Processing of Numerical and Proportional Quantifiers.

Cognitive Science 39 (2015) 1504–1536 Copyright © 2015 Cognitive Science Society, Inc. All rights reserved. ISSN: 0364-0213 print / 1551-6709 online DOI: 10.1111/cogs.12219

Processing of Numerical and Proportional Quantifiers Sailee Shikhare,a Stefan Heim,b,c,d Elise Klein,a,e,f Stefan Huber,f,g Klaus Willmesa,e a

b

Department of Neurology, University Hospital, Aachen Department of Psychiatry, Psychotherapy and Psychosomatics, University Hospital, Aachen c JARA, Translational Brain Medicine d Institute of Neuroscience and Medicine (INM-1), Research Centre Jülich, Germany e Interdisciplinary Center for Clinical Research, University Hospital, Aachen f Knowledge Media Research Center, IWM-KMRC, Tübingen, Germany g Department of Psychology, Eberhard Karls University, Tübingen

Received 9 February 2014; received in revised form 26 June 2014; accepted 5 September 2014

Abstract Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical (at least seven, at least thirteen, at most seven, at most thirteen) or proportional (many, few) quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are fundamental in encoding the meaning of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity (seven or thirteen) is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity (positive vs. negative) biased the response direction (accept/reject). Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers. Keywords: Numerical and proportional quantifiers; Numerical estimation; Numerical comparison; Semantic polarity

Correspondence should be sent to Sailee Shikhare and Klaus Willmes, Section Neuropsychology, Department of Neurology, Uniklinik RWTH Aachen, 52074, Germany. E-mails: [email protected], [email protected] and [email protected]

S. Shikhare et al. / Cognitive Science 39 (2015)

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1. Introduction Linguistic expressions like “every,” “at least,” and “more than half” are part of our day-to-day conversations. Such verbal expressions, referred to as quantifiers, are vital in expressing quantity information about time, magnitude, distance, frequency as well as currency (Troiani, Peelle, Clark, & Grossman, 2009). Barwise and Cooper (1981) defined quantifiers as noun phrases that functionally assert some property of a particular set and assign a truth value to it. For instance, in the quantifier sentence “At least three balls are yellow,” one needs to primarily identify the specific property to be quantified, in this example “balls being yellow.” But, to assign a truth value to the sentence, one must additionally comprehend the semantics of the quantifier “at least” in conjunction with the numerosity “3.” The mapping of quantifiers onto numerical scales has been proposed in logic, formal semantics, syllogistic reasoning as well as linguistic comprehension and processing (Bass, Cascio, & O’Connor, 1974; Clark, 1969; Geurts, 2003; Holyoak & Glass, 1978; Sanford, Moxey, & Paterson, 1994).1 However, the direct link between magnitude knowledge and quantifier comprehension was established in cognitive neuroscience more recently (Halberda, Taing, & Lidz, 2008a; Heim et al., 2012; Lidz, Pietroski, Halberda, & Hunter, 2011; McMillan, Clark, Moore, Devita, & Grossman, 2005; McMillan, Clark, Moore, & Grossman, 2006; Odic, Pietroski, Hunter, Lidz, & Halberda, 2013; Pietroski, Lidz, Hunter, & Halberda, 2009; Szymanik & Zajenkowski, 2010; Troiani et al., 2009; Zajenkowski, Styla, & Szymanik, 2011; Zajenkowski & Szymanik, 2013; Zajenkowski, Szymanik, & Garraffa, 2014). Moreover, the exact role of numerical processing in interpreting quantifiers, namely, in estimation and comparison, is gradually emerging (Halberda et al., 2008a; Heim et al., 2012; Lidz et al., 2011; Odic et al., 2013; Pietroski et al., 2009; Zajenkowski & Szymanik, 2013; Zajenkowski et al., 2014). In this study, we aimed to investigate the interplay between language and core numerical processing aspects in comprehending quantified sentences. Using a semantic truth value judgment task, we made an attempt at identifying the cognitive processes involved in evaluating auditorily presented numerical and proportional quantifier sentences about visual displays comprising two sets of circles with different colors. 1.1. Numerical sense in quantifier comprehension The study of how humans represent numerical information by means of the two core systems, a precise number system for the enumeration of small numbers up to 4 and an approximate number system (ANS) for the estimation of numbers larger than 4, is well established in the numerical cognition literature (Dehaene, 1997; Feigenson, Dehaene, & Spelke, 2004). The behavioral correlates and the corresponding neural signature for counting as well as for estimating numerosities across distinct modalities are well delineated (Dehaene, 1997; Piazza & Izard, 2009; Piazza, Mechelli, Price, & Butterworth, 2006; Pica, Lemer, Izard, & Dehaene, 2004). Researchers have shown that when an array of dots is flashed, enumeration performance is relatively fast and accurate for small

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numbers. This recognition and simultaneous tracking of small numbers up to 4 (i.e., subitizing), therefore, is believed to follow distinct processing properties as compared to larger numbers (Dehaene & Cohen, 1994; Mandler & Shebo, 1982). Much attention in the last few decades has been devoted to reveal the mental representation of cardinalities above 4, governed by the ANS (Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010; Feigenson et al., 2004; Leibovich & Henik, 2014; Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004). The ANS serves to represent sets of numerosities in an approximate fashion, mastered by primates, infants as well as adults (Hubbard et al., 2008). Xu and Spelke (2000) put forth that 6-month old infants’ discrimination performance was subject to the ratio between the two numbers presented. When asked to differentiate between two arrays of dots containing 8 versus 16 dots in comparison with 8 versus 12 dots, their responses to the former array were faster and more accurate as compared to the latter array. Adult’s performance on an analogous task was qualitatively similar but more precise (Barth, Kanwisher, & Spelke, 2003; Whalen, Gallistel, & Gelman, 1999). In both cases, performance was modeled by the ANS, where inherently noisy and approximate magnitude representations of numerosities were encountered on an internal “mental number line” (Brannon, 2002; Dehaene, 1997; Dehaene & Cohen, 1997; Gallistel & Gelman, 2000). The ANS has been extensively investigated using magnitude comparison tasks with varied numerical stimulus material (Arabic digits, number words, dot patterns), as well as linguistic stimuli (days of months, months in a year) across modalities (Ansari, 2008; Halberda, Mazzocco, & Feigenson, 2008b). Two hallmark effects for the ANS, the numerical distance effect (NDE) and the problem size effect, have been reported repeatedly (Moyer & Landauer, 1967). The NDE denotes the finding that the larger the numerical difference between two cardinalities to be compared, the shorter is the time required to decide which number is larger. This NDE is explained in terms of the overlap between mental representations of external numbers on the internal mental number line (MNL) (Moyer & Landauer, 1967). Performance is ratio dependent and is well modeled by Weber’s law (Dehaene, Izard, Spelke, & Pica, 2008; Feigenson et al., 2004; Piazza et al., 2004; Whalen et al., 1999). The NDE is highly reliable and is evident in healthy as well as in clinical populations, e.g., developmental dyscalculia (Ashkenazi, Mark-Zigdon, & Henik, 2009; Banks, Mermelstein, & Yu, 1982; Dehaene, Dupoux, & Mehler, 1990). The problem size effect is defined in terms of increasing response times (RT) and decreasing accuracies for greater numerosities coded on the MNL (conventionally from left to right). Higher variability is expected as the numbers increase linearly on the MNL. Specifically, these effects concentrate on the difference in response latencies when comparing smaller versus larger numbers (Moyer & Landauer, 1967). The capability to understand precise numerical information is directly associated with the ability to comprehend quantifiers like “at least three,” the study of which provides a window on the connections between magnitude and quantifier comprehension (Heim et al., 2012; Lidz et al., 2011; McMillan et al., 2005, 2006; Pietroski et al., 2009; Szymanik & Zajenkowski, 2010; Troiani et al., 2009). Quantifiers have been extensively investigated for several decades, taking into account different theoretical models to trace verification strategies in determining the truth conditionality of quantified sentences


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(Barwise & Cooper, 1981; Bass et al., 1974; van Benthem, 1986; Geurts, 2003; Moxey & Sanford, 1987; Sanford et al., 1994). Empirical studies in the recent past have concentrated on investigating the joint processing of quantifiers like “most” and “some” in combination with non-symbolic numerical patterns (dots, cars, balls) in sentence–picture verification tasks (Heim et al., 2012; McMillan et al., 2005, 2006; Szymanik & Zajenkowski, 2010; Zajenkowski & Szymanik, 2013). In a series of behavioral experiments, the computational model of quantifier verification2 was tested, where participants were asked to respond on logical (all), numerical (at least), parity (an even number of) as well as proportional (more than half) quantifiers against visual scenarios presented for 15 s. van Benthem (1986) in the computational modeling approach highlighted that the cognitive complexity of quantifiers can be characterized by specific abstract automata (meaning “machines”—a term used in computer science) employed for the verification process, adopting an algorithmic approach based on counting (Szymanik & Zajenkowski, 2010; Zajenkowski & Szymanik, 2013; Zajenkowski et al., 2011, 2014). Results obtained in their investigations followed their predictions. Poor performance was observed on proportional quantifiers (more than half) as compared to other quantifier types, in healthy adults (Szymanik & Zajenkowski, 2010; Zajenkowski & Szymanik, 2013; Zajenkowski et al., 2014) as well as in patients with cortico-basal degeneration (CBD), fronto-temporal dementia (FTD), Alzheimer’s disease (McMillan et al., 2005, 2006; Troiani, Clark, & Grossman, 2011), and schizophrenia (Zajenkowski et al., 2011). On the one hand, the deficits observed in patients with CBD and FTD were attributed to the underlying numerical impairments, while quantifier impairments in patients with schizophrenia were attributed to their problems with executive functioning, mainly working memory as well as language impairments. In a few of their studies, the authors highlighted the vital role of the numerical value in the numerical quantifier. Adults took longer to respond to numerical quantifiers in combination with larger numerosities (less than eight, more than seven) in comparison with analogous sentences comprising smaller numbers (less than four, more than five) (Szymanik & Zajenkowski, 2010). The higher the numerical value, the harder is the processing. When a memory task (recall of digits) was used in conjunction with the quantifier verification task, the numerosity in the quantifier expression acted as a predictor of the cognitive load for understanding numerical quantifiers (van Benthem, 1986; Szymanik & Zajenkowski, 2010; Zajenkowski et al., 2011). Additional evidence regarding the involvement of working memory resources, namely, storage and maintenance of information, during verification of proportional quantifiers was proposed in the model (Zajenkowski & Szymanik, 2013; Zajenkowski et al., 2011). The authors suggested that proportional quantifier verification is correlated with short-term working memory to maintain information. Moreover, distance effects were observed (one vs. three), which were explained in terms of the integration process required for numerical comparison (Zajenkowski et al., 2014). In all these studies, approximate (one target item for less than eight in a visual stimulus of 15 objects) versus precise (seven target items for less than eight) judgments were used for the verification of numerical quantifiers. However, results emphasizing the difference between the two (approximate vs. precise judgments) were not reported here (Szymanik & Zajenkowski, 2010; Zajenkowski & Szymanik,

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Table 2 Absolute asymmetry index values for RT, accuracy, and acceptability (ms) for all four numerical quantifiers Quantifier Response time (RT)

Accuracy

Acceptability

At At At At At At At At At At At At

least seven least thirteen most seven most thirteen least seven least thirteen most seven most thirteen least seven least thirteen most seven most thirteen

Asymmetry Index Sign + + + + +

Mean (SD) 60.99 81.67 90.25 30.45 0.01 0.12 0.08 0.02 0.39 0.25 0.33 0.22

(102.21) (127.81) (91.46) (130.37) (0.09) (0.16) (0.09) (0.21) (0.08) (0.10) (0.10) (0.13)

One-Sample t-test t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23)

= 2.92, p = .01 = 3.13, p = .01 = 4.83, p < .001 = 1.14, p = .26 = 0.48, p = .64 = 3.74, p = .001 = 4.84, p < .001 = 0.39, p = .70 = 22.78, p < .001 = 12.20, p < .001 = 15.78, p < .001 = 8.29, p < .001

most” for both numerical references separately (cf. graph C and D in Fig. 4). A detailed analysis of the significant main and interaction effects as well as the post hoc comparisons is explained in Supplementary Material S3. We selected the proportions at a distance of 1 and 2 to the target proportion, similar to the RT analysis (cf. section 3.1.1, paragraph 2). A 2 (Quantifier) 9 2 (Numerical Reference) 9 4 (Proportion) repeated measures ANOVA revealed statistically significant main effects of Quantifier F(1,23) = 6.26, p = .02, Numerical Reference F(1,23) = 40.89, p < .001, and Proportion F(3,69) = 8.05, p = .003. Two-way interactions Quantifier 9 Proportion F(3,69) = 12.66, p < .001, Numerical Reference 9 Proportion F(3,69) = 7.61, p = .004 were observed. The three-way interaction Quantifier 9 Proportion 9 Numerical Reference F(3,69) = 2.40, p = .08 was only marginally significant. 3.1.2.1. Asymmetry index analysis for accuracy: Using an identical coding procedure and asymmetry index formula, we derived asymmetry indices for all four numerical quantifiers for each participant with respect to the accuracy data (cf. Asymmetry index analysis for RT). A 2 (Quantifier) 9 2 (Numerical Reference) 9 2 (Distance) repeated measures ANOVA revealed a main effect of Quantifier F(1,23) = 22.47, p < .001 and Numerical Reference F(1,23) = 6.88, p = .02. The two-way interaction Numerical Reference 9 Distance F(1,23) = 12.15, p = .002 as well as the three-way interaction Quantifier 9 Numerical Reference 9 Distance F(1,23) = 4.38, p = .05 were observed. A graphical representation for the asymmetry index values for each distance is sketched in the Supplementary Material S4. We checked whether asymmetry indices for all four numerical quantifiers were significantly different from zero for distances 1 and 2, respectively, using one-sample t-tests (cf. Table A in Supplementary Material S4). To further assess the interaction effects, post hoc paired t-test comparisons for distances 1 and 2 were computed (cf. Table B in Supplementary Material S4). An average asymmetry index was calculated for each numerical quantifier, compiling data over distances 1 and 2. Main


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1.2. Semantic polarity of quantifiers Semantic polarity of words is a well-addressed research topic in linguistics (Geurts, 2003; Moxey, 2006; Moxey & Sanford, 1987; Oaksford, Roberts, & Chater, 2002; Paterson, Sanford, Moxey, & Dawydiak, 1998). A whole range of word pairs, such as “happy– sad,” “tall–short,” and “up–down,” are investigated in terms of the assigned polarity given to every word in the pair. According to the linguistic markedness account, it is believed that the first word in the pair is positive (unmarked- default), while the second word is negative (marked). It was shown that participants take longer in the encoding stage for the marked (negative) than the unmarked default word (positive), since for the former, one has to additionally process information about negation (Clark, 1969; Clark & Chase, 1972). Studies in syllogistic reasoning (premise-conclusion logic) provided evidence regarding verification of numerical quantifiers. In a behavioral investigation comparing the validity of assessing “at least n,” “some,” and “at most n,” no relevant differences were observed between “at least n” and “some,” suggesting that for both quantifiers, a similar strategy was followed in validating or falsifying arguments (Geurts, 2003). But significant differences were observed between “at least n” and “at most n,” where it was argued that “at most” is more complex to comprehend, since the processing time necessary for negation has to be taken into account (Geurts, 2003). Proctor and Cho (2006) in their polarity correspondence hypothesis suggested that, in binary classification tasks, participants portray a tendency to assign positive or negative polarity to a stimulus as well as to the response alternatives. Therefore, response selection is proposed to be faster when there is congruency between stimulus and response. Nuerk, Iversen, and Willmes (2004) suggested a similar effect in numerical cognition studies, where a property of the (numerical) input stimulus was associated with the response (key press), that is, evenright/odd-left, known as the MARC effect (linguistic markedness of response codes). An alternative approach comes from reading as well as language production tasks, which emphasize the focus patterns associated with quantifier polarity (Moxey, 2006; Moxey & Sanford, 1987; Paterson et al., 1998; Sanford et al., 1994). They propose that when creating an internal representation of a quantified sentence, the positive quantifier instantaneously calls for the reference set, while the negative quantifier focuses on its complement set. For instance, in the sentence “many of the circles are yellow,” participants would invariably focus on the number of yellow circles (reference set) as “many” is a positive quantifier. While in a sentence “few of the circles are yellow,” “few” being a negative quantifier, participants would tend to focus on the non-yellow, that is, blue circles (complement set). 1.3. Numerical and proportional quantifiers and the ANS It is apparent from the above-mentioned literature that linguistic expressions of quantity can be interpreted in terms of the key numerical processing aspects in verification tasks. Two deviating views on the connection between language and number processing in general are debated in the literature. One perspective holds that numerical abilities

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originate from the language faculty (Chomsky, 1988), while the other implies that these two systems are functionally distinct (Dehaene & Cohen, 1997; Gelman & Butterworth, 2005). The latter perspective is also inspired by investigations in brain-damaged patients having either number processing or language deficits (Brannon, 2005; Butterworth, Cappelletti, & Kopelman, 2001; Cappelletti, Butterworth, & Denes, 1991; Cappelletti et al., 2006; Cohen, Dehaene, & Verstichel, 1994). In the present study, we sought to deal with the comprehension and processing of numerical and proportional quantifiers to explore the potential link between language and numbers. In our task, sentences comprising (a) a numerical quantifier with a distinct cardinality (card) and (b) a proportional quantifier “many” (resp. “few”): (a) At least seven of the circles are yellow—True iff card {circles ∩ yellow} ≥ 7. (b) Many of the circles are yellow—True iff card {circles ∩ yellow} > card {circles ∩ blue}. (b’) Many of the circles are yellow—True iff card {circles ∩ yellow} > card {internal numerical criterion}. Prima facie, in the above sentences, we could presuppose that one has to assess the numerical relationship between two sets comprising yellow and blue color circles presented visually, depending on the quantifier semantics (cf. Fig. 1A for the visual stimulus display). In our study, we predict the use of ANS for creating comprehension3 strategies to evaluate numerical and proportional quantifiers. We hypothesized that full comprehension of the quantifier sentence requires the ability to engage the linguistic system (to comprehend the meaning of every word in the sentence), the visual system (detection of the yellow and blue circles), and the ANS (to encode the numerical size of both sets of circles and to further subserve comparison of the numerical values). In addition, shortterm working memory resources would be essential to retain numerical estimates and compare them for making a truth value judgment (Odic et al., 2013). In the present study, we predict that numerical quantifiers and proportional quantifiers would vary on the basis of the numerical comparison process created to verify quantified sentences. We expect that one needs to concurrently encode the linguistic content (at least) and the numerical reference (seven), as well as the target color (yellow) to form a mental representation of the numerical quantifier in sentence (a). To evaluate sentence (a) against a visual display consisting of yellow and blue circles, one needs to estimate the number of yellow circles and hold this numerical estimate in short-term working memory. Furthermore, one needs to carry out the numerical comparison between the estimated cardinality and the numerical reference (seven) to verify the quantifier statement. In principle, depending on the quantifier semantics and the cardinality comparison process, all instances where the cardinality of yellow circles exceeds the numerical reference are true for sentence type (a). Our prediction is different for proportional quantifiers like “many.” Comprehension of the sentence (b) requires one to have the ability to translate (imprecise) verbal information into numerical estimates to judge the relationship between two sets. To evaluate sentence type (b), one has to estimate the number of yellow circles and store the numerical estimate in short-term working memory. Here, the imprecision of representing the meaning


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(A)

(B)

Fig. 1. Experimental design of the study (A). Auditory stimulus sentences included numerical quantifiers (at least seven, at least thirteen, at most seven, and at most thirteen) or proportional quantifiers (many, few) and were of the type “ of the circles are ,” followed by a visual display, showing varied proportions of yellow and blue circles with a constant total (n) of 20. The proportion of yellow circles and blue circles was systematically varied, characterized by the number of circles (c) to be estimated in the target color (TarCol) and ranging from 5 up to 15, as well as the complementary non-target color characterized by the estimation parameter (r) ranging from 15 to 5. Time course of individual trials (B). Each trial starts with a fixation cross, followed by the auditory sentence for 2.6 s. Then a visual display with the parametrically varied proportions is presented for 1 s, followed by a visual mask for 2 s. Participants are asked to respond per trial, if the auditory sentence matches the visual display or not, via a button press on one of two response keys. RTs are recorded from the onset of the visual display until the offset of the visual mask (maximum time for response: 3 s). The overall duration of a trial is 6.6 s.

of the quantifier “many,” for example, as compared to its close relatives like “more than half,” is greater in terms of interindividual variability. Hackl (2009) in his investigation put forth that the proportional quantifier “most” triggers a distinct behavioral strategy when compared to “more than half,” which can be attributed to the semantic differences between them. “Most” can be assumed to be the superlative form of “many” while “more than half” is its comparative form. From a numerical perspective, for “more than half” there is a fixed reference to compare between sets, namely, “half.” Therefore, although the comprehension strategy for “more than half” triggers complex strategies, one could assume similarity in the processing steps across individuals. However, for “many” no such reference is provided externally and thus could depend on the subjective interpretation of each individual regarding its meaning. It is conceivable that participants might adopt the most common strategy to focus on the reference set, that is, the target color mentioned in the quantifier


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Odic, D., Pietroski, P., Hunter, T., Lidz, J., & Halberda, J. (2013). Young children’s understanding of “more” and discrimination of number and surface area. Journal of Experimental Psychology. Learning, Memory and Cognition, 39(2), 451–461. Paterson, K. B., Sanford, A. J., Moxey, L. M., & Dawydiak, E. (1998). Quantifier polarity and referential focus during reading. Journal of Memory and Language, 39(2), 290–306. Piazza, M., & Izard, V. (2009). How humans count: Numerosity and the parietal cortex. Neuroscientist, 15 (3), 261–273. Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004). Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron, 44(3), 547–555. Piazza, M., Mechelli, A., Price, C. J., & Butterworth, B. (2006). Exact and approximate judgements of visual and auditory numerosity: An fMRI study. Brain Research, 1106, 177–188. Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306(5695), 499–503. Pietroski, P., Lidz, J., Hunter, T., & Halberda, J. (2009). The meaning of ‘most’: Semantics, numerosity and psychology. Mind & Language, 24(5), 554–585. Proctor, R. W., & Cho, Y. S. (2006). Polarity correspondence: A general principle for performance of speeded binary classification tasks. Psychological Bulletin, 132(3), 416–442. Routh, D. A. (1994). On representations of quantifiers. Journal of Semantics, 11(3), 199–214. Sanford, A. J., Moxey, L. M., & Paterson, K. (1994). Psychological studies of quantifiers. Journal of Semantics, 11(3), 153–170. Szymanik, J. (2007). A comment on a neuroimaging study of natural language quantifier comprehension. Neuropsychologia, 45(9), 2158–2160. Szymanik, J. (2009). Quantifiers in TIME and SPACE. Computational complexity of generalized quantifiers in natural language. PhD thesis, University of Amsterdam. Szymanik, J., & Zajenkowski, M. (2009). Improving methodology of quantifier comprehension experiments. Neuropsychologia, 47(12), 2682–2683. Szymanik, J., & Zajenkowski, M. (2010). Comprehension of simple quantifiers: Empirical evaluation of a computational model. Cognitive Science, 34(3), 521–532. Szymanik, J., & Zajenkowski, M. (2013). Monotonicity has only a relative effect on the complexity of quantifier verification. In M. Aloni, M. Franke & F. Roelofsen (Eds.), Proceedings of the 19th Amsterdam colloquium (pp. 219–225). Amsterdam: Institute for Logic, Language, and Computation (ILLC) at the University of Amsterdam. Troiani, V., Clark, R., & Grossman, M. (2011). Impaired verbal comprehension of quantifiers in corticobasal syndrome. Neuropsychology, 25(2), 159–165. Troiani, V., Peelle, J. E., Clark, R., & Grossman, M. (2009). Is it logical to count on quantifiers? Dissociable neural networks underlying numerical and logical quantifiers. Neuropsychologia, 47(1), 104–111. Whalen, J., Gallistel, C. R., & Gelman, R. (1999). Nonverbal counting in humans: The psychophysics of number representation. Psychological Science, 10(2), 130–137. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11. Zajenkowski, M., Styla, R., & Szymanik, J. (2011). A computational approach to quantifiers as an explanation for some language impairments in schizophrenia. Journal of Communication Disorders, 44(6), 595–600. Zajenkowski, M., & Szymanik, J. (2013). MOST intelligent people are accurate and SOME fast people are intelligent. Intelligence, working memory, and semantic processing of quantifiers from a computational perspective. Intelligence, 41(5), 456–466. Zajenkowski, M., Szymanik, J., & Garraffa, M. (2014). Working memory mechanism in proportional quantifier verification. Journal of Psycholinguistic Research, 43(6), 839–853.


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comprehension of quantifiers. On the one hand, we predicted that participants would find it harder to evaluate numerical quantifier sentences about visual displays with proportions closer to the numerical reference as compared to proportions farther away. We hypothesized an increase in RTs and a decrease in accuracy for proportions closer to the numerical reference for “at least” and “at most” (NDE). We expected participants to take longer RT and make more errors on proportions closer to the target proportion 7:13, that is, with the numerical reference “seven” (5:15, 6:14, 8:12, 9:11) when evaluating sentences with “at least seven” and “at most seven.” A similar performance pattern was expected for proportions closer to the target proportion 13:7 (11:9, 12:8, 14:6, 15:5) for numerical quantifiers “at least thirteen” and “at most thirteen.” Moreover, we expected an overall increase in RTs for conditions comprising “thirteen” as compared to “seven” because numerical estimation tends to be more demanding with higher cardinalities (problem size effect). On the other hand, we predicted that participants would find it harder to evaluate proportional quantifier sentences about visual displays with proportions closer to one (NDE). We hypothesized that for comprehension of proportional quantifiers, participants will take longer for proportions closer to one (8:12, 9:11, 11:9, 12:8) as compared to proportions further away from one. Lastly, taking into account the semantic polarity aspect, we predicted that performance on positive quantifiers (many, at least) would be faster as compared to those on negative quantifiers (few, at most) because an additional mental negation operation would be required for the latter group of quantifiers in terms of processing time.

2. Methods 2.1. Participants Twenty-five German native speakers were recruited from the RWTH Aachen University, Germany. Accuracy for the whole task was expected to reach a minimum of at least 75% for inclusion in the final statistical analysis. All except one participant fulfilled this criterion and therefore 24 right-handed healthy adults (11 males: mean age 26.2 years, SD = 3.3; 13 females: mean age 28.5 years, SD = 3.2) were included. The study was approved by the local Ethics Committee of the Medical Faculty at RWTH Aachen University, Germany. 2.2. Semantic truth value judgment task The semantic truth value judgment task from Heim et al. (2012) was adapted in the current study. This task involved a sequence of two phases per trial as shown in Fig. 1A. Short quantifier sentences containing numerical quantifiers (at least seven, at least thirteen, at most seven, at most thirteen) and proportional quantifiers (many, few) were presented (see Table 1 for stimulus material) in the first auditory phase of an item. All auditorily presented sentences had an identical structure “ of the circles are

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Table 1 German quantifiers with English translations (auditory stimulus material) used in the study including their polarity Quantifier Type

German Quantifiers

English Quantifiers

Polarity

Proportional

viele wenige mindestens sieben mindestens dreizehn hoechstens sieben hoechstens dreizehn

Many Few At least seven At least thirteen At most seven At most thirteen

Positive Negative Positive Positive Negative Negative

Numerical

,” where a quantifier was presented in the subject position, followed by a TarCol (yellow or blue) in the predicate position. For the numerical quantifiers, one positive quantifier “at least” and one negative quantifier “at most” were chosen. The two numerical referents “seven” and “thirteen” were used for these numerical quantifiers, thus including four numerical quantifiers in the experimental task (at least seven, at least thirteen, at most seven, at most thirteen). For the proportional quantifiers, a positive quantifier “many” and a negative quantifier “few” were selected. All together, half of the quantifiers included in the task were positive (many, at least seven, at least thirteen) while the other half were negative (few, at most seven, at most thirteen). The subsequent phase consisted of a visual display containing some random allotment of yellow and blue circles against a gray background (Fig. 1A). A total of 20 circles comprised the visual display, where proportions between yellow and blue circles were systematically varied as explained in the introduction section 1.4 and graphically outlined in Fig. 1A. 2.3. Experimental procedure Participants were administered a total of 448 trials. Every trial lasted for 6.6 s, where the auditory sentence took 2.6 s and the visual display was shown for 1 s as sketched in Fig. 1B. A visual mask consisting of a pixilated image using yellow, blue, and gray colors followed the visual display for 2 s. Participants had to decide whether the auditory sentence matched the visual display and respond either “Accept” by pressing the right control key (right Strg on a German keyboard) with the index finger of the right hand or respond “Reject” by pressing the left control key (left Strg on a German keyboard) with the index finger of the left hand. RT was recorded from the onset of the visual display until the offset of the visual mask, thereby giving participants 3 s to respond. Task instructions were given in oral and written format. Twelve practice trials were presented to make sure that participants understood the task. Trials were presented in a pseudo-random sequence that was fixed for all participants. Two major constraints were used in pseudo-randomizing trial order. First, the quantifier presented in the auditory sentence in a trial was not allowed to be followed by the same quantifier in the consecutive trial. Second, experimental trials with no more than three identical responses, either “Accept” or “Reject” were presented in a row. Four replications of each experimental condition were


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included. Participants were presented four blocks, each block consisting of 112 trials lasting for 13 min each. This resulted in approximately 1 h for the experimental session including breaks. At the end of the entire experiment, participants received 10 € for participation in the study. 2.4. Stimulus material There were 6 quantifiers 9 2 colors = 12 different quantifier sentences, as either of the two colors could be the target color. These sentences were professionally spoken by a female German native speaker and were recorded at the Audio-Visual Media Center of the Medical Faculty of RWTH Aachen University (Audiovisuelles Medienzentrum, AVMZ). The visual displays comprising 10 proportions consisting of varying yellow and blue circles were drawn using the Java Abstract Window Toolkit. Target proportions (7:13 and 13:7) exactly matched the numerical information presented in the quantifier sentence (at least seven, at least thirteen, at most seven, at most thirteen). This would have created an imbalance with respect to accuracy performance (namely, for numerical quantifiers) in the total number of “true” versus “false” judgments. Therefore, to include 50% “true” and “false” experimental trials for all quantifier sentences, eight trials with these two target proportions were excluded in every session only for quantifiers many, few, at least seven, and at most thirteen either for the TarCol yellow or blue. Two sets of visual stimuli were included. One stimulus set contained circles matched for size, while the rest were matched for area. Size-matched displays consisted of circles with varied individual size, but the size of the average blue circle was equal to the size of the average yellow circle. Area-matched displays also consisted of circles with varied individual size, but the area covered by all the blue circles was equal to the area covered by the yellow circles. In all displays, the overall area covered by the number of circles was one third of the entire image. Luminance of yellow and blue, as well as the background color gray, was controlled for all images. The main reason to either balance size or area was to discourage participants to adopt a specific strategy using either circle “size” or “area” as a proxy for numerical estimation and comparison. Stimulus presentation was carried out on a DELL computer, where the participant was seated at a comfortable distance from the monitor (eyes to the monitor: 50–60 cm). The experiment was programmed and presented using Presentation software version 14.0 (Neurobehavioral Systems, Albany, CA, USA). 2.5. Data analysis Data analysis was carried out using IBM SPSS Statistics for Windows (version 20.0; IBM Corp., Armonk, NY, USA). RT for all four blocks were merged together and submitted to a data trimming procedure using a 3 standard deviation (mean 3 SD) cut-off value for each participant. In this trimming procedure, 3.2% of data were excluded. In addition, non-responses (1.3%) were excluded for further analysis. In addition to RT, we chose two additional response measures, namely, accuracy and acceptability.

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Accuracy was defined only for numerical quantifiers, where we coded every participant’s accuracy score as 1 only if the “Accept” response coincided with the “true” response for the respective quantifier–proportion pair. Otherwise, the trial response was regarded as an error and excluded from the analysis of RTs. For acceptability, each participant’s binary response on every trial was coded. All “Accept” responses were coded as “1” while all “Reject” responses were coded as “0” for each quantifier–proportion pair for both, numerical and proportional quantifiers. For proportional quantifiers, only the acceptability measure was computed because for “many” and “few” it is the subjective decision of the participants to accept or reject the sentence (as described in the introduction section), as opposed to an objective decision for “at least” and “at most.” In addition, to be consistent across all measures reported, we used only eight proportions, excluding the target proportions for numerical as well as proportional quantifiers. Each of the 12 quantifier sentences was scored individually. This resulted in a total of 96 experimental conditions (12 quantifiers 9 8 proportions) for further analysis. To evaluate the impact of color and stimulus set in the task, we submitted the correct RT data to a repeated measures ANOVA with Quantifier (6 levels) 9 Proportion (8 levels) 9 Color (2 levels) 9 Stimulus Set (2 levels) as within subject factors. Here, we excluded the RTs of incorrect responses (16% overall). The main effects of Color F(1,23) = 1.53, p = .23 and Stimulus Set F(1,23) = 2.31, p = .14 were not significant. Therefore, we collapsed the data over stimulus set and color for each quantifier. Here, the auditory sentences with “blue” as TarCol were averaged together with their complementary proportions with sentences containing “yellow” as TarCol. Therefore, each participant was characterized by one mean RT value per quantifier per proportion. The final analysis thus included six quantifiers (4 numerical quantifiers and 2 proportional quantifiers) and eight proportions.

3. Results 3.1. Numerical quantifiers: “At least” and “at most” 3.1.1. Response times To examine the impact of the three experimental factors on RT, we computed a 2 (Quantifier—at least, at most) 9 2 (Numerical Reference—seven, thirteen) 9 8 (Proportion—5:15, 6:14, 8:12, 9:11, 11:9, 12:8, 14:6, 15:5) repeated measures ANOVA. The analysis of RT showed that participants’ mean RT varied depending on the relationship between the proportion (c, presented in the visual display) and the numerical reference, that is, close or far away from the target proportion. A graphical representation for the group average RTs for numerical quantifiers is outlined in Fig. 2. RT differences between quantifiers “at least” and “at most” for numerical references “seven” and “thirteen” are displayed in parts A and B in Fig. 2, as well as RT differences between the two numerical references, that is, “seven” and “thirteen” for each quantifier “at least” and “at most” separately are sketched in graphs C and D in Fig. 2. The detailed analysis of the main


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Fig. 2. Numerical Quantifiers (RT): Mean response time (RT) (with standard deviation) for each numerical reference “seven” (A) and “thirteen” (B) separately for both quantifiers, as well as Mean RT for each quantifier “at least” (C) and “at most” (D) separately. Statistically significant effects are marked with * in the graphs.

and interaction effects as well as the post hoc comparisons are described in Supplementary Material S1 and S2. The overall RT pattern showed an asymptote for proportions further away from the numerical reference “seven” and “thirteen” as compared to the proportions closer to the numerical reference. Participants took longer, on average, to respond to proportion 8:12 (c = 8) as compared to proportion 6:14 (c = 6) for quantifiers with numerical reference “seven” (cf. graph A in Fig. 2). However, this pattern was reversed for numerical reference “thirteen,” where participants took slightly longer on average, for proportion

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12:8 (c = 12) than proportion 14:6 (c = 14), namely, on “at least” as compared to “at most” (cf. graph B in Fig. 2). Therefore, to precisely investigate the influence of proportions closer to the numerical reference while evaluating numerical quantifier sentences, we selected the proportions at a distance of 1 and 2 to the target proportions (7:13 and 13:7). A repeated measures ANOVA with Quantifier (2 levels: at least, at most), Numerical Reference (2 levels: seven, thirteen), and Proportion (4 levels: 5:15, 6:14, 8:12, 9:11 for the numerical reference “seven” and 11:9, 12:8, 14:6, 15:5 for the numerical reference “thirteen”) as within subject factors was computed for the RT data. Statistically significant main effects of Numerical Reference F(1,23) = 5.85, p = .02 and Proportion F(3,69) = 9.92, p < .001 were observed. The main effect of Quantifier F (1,23) = 3.45, p = .08 was only marginally significant. The two-way interaction, Numerical Reference 9 Proportion F(3,69) = 9.91, p < .001 as well as the triple interaction effect, Quantifier 9 Numerical Reference 9 Proportion F(3,69) = 7.07, p = .001, revealed statistical significance. All other two-way interactions were not significant (p > .05). 3.1.1.1. Asymmetry index analysis for RT: An asymmetry was observed in the pattern of average RT for “at least” and “at most” with respect to close distances to both numerical references “seven” and “thirteen” (cf. Fig. 2). To examine this asymmetry, the “Proportion” factor was recoded into “Distance from the Numerical Reference,” further referred to as the “Distance” factor. This Distance factor was individually coded for the numerical references “seven” and “thirteen.” For the numerical reference “seven,” RTs for proportions closer to the numerical reference, that is, 5:15, 6:14, 8:12, and 9:11 (highlighted section of graph A in Fig. 2) were coded as distance 2, 1, +1, and +2, respectively. Similarly, for numerical reference “thirteen,” RTs for proportions closer to the numerical reference, that is, 11:9, 12:8, 14:6, and 15:5 (highlighted section of graph B in Fig. 2) were coded as distance 2, 1, +1, and +2, respectively. The RT difference between distance +1 and distance 1 as well as between distance +2 and distance 2 was calculated separately for each participant for all four numerical quantifiers (at least seven, at least thirteen, at most seven, at most thirteen). These asymmetry indices were obtained for both distances (1 and 2) using the following formula: asymmetry index ¼ 0:5 ½ðMean RT of þ dÞ ðMean RT of dÞ

for d ¼ 1; 2

A three-way repeated measures ANOVA using 2 (Quantifier: at least, at most) 9 2 (Numerical Reference: seven, thirteen) 9 2 (Distance: 1 and 2) as within subject factors and the asymmetry index as the dependent variable was carried out. Significant main effects of Quantifier F(1,23) = 6.71, p = .02 and Numerical Reference F(1,23) = 13.04, p = .001 were revealed, however, not of Distance F(1,23) = .01, p = .91. Two-way and three-way interactions were not present (all p > .05). This indicated that the bias away from a symmetric NDE, introduced by the numerical quantifier semantics, was not affected by increasing distance from the reference numerosity. Furthermore, we calculated an average asymmetry index by taking the average of the asymmetry indices of distances

S. Shikhare et al. / Cognitive Science 39 (2015) (A)

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Fig. 3. Mean asymmetry index (with standard deviation) for “at least” and “at most” for numerical references “seven” and “thirteen.” An individual asymmetry index was calculated using the formula “0.5 9 [(Mean RT of +d) (Mean RT of d) for d = 1, 2]” for each Numerical Quantifier (at least and at most) per Distance (1 and 2) for each Numerical Reference (seven and thirteen), displayed in A and B. Furthermore, the average asymmetry index across distances 1 and 2 was calculated for C.

1 and 2 for each quantifier per participant, since the distance factor was not statistically significant. Fig. 3C illustrates the average asymmetry index for the numerical quantifiers “at least” and “at most” against numerical reference “seven” and “thirteen.” We used one-sample t-tests to check whether there are differential biases in the asymmetry indices. The average asymmetry index was significantly different from zero after Bonferroni correction for all quantifiers except “at most thirteen” (cf. Table 2 for descriptive data). In addition, to examine whether the degree of the observed asymmetry was comparable, we reversed the sign of the individual participant’s asymmetry indices for quantifiers “at least thirteen” and “at most thirteen” because both quantifiers had negative average values (cf. Table 2 and Fig. 3C). With these values, we computed a 2 (Quantifier: at least, at most) 9 2 (Numerical Reference: seven, thirteen) repeated measures ANOVA. A statistically significant interaction effect Quantifier 9 Numerical Reference F(1,23) = 6.71, p = .02 was observed; however, the main effects were not significant (p > .05). Paired ttests showed significant differences only between “at most seven” and the reversed sign asymmetry index value for “at most thirteen” t(23) = 2.17, p = .01, and between the reversed sign asymmetry index values of “at least thirteen” and “at most thirteen” t(23) = 2.86, p = .01. 3.1.2. Accuracy A similar 2 9 2 9 8 repeated measures ANOVA using Quantifier, Numerical Reference, and Proportion as within subject factors was computed to assess participants’ success rate on numerical quantifiers. Accuracy results were in total agreement with the RT results. Participants’ accuracy, for both numerical references “seven” and “thirteen,” was affected by the proportions closer to the target proportions (7:13 or 13:7) compared to those farther away from the target proportions. Fig. 4 outlines the accuracy scores for each numerical reference “seven” and “thirteen,” highlighting the accuracy differences between quantifiers (cf. graph A and B in Fig. 4), as well as for each quantifier “at least” and “at

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Table 2 Absolute asymmetry index values for RT, accuracy, and acceptability (ms) for all four numerical quantifiers Quantifier Response time (RT)

Accuracy

Acceptability

At At At At At At At At At At At At

least seven least thirteen most seven most thirteen least seven least thirteen most seven most thirteen least seven least thirteen most seven most thirteen

Asymmetry Index Sign + + + + +

Mean (SD) 60.99 81.67 90.25 30.45 0.01 0.12 0.08 0.02 0.39 0.25 0.33 0.22

(102.21) (127.81) (91.46) (130.37) (0.09) (0.16) (0.09) (0.21) (0.08) (0.10) (0.10) (0.13)

One-Sample t-test t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23) t(23)

= 2.92, p = .01 = 3.13, p = .01 = 4.83, p < .001 = 1.14, p = .26 = 0.48, p = .64 = 3.74, p = .001 = 4.84, p < .001 = 0.39, p = .70 = 22.78, p < .001 = 12.20, p < .001 = 15.78, p < .001 = 8.29, p < .001

most” for both numerical references separately (cf. graph C and D in Fig. 4). A detailed analysis of the significant main and interaction effects as well as the post hoc comparisons is explained in Supplementary Material S3. We selected the proportions at a distance of 1 and 2 to the target proportion, similar to the RT analysis (cf. section 3.1.1, paragraph 2). A 2 (Quantifier) 9 2 (Numerical Reference) 9 4 (Proportion) repeated measures ANOVA revealed statistically significant main effects of Quantifier F(1,23) = 6.26, p = .02, Numerical Reference F(1,23) = 40.89, p < .001, and Proportion F(3,69) = 8.05, p = .003. Two-way interactions Quantifier 9 Proportion F(3,69) = 12.66, p < .001, Numerical Reference 9 Proportion F(3,69) = 7.61, p = .004 were observed. The three-way interaction Quantifier 9 Proportion 9 Numerical Reference F(3,69) = 2.40, p = .08 was only marginally significant. 3.1.2.1. Asymmetry index analysis for accuracy: Using an identical coding procedure and asymmetry index formula, we derived asymmetry indices for all four numerical quantifiers for each participant with respect to the accuracy data (cf. Asymmetry index analysis for RT). A 2 (Quantifier) 9 2 (Numerical Reference) 9 2 (Distance) repeated measures ANOVA revealed a main effect of Quantifier F(1,23) = 22.47, p < .001 and Numerical Reference F(1,23) = 6.88, p = .02. The two-way interaction Numerical Reference 9 Distance F(1,23) = 12.15, p = .002 as well as the three-way interaction Quantifier 9 Numerical Reference 9 Distance F(1,23) = 4.38, p = .05 were observed. A graphical representation for the asymmetry index values for each distance is sketched in the Supplementary Material S4. We checked whether asymmetry indices for all four numerical quantifiers were significantly different from zero for distances 1 and 2, respectively, using one-sample t-tests (cf. Table A in Supplementary Material S4). To further assess the interaction effects, post hoc paired t-test comparisons for distances 1 and 2 were computed (cf. Table B in Supplementary Material S4). An average asymmetry index was calculated for each numerical quantifier, compiling data over distances 1 and 2. Main


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Fig. 4. Numerical Quantifiers (Accuracy): Mean accuracy (with standard deviation) for each numerical reference “seven” (A) and “thirteen” (B) separately for both quantifiers as well as mean RT for each quantifier “at least” (C) and “at most” (D) separately. Statistically significant effects are marked with * in the graphs.

effects of Quantifier and Numerical Reference were statistically significant (p < .05), while the interaction effect revealed only marginal significance (p < .07) in a 2 (Quantifier) 9 2 (Numerical Reference) repeated measures ANOVA. Table 2 reports the descriptive statistics for the average asymmetry indices as well as the one-sample t-test results. 3.1.3. Acceptability To explore the responses given (accept or reject) for each numerical quantifier sentence, we computed a repeated measures ANOVA using Quantifier (2 levels: at least, at

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most), Numerical Reference (2 levels: seven, thirteen), and Proportion (8 levels: 5:15, 6:14, 8:12, 9:11, 11:9, 12:8, 14:6, 15:5). This analysis revealed a statistically significant main effect of Proportion F(7,161) = 4.45, p < .001, as well as two-way and a three-way interaction effects (all p < .05). The details about simple main and interaction effects are mentioned in Supplementary Material S5. A crossover pattern in the acceptability ratings at both target proportions “seven” and “thirteen” was observed. Participants’ level of mean acceptability for “at least seven” and “at most seven” crossed over at the target proportion 7:13 (c = 7), while a complementary, reversed crossover effect was observed at the target proportion 13:7 for “at least thirteen” and “at most thirteen.” A graphical illustration of acceptability ratings for numerical quantifiers is displayed in Fig. 5. 3.1.3.1. Asymmetry index for acceptability: We derived asymmetry indices for acceptability ratings for both distances (1 and 2) for all four quantifiers, similar to RT and accuracy analyses. A three-way repeated measures ANOVA using 2 (Quantifier) 9 2 (Numerical Reference) 9 2 (Distance) as within subject factors was carried out. Significant main effects of Quantifier F(1,23) = 464.68, p < .001 were observed but not of Numerical Reference F(1,23) = 0.52, p = 0.48 and Distance F(1,23) = 2.25, p = 0.15. Two-way interactions Quantifier 9 Numerical Reference F(1,23) = 40.87, p < .001 and Quantifier 9 Distance F(1,23) = 83.49, p < .001 were significant but not Numerical

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Fig. 5. Mean acceptability ratings (with standard deviation) for “at least seven” versus “at most seven” are shown in (A) and “at least thirteen” versus “at most thirteen” are shown in (B). Complementary acceptability curves are observed for numerical quantifiers; participants when accepting “at least seven” tend to reject “at most seven” which is an identical pattern for quantifiers with numerical reference “thirteen.” With respect to polarity, “at least seven” in (A) shows the mirror image of “at most thirteen” in (B). Similarly, “at most seven” in (A) shows a mirror pattern of “at least thirteen” in (B). Statistically significant effects are marked with * in the graphs.


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Reference 9 Distance F(1,23) = 0.01, p = .92. The three-way interaction Quantifier 9 Numerical Reference 9 Distance F(1,23) = 5.99, p = .02 revealed significance. First, we checked if the asymmetry indices for acceptability decisions were significantly different from zero for both distances using one-sample t-tests. All quantifiers showed statistical significance for both distances 1 and 2 (cf. Table A in Supplementary Material S6). Post hoc t-tests comparisons (Bonferroni corrected) were computed to test for differences between the asymmetry indices for distances 1 and 2 (cf. Table B in Supplementary Material S6). We further calculated an average asymmetry index by taking the average of the asymmetry indices of distances 1 and 2 for each quantifier per participant. The main effect of Quantifier F(1,23) = 462.99, p < .001 and the interaction effect of Quantifier 9 Numerical Reference F(1,23) = 40.86, p < .001 were statistically significant in a 2 (Quantifier) 9 2 (Numerical Reference) repeated measures ANOVA. One-sample t-tests confirmed that the average asymmetry indices for all quantifiers were significantly different from zero (cf. Table 2). The asymmetry indices for distances 1 and 2 as well as the average asymmetry index are graphically outlined in Supplementary Material S6. 3.2. Proportional quantifiers: “Many” and “few” 3.2.1. Response times To evaluate the specific effects of the proportional quantifiers in case of systematically varied proportions, we computed a repeated measures ANOVA with Quantifier (2 levels: many, few) and Proportion (8 levels: 5:15, 6:14, 8:12, 9:11, 11:9, 12:8, 14:6, 15:5) as within-subject factors for the RT data. Greenhouse-Geisser adjustment was applied wherever necessary. This analysis revealed a significant main effect of Quantifier F(1,23) = 32.64, p < .001 as well as Proportion F(7,161) = 35.92, p < .001. The interaction effect Quantifier 9 Proportion was not significant F(7,161) = 1.17, p = .33. Slower responses were given for proportions closer to one (8:12 and 9:11), however, not for the inverse proportions (11:9 and 12:8). Both “many” and “few” followed a similar pattern but responses to “few” took longer as compared to “many” for all eight proportions. An overview is displayed in Fig. 6A, where average RT is plotted against the number of yellow (TarCol) circles. 3.2.2. Acceptability To investigate the acceptability pattern for “many” and “few,” we computed a repeated measures ANOVA with Quantifier (2 levels: many, few) and Proportion (8 levels: 5:15, 6:14, 8:12, 9:11, 11:9, 12:8, 14:6, 15:5) as within subject factors and frequency of accept responses as the dependent measure. Main effects of both Quantifier F(1,23) = 77.75, p < .001 and Proportion F(7,161) = 3.12, p = 0.04, as well as the interaction effect Quantifier 9 Proportion F(7,161) = 281.49, p < .001, revealed statistical significance. Participants’ responses are graphically outlined in Fig. 6B. Post hoc paired t-tests comparisons showed significant differences between “many” and ‘few’ at all eight levels of proportions except 8:12 and 9:11 (p < .001). Proportion Accept in Fig. 6B indicates that participants agreed that the sentence and the visual display were matched. Mirror images

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Fig. 6. Mean response times (RT with standard deviation) for proportional quantifiers “many” and “few” are shown against the number of target color (TarCol) circles in (A); “few” takes longer than “many” for all eight estimation parameters c; Higher RTs for proportions (yellow: blue) closer to 1. Mean acceptability ratings (with standard deviation) for “many” and “few” are shown in (B). Mirror response patterns are revealed for positive quantifier “many” in comparison with negative quantifier “few.” Statistically significant effects are marked with * in graph B.

of results were observed for “many” and “few,” showing a crossover pattern around proportions 8:12 and 9:11. This suggests that participants started accepting “many” already at proportion 8:12 while rejecting “few” from the same proportion on.

4. Discussion The present study aimed to uncover the processing of numerical and proportional quantifiers via the ANS in a semantic truth value judgment task. In the experimental task, adults were asked to evaluate short quantifier sentences about visual displays comprising systematically varied proportions of yellow and blue circles. Our study has three main findings. First, numerical estimation and comparison are crucial in the evaluation of both numerical and proportional quantifier sentences. Second, numerical reference information, explicitly contained in the semantic expression, aids in selecting the magnitude comparison strategy when processing both types of quantifiers. And third, quantifier semantics per se bias the response direction (either “accept” or “reject”) during the evaluation process. To address our question of quantifier comprehension on the basis of the results obtained, the following discussion will concentrate on the core numerical processing aspects necessary for understanding and processing quantifiers. We will also elaborate on the influence of quantifier semantics and their implicated polarity in the evaluation process.


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4.1. Numerical distance effect in quantifier comprehension The results revealed the presence of the NDE while evaluating numerical and proportional quantifier sentences. This effect indicates the employment of the ANS. Earlier empirical studies in the quantification literature have linked precise magnitude knowledge to quantifier comprehension using the computational approach of quantifier verification (Hackl, 2009; McMillan et al., 2005, 2006; Szymanik & Zajenkowski, 2010; Zajenkowski et al., 2011). Only a few studies proposed the default involvement of the ANS in evaluating quantifiers (Heim et al., 2012; Lidz et al., 2011; Pietroski et al., 2009). Our results are in agreement with this latter approach, considering approximation strategies to verify linguistic sentences about quantity. They also corroborate the results put forth in Heim et al. (2012), who suggested the utilization of a similar strategy in an fMRI setting. Pietroski et al. (2009) manipulated the visual stimulus composed of yellow and blue circles, making it easy for participants to select simple verification strategies like one-to-one correspondence. Despite this visual cue manipulation, participants’ performance was affected by the ratio between two numerosities to be compared, resulting in a Weber fraction of w = 0.3 in adults. In our task, the NDE was observed in both types of quantifier sentences, endorsing the ANS model. Participants seem to have activated the ANS to use the cardinality estimation and comparison strategy for comprehension and further processing of quantifiers. In addition to varying the visual display with respect to proportion, we also manipulated the linguistic input with respect to cognitive processing difficulty in the current task. Therefore, carrying out the current task was more demanding than in the above-mentioned study by Pietroski et al. (2009). However, the obtained results are reassuring as far as the use of ANS for quantifier comprehension is concerned. Let us consider the numerical quantifiers first. The statistical analysis revealed a NDE: Participants’ processing demands increased when they had to respond to quantifier sentences with “at least seven” and “at most seven” for proportions closer to the reference, that is, 5:15, 6:14, 8:12, and 9:11, as compared to proportions far away from the reference, that is, 11:9, 12:8, 14:6, and 15:5. An identical pattern, that is, increased RTs and lowered accuracies, was observed for sentences with “at least thirteen” and “at most thirteen” (cf. Fig. 2, Fig. 4 and Supplementary Material S2). The NDE is a key property for magnitude comparison, observed in several investigations using non-symbolic numerical stimulus material (Feigenson et al., 2004). So far, no studies in quantification have put forth evidence of a NDE observed, while evaluating numerical quantifiers with higher cardinalities. Precise versus approximate judgments were targeted in a few studies (McMillan et al., 2005; Szymanik & Zajenkowski, 2010); however, no study has reported results for the individual proportions involved but has aggregated over proportions instead to report on overall RT or accuracy for the quantified sentence. Our study has a different focus on the NDE in evaluating numerical quantifiers with larger cardinalities—beyond the subitizing range—like “seven” and “thirteen,” proposing that participants used numerical estimation and comparison as reliable strategies to evaluate quantified sentences. McMillan et al. (2005) demonstrated neural differences between precise versus approxi-

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mate judgments while verifying the numerical quantifier “at least three.” However, as explained in the introduction section 1.1, the numerosity “three” belongs to the subitizing range, which makes the translation of the cardinality to its internal representation effortless and easy. Their results, together with ours, imply that indeed the comprehension process for numerical quantifiers with small numerosities is distinct from the one for numerical quantifiers with larger numbers, at least as far as estimation processes are involved. Our findings demonstrate the saliency of higher numerosities referred to in the numerical quantifier expressions in a time-constrained verification task. Here, we want to argue that the magnitude information provided in the numerical quantifier acts as a cue to access to the internal MNL. We believe that when participants attended to the auditory sentence, they to used the numerical reference information, that is, “seven” or “thirteen,” to gain access to the MNL. Both these numerosities are relatively large. Therefore, under time pressure, participants need to approximate the cardinality to generate a symmetric activation function around the reference number. Furthermore, when participants encountered the visual display with two sets of numerosities (yellow and blue circles), they needed to estimate the cardinality of target color circles by activating the ANS and represent that cardinality in an approximate fashion on the internal MNL. We observed that when c was closer to the numerical reference (seven or thirteen), performance deteriorated, since there was more overlap between the internal representations of these two numerical magnitudes (c and numerical reference). Conversely, when c was more distant from the numerical reference, performance improved, as the internal distributions of the two numbers were well separated from each other (Moyer & Landauer, 1967). An alternative explanation regarding the impact of cardinality used in numerical quantifiers is provided in the literature based on computational modeling (Szymanik & Zajenkowski, 2010) as well as self-paced counting studies using numerical quantifiers, also referred to as superlative quantifiers (Jorie, Varvoutis, & Hackl, 2008).4 In both these approaches, the number provided in the quantifier sentence affects the counting stage rather than the total number of elements presented in the stimulus: the higher the number, the longer it takes to count. A less robust finding, namely, the problem size effect, was observed in the numerical comparison used in the evaluation of numerical quantifier sentences. Performance is hampered with higher magnitudes, resulting in longer processing time and lower accuracies (Moyer & Landauer, 1967). Participants performed less well when having to discriminate between two greater numerosities (12 and 13) as compared to smaller numerosities (6 and 7) while verifying numerical quantifiers. This effect manifested itself more consistently in the accuracies than in the RT data (cf. graph A and B in Fig. 4). This could be explained in terms of “thirteen” being a larger magnitude; the initial encoding and translating into the internal representation might be more demanding. Consequently, the subsequent numerical discrimination process would be harder when the cardinality was closer to “thirteen” as compared to far away cardinality. We also observed a NDE for proportional quantifiers “many” and “few.” Participants took longer on proportions closer to one (8:12, 9:11, 11:9 and 12:8) where the


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numerical distance between two numbers was small as compared to proportions farther away from one (5:15, 6:14, 14:6, 15:5) (cf. Fig. 6A and Supplementary Material S2). Our results are in agreement with a few studies using proportional quantifiers “many,” “most,” and “more than half” (Heim et al., 2012; Pietroski et al., 2009; Zajenkowski et al., 2013). It was shown that proportional quantifiers led to a similar effect, despite the use of a much larger total number of 50 circles with varying proportions between yellow and blue circles (Heim et al., 2012). Zajenkowski et al. (2014) reported distance effects, when evaluating sentences with “more than half” from the computational perspective of quantifier verification. The presence of distance effects even in case of counting suggests that the integration of numerical information required for comparing two numerosities may lead to these effects. In the current study, poor performance was observed for proportions closer to but only below one (8:12 and 9:11). Performance improved again quite quickly for proportions larger than 1 (11:9 and 12:8). One intriguing question related to the nature of performance for proportions closer to but beyond one (11:9 and 12:8) is raised here. A possible explanation for this change in processing times would relate to our tendency of comprehending semantic information about quantity. We tend to translate the linguistic information into some internal numerical quantity representation. Therefore, as hypothesized, we think that participants use one of the possible strategies, namely, generating a subjective numerical criterion to judge the truth value of the sentence with proportional quantifiers like “many” or “few.” This would imply that each individual may implement a (slightly) different internal numerical criterion to verify these expressions accurately, depending on the individual concept of the quantifier. One proposition might be that this internal reference corresponds to an activation distribution centered closer to half of the overall number of circles (20) in the current task (namely, between 8 and 10) across all participants. To illustrate, in the sentence “Many of the circles are yellow,” if c = 5, participants might compare this to an internally generated more or less stable numerical criterion to respond to the quantified sentence. Although the view that participants establish an internal numerical criterion for the comparison and evaluation of proportional quantifiers may be a simplification, supporting evidence comes from the pattern of mean RTs and acceptability (cf. Fig. 6). Higher variability is observed for the proportions 8:12 and 9:11, supporting our interpretation that participants tend to compare the estimated numerosity to an internal criterion while evaluating “many” and “few.” Since the numerosities 8 and 9 are close to the established internal criterion, the numerical comparison process is more challenging (cf. Supplementary Material S2). Supporting evidence for our speculation comes from ongoing work in healthy controls, who were assessed in terms of the internal numerical criterion for proportional quantifier “many” and “few.” Then, participants were reinforced with an internal criterion only on one proportional quantifier (e.g., “many”). But, when the same individuals were asked to evaluate sentences with “few,” performance was influenced by the numerical criterion learned and established for “many” (Heim et al., 2013).

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4.2. Impact of quantifier semantics and polarity So far, we have discussed the NDE observed in both numerical and proportional quantifiers, but the data were also suggestive of an asymmetric distance effect. Here, we propose that quantifier semantics per se bias the overall numerical comparison process while evaluating quantified sentences (Lidz et al., 2011). Distance effects suggested that we represent the numerical reference (provided externally or generated internally) and the estimated numerosity on the internal MNL. For the further comparison process, we propose that the quantifier semantics introduce a bias in accessing the MNL. By this, we mean that the quantifier expression specifically contains information about where to access the MNL, that is, from left to right or vice versa during the evaluation process. We illustrate our interpretation for numerical quantifiers. One primary explanation is the congruency effect between quantifier semantics and their associated numerical representation, that is, we tend to relate “at least” with “large” numbers (Nuerk et al., 2004). The colloquial meaning of “at least seven” is “a minimum of seven or more.” This guides us to concentrate on numerosities larger than the numerical reference and therefore access the MNL from left to right to accept the quantified sentence. Let us consider an example from our experimental task. If c = 5, then participants need to compare two internal distributions, that is, the internal representation of the numerosity 5 to the internal representation of numerical reference “seven.” It is evident that distributed representation for 5 is not so far away from the externally provided reference “seven.” But the quantifier semantics “at least” biases the internally distributed representation of the reference “seven” toward larger numbers (resulting in a left skewed activation distribution) in line with the general congruency tendency. Eventually, the symmetric activation density distribution for 5 has less overlap with the left skewed distribution of the reference “seven,” which makes the decision easier. Interestingly, we observed similar asymmetry deviations from zero for both “at least seven” and “at least thirteen” (cf. Fig. 3), suggesting that participants maintain a similar strategy for “at least.” To the contrary, a similar congruency effect is evident for “at most,” that is, we tend to associate “at most” with “small” numbers (Nuerk et al., 2004). The numerical quantifier “at most seven” corresponds to “a maximum of seven or less.” Hence, for “at most seven,” we focus attention on smaller numbers, below the numerical reference, and thus there is biased access to the MNL in the opposite direction (from right to left) from the numerical reference. Performance is better when c is lower than the numerical reference. This explains the larger asymmetry observed for “at most seven” but is inadequate for interpreting “at most thirteen” (cf. Fig. 3). The absence of significant asymmetry for “at most thirteen” in comparison with “at most seven” can be explained in the following way: As mentioned above, “at most” is more congruent with “small,” therefore the general tendency is to tune in for smaller numerosities for “at most” sentences. When sentences with “at most” are combined with the higher numerosity “thirteen,” incongruency is created between quantifier semantics and numerical magnitude information. From the numerical perspective, a problem size effect is already present when accurately evaluating these sentences. Therefore, there might be interference at play between the quantifier


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semantic bias and the problem size effect for “at most thirteen.” Thus, there was a reduced degree of asymmetry for “at most thirteen” as compared to “at most seven.” Another intriguing finding is the crossover effect found for acceptability ratings at the numerical references “seven” and “thirteen” (cf. Fig. 5). Here, participants’ response direction is biased by the quantifier semantics either to verify or falsify the judgment. This result indicates that quantifier semantics additionally aid in searching the required response set to evaluate the quantifier expression. This interpretation requires further explanation. We observed complementary acceptability patterns for “at least seven” and “at most thirteen” as well as for “at most seven” and “at least thirteen.” Let us consider the number range from 5 up to 15 used to create proportions between yellow and blue circles. To evaluate sentences with “at least seven,” one needs to search a larger set for accepting the sentence. In the current task, participants are provided with a larger set of cardinalities above the numerical reference “seven” to accept sentences with “at least seven.” In contrast, participants are provided with a larger set of cardinalities below the numerical reference “thirteen” to accept sentences with “at most thirteen.” Therefore, in both sentences “at least seven” and “at most thirteen,” participants need to search a larger set to verify judgments, which is clearly evident in our results. Following the similar interpretation, participants are provided with a larger rejection set beyond the numerical reference for “at most seven” and a larger rejection set below the numerical reference for “at least thirteen.” Participants need to focus on searching instances to reject, which is observed in our data for “at most seven” and “at least thirteen.” Therefore, in these two cases, participants search a larger set to falsify judgments. Another possible explanation is put forth in the literature referring to the monotonicity aspect of quantifiers. It is suggested that decreasing monotonicity quantifiers like “at most” are harder than increasing monotonicity quantifiers like “at least.” Precisely, in a truth conditionality experiment, authors put forth that quantifiers like “at least” are faster to verify than falsify which is the converse for quantifiers like “at most.” Monotone decreasing quantifiers are easier to falsify than to verify (Jorie et al., 2008). In a similar fashion, we propose that for proportional quantifiers “many” and “few” the quantifier semantics bias the response direction, that is, aid in deciding whether to accept or reject a specific judgment. From a numerical perspective, participants have to represent the quantifier on an internal MNL with “many” resp. “few” coupled with larger resp. smaller numbers in a given number range on the MNL. A congruency effect is observed, that is, participants associate the quantifier “many” with large numbers (rightward numbers) to accept the quantifier sentence. Supporting evidence is provided by the acceptability data where participants show a tendency to accept “many” already when c exceeds 8 (proportion 8:12). Similarly, for “few,” the semantics bias our attention toward small numbers. Thus, participants associate the verbal expression “few” with the leftmost cardinalities on the MNL. The acceptability pattern for “few” supports our interpretation because participants accept the leftmost cardinalities but reject the remaining ones (cf. Fig. 6B) (Nuerk et al., 2004). Our results are in line with the effects observed in visual attention studies focusing on natural language quantifiers. They observed correspondence between high numerosity quantifiers (many) with large numbers of focus objects presented in the stimuli (Coventry, Cangelosi, Newstead, Bacon, & Rajapakse, 2005).

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Interestingly, the crossover effect was also evident for proportional quantifiers “many”–“few” at the proportion 8:12, where the acceptability rating profile for “many” intersects with the one for “few” to follow an opposite acceptability pattern. For proportions 5:15 and 6:14, participants seem to be quite comfortable in rejecting “many” and accepting “few” judgments, respectively. However, when the proportion gets closer to one, that is, the number of yellow circles gets closer to the number of blue circles, processing demands are raised whether to accept or reject the quantified sentence (cf. Fig. 6A). Although proportional quantifiers lend themselves to quite subjective interpretations with respect to the exact internal representation of a numerosity, the current acceptability data are supportive of our interpretation that participants adopt an internal numerical criterion for comparing numerical estimates and thus take longer for proportions closer to one, that is, 8:12 and 9:11. An interesting finding related to the evaluation of quantifiers can be explained in terms of the polarity construct. First, our results highlight that there is correspondence between the polarity adjectives “many-right” (both linguistically unmarked) as well as “few-left” (both linguistically marked). Similarly, there is correspondence for numerical quantifiers “at least-large” as well as “at most-left” (Nuerk et al., 2004). This markedness correspondence was also evident in the response hand assignment because participants were asked to give “Accept” judgments with the right hand, but “Reject” judgments with the left hand (Nuerk et al., 2004). However, in our data, it was also evident that positive quantifiers (many, at least) were comparatively easier to process than their negative counterparts (few, at most). The accuracy data for numerical quantifiers indeed suggest that for “at most,” participants found it difficult to precisely verify the judgments in comparison with “at least” for both numerical references. Because “at most” comprises a negation in its meaning, it requires an extra processing step of dealing with that negation before deciding on the response (Geurts, 2003; Routh, 1994). This interpretation is corroborated by several psycholinguistic studies, exploring the polarity of semantic expressions (Geurts, 2003; Proctor & Cho, 2006). With respect to proportional quantifiers, it could be possible that participants process “few” as “not many,” suggesting that they might tend to negate the positive expression, which will then take more time (Geurts, 2003; Heim et al., 2012; Just & Carpenter, 1971). Another argument comes from the linguistic comprehension literature, where it is proposed that participants focus on the reference set for “many” (positive quantifier) and on the complement set for “few” (negative quantifier). For instance, in the sentence “many of the circles are yellow,” participants would invariably focus on the number of yellow circles (reference set) while in a sentence “few of the circles are yellow,” participants would tend to focus on the non-yellow, that is, blue circles (complement set). However, this seems less likely for our study, because if participants used such a strategy, they would follow a subtraction operation (reference set–complement set), which might be very demanding in a time-constrained task. Future investigations evaluating comparative polarity pairs, namely, “at least seven–at most six” as well as “many–few” in contrast to “more than half–less than half,” would be interesting to further pinpoint potential differences in numerical estimation and comparison when linking numbers and language in quantifier expressions.


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5. Conclusion This paper explored how adult participants quickly evaluate auditorily presented numerical and proportional quantifier sentences about visually presented numerosities. We showed that numerical estimation and comparison strategies are used, which are biased by the quantifier semantics involved. With respect to quantification, our results highlight numerical estimation and comparison processes being at work in case of fast quantifier sentence judgments, suggesting a link between language and number processing for accurate quantifier comprehension and evaluation. We showed that numerical estimation is crucial in evaluating quantifier sentences under time pressure. In addition, we propose that, for numerical quantifiers, numerosity comparison depends on a fixed externally specified numerical reference, whereas comparisons relied on an internal subjective numerical criterion for proportional quantifiers. However, the NDEs observed for both types of quantifiers are also biased by semantic (polarity) aspects of the quantifiers while relating the linguistic information to the internal MNL. In both types of quantifiers— numerical and proportional—reference information was used to access the MNL with this access being further biased by quantifier semantics, determining the direction of the response bias.

Acknowledgments This study was supported by the Deutsche Forschungsgemeinschaft (DFG, IRTG 1328) and a RWTH scholarship supporting Sailee Shikhare. We are grateful to Anjan Chatterjee at the University of Pennsylvania, Philadelphia, for helpful discussions on the results. We thank Andre Knops for advice with the experimental design, Uli Heuter, from the Audiovisuelles Medienzentrum [AVMZ] at the RWTH University Hospital, and Stefanie Jung for their assistance in recording and preparing auditory stimuli for the experimental task. We are also very grateful to the reviewers of the paper for many very helpful suggestions.

Conflict of interest statement We confirm that all authors declare no financial or commercial conflict of interest associated with the publication of this article.

Notes 1. A large amount of literature is available on quantification, referring to quantifier terms in several distinct terminologies. Cliff (1959), Erev and Cohen (1990), and Brun and Teigen (1988) elaborate on context dependency and ambiguity as well as

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distinctions between verbal and numerical probabilities, topics outside the scope of the current study. 2. The computational model of quantifier verification assumes that the cognitive complexity of quantifiers can be directly assessed depending on the automata used. It was suggested that logical quantifiers (all, some) rely on an acyclic two-state finite automaton, the numerical quantifiers (at least n) depend upon an acyclic finite automaton with the number of states depending on n and the parity quantifiers (an even number of) depend upon two-state finite automata with loops. Only the proportional quantifiers (more than half) depend on a pushdown automaton, which additionally require working memory resources. These models are well explained in Szymanik (2007) as well as empirically studied in the dissertation by Szymanik (2009) and in Szymanik and Zajenkowski (2009, 2010) as well as Zajenkowski et al. (2013). However, because this model relies on a counting-based algorithmic strategy, our experimental design calling for fast responses and furthermore our own results are not entirely in agreement with this model. 3. It is evident that verification tasks provide some information about comprehension but not all of it. For instance, in the case of comparative (more than) versus superlative (at least) quantifiers, people might use similar verification strategies but the process of comprehension might be more complex for superlative quantifiers. In addition, recent evidence from Szymanik and Zajenkowski (2013) suggests that monotonicity effects go in diverging directions with respect to comprehension and verification tasks, depending on the cognitive task. Therefore, future scientific investigations should clarify the contribution of various cognitive processes in similar experimental tasks. 4. There are some publications regarding the differences between comparative (more than) versus superlative (at least) quantifiers in verification tasks (cf. Hackl, 2000; PhD dissertation, Geurts and Nouwen, 2007; and Geurts, Katsos, Cummins, Moons, and Noordman, 2010). In the current study however, we do not concentrate on these differences but are primarily interested in studying these scalar quantifiers from a numerical cognition perspective.

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Supporting Information Additional Supporting Information may be found in the online version of this article: Supplementary Material S1. Response time. Supplementary Material S2. Depiction of differences among proportions. Supplementary Material S3. Accuracy. Supplementary Material S4. Asymmetry index for accuracy. Supplementary Material S5. Acceptability. Supplementary Material S6. Asymmetry index for acceptability.

The processing of polar quantifiers, and numerosity perception.

Children's interpretations of general quantifiers, specific quantifiers, and generics.

Energetic arousal and language: predictions from the computational theory of quantifiers processing.

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Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats.

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