4.136, p = 0.0004), as manifested by the fact that the mean RTs of cross-domain not were on average higher than those of intra-domain not. Secondly, there was a strong significant interaction effect between Quantifier Type and Sentence Type (t = −6.033, p < 0.0001), indicating that the two double negative structures had very different patterns when compared to their more counterpart, respectively.¹²

Summary of the mixed-effect regression model for the data of Experiment I.

From the averaged data, we observe that |$\overline{RT}$|_less...not > |$\overline{RT}$|_more...not, and |$\overline{RT}$|_{not less} < |$\overline{RT}$|_{not more}. From the results of the mixed effect model, we learned that processing of the pair of intra-domain not differs from the processing of the pair of cross-domain not, in terms of RT¹³.

4.5. Discussion

Experiment I tested four Sentence Types with two Comparative Quantifiers in Hebrew, resulting in eight conditions in total, including more/less (plain), not more/less (intra-domain not), more/less...not (cross-domain not) and it is true that more/less (that-clause). We were particularly interested in the contrast between participants’ RTs in the cross-domain not condition and those in intra-domain not. Even though both less...not and not less contain two DE operators, the processing difficulty of the two conditions is very different. Our results show that less...not had significantly longer RTs and lower accuracy rate than more...not, indicating that it was more taxing.¹⁴ By contrast, the not less condition yielded slightly shorter RTs and slightly lower accuracy rates than not more, and was hence not more difficult to process (perhaps even easier).

The difference between less…not and not less is further corroborated by the results of the linear mixed-effects model when comparing cross-domain not and intra-domain not, which is also in line with the results of Sherman’s study (1976). A related study (Tan et al., 2023) also showed two DE operators in the same domain (i.e., not less) facilitate the processing of a sentence, suggesting that the two DE operators are counted as a single UE one.

Thus, the processing difference between the intra-domain (not less) and cross-domain (less...not) conditions indicates that the processing of two DE operators is not determined by the number of DE operators, suggesting that it is domain-based. In the present context, the notion domain is defined syntactically. As shown elsewhere (e.g., Crnič, 2019; Bar-Lev & Fox, 2020), this definition has much semantic and pragmatic relevance. Yet, the nature of this definition leads to the conclusion that syntax is an important determinant of the processing cost of monotonicity. That is, when two DE operators are in the same domain, their processing cost becomes UE (as DE*DE = UE); but when they are not in the same domain, they cannot be integrated into one during processing and cause MRD, which manifests in elevated cost.

4.6. Learning effect?

The account above works, but alternatives must also be considered. One possibility we explored was that the effects obtained were due to selective learning. That is, the differences we observed may have been due to participants’ improved performance over the course of the testing session, in a manner that differentiated between the conditions. RTs typically go down in the course of a testing session, and one possibility is that the effect we obtained is due to differential learning (some participants reported that they realized that responding is faster if not less is converted into more during a trial). To see whether this response strategy was indeed used and had a substantial effect on our results, we fitted four regression models for four relevant comparisons, including <more, less>, <not more, not less>, <more…not, less…not > and <not less, less…not>.¹⁵ We first calculated the cross-participant mean RT_n in trial n (1 ≤ n ≤ 32) in each condition (Fig. 4). Then, we fitted a regression model to each comparison, which included RT as the dependent variable, Trial Order as one predictor, either Sentence Type (for <not less, less…not>) or Quantifier Type (for the other three comparisons) as a second predictor, as well as their respective interaction terms. If differential learning does account for the difference in means within a pair, we expect to observe an interaction effect between Trial Order and one of these pairs. For example, if the result of RT_{not more} > RT_{not less} is actually a consequence of differential learning, we expect to find an interaction effect in the model for Trial Order and <not more, not less>. After adjusting for multiple comparisons by Bonferroni corrections (Bonferroni-adjusted p-values = 0.05/4 = 0.0125), we found no significant interaction effects.^16,17 Therefore, learning cannot explain why the pair of intra-domain not conditions exhibited a different pattern from the pair of cross-domain not, namely, RT_{not more} > RT_{not less} but RT_more…not < RT_less…not.

5. EXPERIMENT II

Experiment II was a replication attempt of Experiment I, using the exact same conditions and structure, with three differences: (a) participants were recruited on the Internet and the experiment was conducted online; (b) they were native speakers of English, not Hebrew; (c) they constituted a larger group. As will become clear, this experiment further solidifies our results with a large number of participants.

5.1. Participant recruitment

We recruited native English speakers online from Prolific’s participant pool. Participants were redirected from Prolific.ac to an online experiment hosted on PCIbex Farm. They were promised a monetary reward of £8.00 per hour (which was almost twice the time the test took on average, resulting in an average reward of £4.7). They were also told that completing the study at an overall accuracy of ≥95% would award them a bonus—a completion of their reward to £8.

5.2. Accuracy-based screening

• 1)

  Upon registration to the study, as well as while signing the consent form, participants were told: “you are required to meet a 90% accuracy threshold. Keep in mind that you may be rejected in the middle of the experiment, due to unsatisfactory accuracy levels. You have 50 minutes to complete the experiment”.

• 2)

  At the beginning of each run, participants were reminded of the bonus and its requirements.

• 3)

  At the experiment’s half point, participants got feedback containing their response accuracy rates, along with a comment on how much closer it brings them to the bonus. Participants whose accuracy rate was < 85% were excluded at this point. Eighty-four participants moved past the half point at the required pace and accuracy, completing the test at an average of 35 minutes. Of these, six exceeded the time limit, or were below the overall 90% accuracy threshold at the experiment’s end; they were excluded, but paid. Thirty-three (~ 40%) performed at a level that won them a bonus.

5.3. Procedures

After a general explanation about the experiment, each of the two experimental blocks began with a screen containing a graphic representation of a trial’s time-course, accompanied by an explanation of the task, and an invitation to perform a single trial. On the next screen, a table with all 8 sentences to be heard in this block were displayed. This was followed by a practice session containing 16 trials (equally representing all conditions) with feedback (“Correct!”, “Wrong!”, and “Too slow… please try to respond faster”). Each trial in both the practice and the experimental sessions was accompanied by a display of the keys representing match and nonmatch. This long preparation phase helped reduce errors, with no bias.

The trial structure was very similar to the one in Experiment I, only with slight time differences in order to adapt to the language differences. Details are depicted in Fig. 5.

After receiving feedback at the experiment’s middle point, there was a forced 2-minute break (which the participants could choose to extend to 5 minutes). A forced 1-minute break was also given between the two experimental sessions in each block.

5.4. Results and analysis

The data of the 78 participants who completed the experiment were now subjected to more stringent screening criteria: admitted to analysis were only those participants who not only performed at 90% and below 50 minutes, but also, at a level of 75% correct on each condition (overall mean accuracy: 95.21%). There were no misses. This screening left us with 70 participants. We show the accuracy data of intra-domain not and cross-domain not in Fig. 6 (after screening), where the mean accuracy of each condition is marked in dark red in each boxplot.¹⁸ The accuracy level in every condition is above 90%.

Next, we moved from the error domain to the time domain. We analyzed the RTs of the surviving 70 participants, omitting incorrect responses (4.49% of the data). We found that the less condition in each Sentence Type took longer to process than its corresponding more condition, except in intra-domain not, where the RTs of the two Quantifier Types approximate to each other. The mean RTs in intra-domain not and cross-domain not are exhibited in Fig. 7 (the mean RTs are marked by a dark red diamond in each boxplot.).¹⁹ Despite the difference in experimental method and language, the pattern between the Quantifier Types in each Sentence Type resembles what we found in Experiment I, as shown in Fig. 7. Regarding the two double-DE operator conditions, it seems that less...not is much more taxing to process than not less because both of the mean and the median of the former are much higher than those of the latter.

To test the significance of the difference between not less and less…not, we fitted a linear mixed effects model to the data of intra-domain not and cross-domain not with the logarithmic transformation of RT as the dependent variable (using R and lme4, Bates et al., 2015). Quantifier Type, Sentence Type and Truth Value and their interaction were used as fixed effects. The two random-effect terms include an intercept of subjects and a random slope of Sentence Type. P-values were derived via the lmerTest package in R (Kuznetsova et al., 2017). Similar to the result in Experiment I, there was a significant main effect of Sentence Type (t = −4.719, p < 0.0001) as well as an interaction effect between Quantifier Type and Sentence Type (t = −11.486, p < 0.0001) (see Table 3). The results here replicate the results from Experiment I, indicating that the pattern of cross-domain not exhibits a quality different from that of intra-domain not.

Summary of the mixed-effect regression model for the data of Experiment II.

The online experiment along with the meticulous reward mechanism enabled us to collect a large amount of high-quality data, providing us cross-language evidence which supports MPH + EnvBA. We observed the same contrast between not less and less…not among the English speakers as among the Hebrew speakers that less...not is much more taxing than not less. The fact that we found the same effect in English reinforces our argument that the configuration of a sentence which contains two DE operators determines the way how we process it.

5.5. Learning effect

In the same manner as in Experiment I, we calculated the cross-participant mean RT in each trial in each condition for the data in Experiment II, as shown in Fig. 8. Likewise, we fitted four regression models to check the difference between the learning rates in four comparisons, including <more, less>, <not more, not less>, <more…not, less…not> and <not less, less…not>. In each model, RT was the dependent variable, with Trial Order as one predictor, either Sentence Type or Quantifier Type as another predictor, and their interaction term. Among the four comparisons, we found no significant interaction effect, indicating that the learning rates were not distinguishable in each pair.²⁰ In sum, it was manifested in both Hebrew and English data, that there were no learning rate differences between more and less conditions in each Sentence Type.

6. GENERAL DISCUSSION

We started by exploring two approaches to NPI licensing, as contrasted by Homer (2010, 2021) in his study of flip-flop phenomena in French. He argued for the EnvBA by showing that only EnvBA could provide a reasonable account of the sentences with two DE operators which do not license NPI. Parallel to Homer’s example in French, we showed that there is also a flip-flop of NPI licensing in Hebrew. The Hebrew NPI ey pa’am ‘ever’ is licensed when the two DE operators—lo ‘not’ and paxot ‘less’—are in different domains, but not when the two sit in the same domain. We then devised the Monotonicity Processing Hypothesis (MPH), which, coupled with EnvBA, predicts that during processing, if domain-wise monotonicity reverses when a lower domain integrates into a higher domain, extra processing cost is induced. In contrast, the MPH + OpBA predicts no effect of syntactic structure on processing. We then provided experimental evidence substantiating the MPH + EnvBA through the measurement of how syntactic structure affects the processing of monotonicity. We demonstrated that, all else being equal, the RT for verifying a sentence containing two DE operators depends on the syntactic relationship between the two. First, compared with their one DE-operator counterparts, respectively, we observed RT_{not less} ≤ RT_{not more} but RT_less...not > RT_more...not in both Hebrew and English. Secondly, even though the two double DE-operator structures comprise exactly the same words, it took much longer to process the sentence “less than half of the circles are not blue” than “not less than half of the circles are blue”. We suggested that participants were able to integrate two DE operators as one UE operator only when the two DE operators were situated in the same domain.

To conclude, our novel findings—that come from the world of speeded behavior in which the RT, the variable of interest, is continuous—show remarkable convergence with the judgment data that Homer (2021) presented. They therefore provide further evidence to the Environment-based Licensing Condition for NPIs he proposed: NPIs are sensitive to monotonicity of their syntactic environment on the basis of domain because, cognitively, the processing of monotonicity is domain-based. In other words, when two DE operators occur in the same domain, we perceive the domain as one UE domain, resulting in the anti-licensing of NPI in such cases.

Acknowledgements

This research has received funding from an internal grant from Edmond & Lily Safra Center of Brain Sciences, and funding from the European Union’s Horizon Europe Programme under the Specific Grant Agreement No. 101147319 (EBRAINS 2.0 Project). This research was supported by the Joint Lab “Supercomputing and Modeling for the Human Brain”. We are very grateful to Bernhard Schwarz, Luka Crnič and Emmanuel Chemla for their critical help at different stages of this project. We also thank Jakub Dotlačil and 3 anonymous reviewers for their most helpful comments.

Footnotes

References

A. Appendix 1: French “flip flop” and our Hebrew experimental materials

This appendix shows the direct connection between the sentence materials of our experiment and the environmental characterization of NPI licensing domains as presented by Homer (2021). Our experiment was conducted in Hebrew main clauses (containing zero, one or two DE operators in subject position). A demonstration that Hebrew is like English and French is thus in order.

We begin with the “flip-flop asymmetry” Homer discusses, inspired by Chierchia (2004), Gajewski (2005) and Guerzoni (2006): a weak NPI is licensed by the DE operator in sentence (A1), but not in (A2). Homer account for this asymmetry by assuming that NPI licensing is syntactically constrained. He posits a Polarity Phrase (PolP) as a domain of NPIs, which DE operators are said to be part of. A weak NPI requires a DE licensor (A1). An NPI is licensed if contained within a PolP with an odd number of DE operators (A2); otherwise, it is anti-licensed (A3).²¹

• (A1) *…[_TP T [_PolP possible [_CP que Jean [_TPT [_PolP [quoi que ce soit]₁ faire t₁]]]]].

• (A2) …[_TP T [_PolP  impossible [_CP que Jean [_TPT [_PolP [quoi que ce soit]₁ faire t₁]]]]].

• (A3) *…[_TP T [_PolP  pas impossible [_CP que Jean [_TPT [_PolP [quoi que ce soit]₁ faire t₁]]]]].

• (A4) …[_TP T [_PolP  impossible [_CP que Jean [_TPT [_PolP  pas [quoi que ce soit]₁ faire t₁]]]]].

Example (A5), where DE-ness is induced in the antecedent clause of a conditional sentence, confirms Homer’s hypothesis. In (A5) a PolP is found, that contains the NPI, as well as a single DE operator, impossible. Thus, the second DE operator in the antecedent of the conditional, si, is above the licensing domain, unlike in (A3), where both are within this domain:

• (A5) [_TP [_CP  Si  … T [_PolP  impossible [_CP que Jean [T [_PolP [quoi que ce soit]₁ faire t₁]]]]] T]…

In addition to PolP, Homer posits TP as an another domain of NPIs, based on (A6), which contains two DE operators, si “if” and au plus cinq personnes “at most five people”. Assuming the subject au plus cinq personnes sits at [Spec, TP], for quoi que ce soit to be licensed, there must be a domain between CP and PolP, namely, TP, as Homer proposes.

• (A6) [_TP [_CP  Si  [_TP [au plus cinq personnes] T [_PolP ont [quoi que ce soit]₁ faire t₁]]] T]…

Hebrew offers a perfect replication of the examples that get Homer’s theory off the ground (with the weak NPI ‘ey-pa’am = ever):

• (A1-heb) ^*…[_TP T [_PolP efšari [_CP še Dani [_TP T [_PolP nirdam ‘ey-pa’am possible that Dani fell.asleep ever be-šmira]]]]] while.on.guard ‘… possible that Dani ever fell asleep while on guard’.

• (A2-heb) …[_TP T [_PolP  bilti-efšari [_CP še Dani [_TP T [_PolP nirdam ‘ey-pa’am im-possible that Dani fell.asleep ever be-šmira]]]]] while.on.guard

  ‘… impossible that Dani ever fell asleep while on guard’

• (A3-heb) ^*…[_TP T [_PolP  lo bilti-efšari [_CP še Dani [_TP T [_PolP nirdam ‘ey-pa’am. not im-possible that Dani fell.asleep ever be-šmira]]]]]. while.on.guard. ‘… not impossible that Dani ever fell asleep while on guard’

• (A4-heb) …[_TP T [_PolP  bilti-efšari [_CP še Dani [_TPT [_PolP  lo nirdam ‘ey-pa’am. im-possible that Dani not fell.asleep ever be-šmira]]]]]. while.on.guard. ‘… impossible that Dani did not ever fall asleep while on guard’

• (A5-heb) …[_TP [_CP  ‘im  [_PolP  bilti-efšari [_CP še Dani [_TP T [_PolP nirdam ‘ey-pa’am. if im-possible that Dani fell.asleep ever be-šmira]]]]] T]… while.on.guard. ‘If it is impossible that Dani ever fell asleep while on guard, …’

• (A6-heb) …[_TP [_CP  ‘im [_TP  [le-xol-ha-yoter xamiša ‘anašim] T [_PolP siy’uif at.most five people assisted ‘ey-pa’am la mafia]]] T]… ever to.the. mafia. ‘If at most five people ever assisted the mafia, …’

B. Appendix 2: extension of “flip flop” to simple, monoclausal sentences

Embedded clauses and conditionals turn out to not be the only flip-flop environments. Homer presents (A7), a monoclausal sentence containing two DE operators, au plus cinq and n’…pas. Again, assuming the subject au plus cinq is at [Spec, TP], there must be a DE domain below TP, which licenses the NPI, namely, PolP.

• (A7) Au plus cinq personnes n’ ont pas fait quoi que ce soit at most five people ne have neg done what that this be.subj pour aider la Mafia.to help the Mafia. ‘At most five people didn’t do anything to help the Mafia.’

Unfortunately, Homer does not provide a minimal contrast where at most combines with a constituent negation. This is important to our experiment, as our materials were simple double-negative sentences with either constituent negation or sentential negation. Below we show that Homer’s analysis holds for simple sentences, as a “flip-flop asymmetry” is found elsewhere.

We used mono-clausal sentences whose subjects are generalized quantifiers with modified numerals (yoter/paxot me-xamiša = more/less than five), and their negated counterparts (lo yoter/paxot me-xamiša = not more/less than five). A weak NPI (‘ey pa’am = ever) in object position is not licensed by a UE-quantifier (A8); yet it is licensed by the DE-quantifier paxot (A9). As in (A7), when a sentential negation lo is added to a sentence with a DE operator in the subject position, an NPI is still licensed (A10). The flip-flop happens when a constituent negation is added to (A9), as no DE domains are available in this case (A10).

• (A8) ^*[_TP [Yoter me- xamiša ratzim][T [_PolP [higi’u ‘ey pa’am la-gmar]]]]. more than-five runners reached ever to.the-finish.linet ‘More than five runners have ever passed the finish line.’

• (A9) [_TP [paxot me-xamiša ratzim][T [_PolP [higi’u ‘ey pa’am la-gmar]]]]. less than-five runners reached ever to.the-finish.linet ‘Less than five runners ever reached the-finish-line.’

• (A10) [_TP [paxot me-xamiša ratzim] [T [_PolP  lo [higi’u ‘ey pa’am la-gmar]]]]. less than-five runners not reached ever to.the-finish.line.

• (A11) ^*[_TP [lo paxot me-xamiša ratzim] [T [_PolP [higi’u ‘ey pa’am la-gmar]]]]. not less than-five runners reached ever to.the-finish.line

These examples indicate that Homer’s effects can be extended to monoclausal sentences. The two DE operators in (A11) integrate and together provide a UE environment to ey pa’am, whereas in (A10), the not itself may provide a DE domain to ey pa’am. The key point here is how the integration of two DE operators occurs, since eventually, in the maximal domain, i.e. the sentence, ey pa’am is in a UE domain in both (A10) and (A11). The contrast between (A10) and (A11) suggest that the integration happens stage by stage (cf.   Papeo et al., 2016).

These Hebrew data align with Homer’s hypothesis that PolP and TP are both valid domains for NPIs, showing that our experimental materials are well-suited to distinguish between the hypotheses.