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On the Identification of Quantifiers’ Witness Sets: A Study of Multi-quantifier Sentences

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Abstract

Natural language sentences that talk about two or more sets of entities can be assigned various readings. The ones in which the sets are independent of one another are particularly challenging from the formal point of view. In this paper we will call them ‘Independent Set (IS) readings’. Cumulative and collective readings are paradigmatic examples of IS readings. Most approaches aiming at representing the meaning of IS readings implement some kind of maximality conditions on the witness sets involved. Two kinds of maximization have been proposed in the literature: ‘Local’ and ‘Global’ maximization. In this paper, we present an online questionnaire whose results appear to support Local maximization. The latter seems to capture the proper interplay between the semantics and the pragmatics of multi-quantifier sentences, provided that witness sets are selected on pragmatic grounds.

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Notes

  1. See Peters and Westerståhl (2006) for a survey on possible monotonicities featured by GQs and Szymanik and Zajenkowski (2013) for a recent computational account.

  2. That formula is obtained from (9) by simply replacing the quantifier condition “\(3!_y\)(star’(\(y\)), \(P_2\)(\(y\)))” with “\(2!_y\)(star’(\(y\)), \(P_2\)(\(y\)))”.

    Fig. 1
    figure 1

    A model for sentences (13.a–b)

    Full size image

  3. Landman (2000) does not consider Branching Quantifier readings. (14) is a cumulative reading among a set of exactly two dots and a set of exactly two stars. Cartesian products are only special instantiations of CONNECT’s extension.

  4. Fillers with obvious truth values (e.g., “In the figure, there are eight boys”) were used to prevent subjects from using some simplified strategy that could only work with specific experimental target items. The full list of fillers is not reported in this paper, because their results were not stored in the database.

  5. We privileged the pragmatic factor about color over the other one. In our view, the former mostly favor Local reading. Thus, it is more important to trial its effect. In the light of this, we think it is fine to leave tuples that include only trials without relevant arrangements of the items.

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Acknowledgments

The authors would like to thank Lucas Champollion for suggestions to the previous versions of the online questionnaire and an anonymous reviewer for fruitful comments. BM and JS were supported by a Vici Grant NWO-277-80-001. JS also acknowledges NWO Veni Grant 639.021.232.

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Authors and Affiliations

  1. Department of Computer Science, University of Turin, Turin, Italy

    Livio Robaldo

  2. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherland

    Jakub Szymanik

  3. Institute of Artificial Intelligence, University of Groningen, Groningen, The Netherland

    Ben Meijering

Authors
  1. Livio Robaldo
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  2. Jakub Szymanik
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  3. Ben Meijering
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Correspondence to Livio Robaldo.

Results of the Questionnaire in Polish, English, and German

Results of the Questionnaire in Polish, English, and German

As mentioned at the beginning of Sect. 4, in order to check whether the results of the questionnaire apply to languages other than Italian, we translated the questionnaire into three other languages (Polish, English, and German). 809 additional participants answered to the questionnaire: 415 Polish, 305 English, and 89 German native speakers.

Table 9 reports the answers of the 809 non-Italian subjects.

Table 9 Evaluation of the eight trials by Polish, English, and German native speakers
Full size table

Given the results shown in Table 9, it seems that language does not affect the interpretation of the sentences in our questionnaire. The proportions are by and large the same found for Italian. Nevertheless, we cannot ignore the fact that the meaning of some quantifiers differ across languages and that we cannot expect to obtain the same meaning by translating quantifiers literally.

For instance, with respect to the sentence in Trial 4 (“Più di tre ragazzi hanno mangiato la maggior parte delle pizze”, translated in the English version of the questionnaire as “More than three boys ate most pizzas”), an anonymous reviewer suggested that in English “most” may have a proportional or a relative reading.

According to our informants and the available literature, e.g. (Hackl 2009; Krasikova 2011; Szabolcsi 2013), the proportional-relative ambiguity does not exist in English. The literature assumes that “most books” by itself is always proportional and “the most books” is always relative.

Indeed, such ambiguity does occur in German, but not in Italian or Polish. In our view, such potential ambiguity could not significantly affect our overall results. It seems most likely that English subjects interpreted “most” as a proportional determiner, especially, as it was explained very clearly in the instructions that trial sentences have the structure “X boys ate Y pizzas” and must be interpreted as “a group of X boys ate a group of Y pizzas”. Moreover, this is the fourth trial, which took place before subjects had evaluated three syntactically similar trials with clearly proportional interpretation.

To conclude, although our findings appear to be language-independent, in this paper we should consider reliable only the results for the Italian version of the questionnaire, while the evaluation of the trials in other language would need further empirical analyses.

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Robaldo, L., Szymanik, J. & Meijering, B. On the Identification of Quantifiers’ Witness Sets: A Study of Multi-quantifier Sentences. J of Log Lang and Inf 23, 53–81 (2014). https://doi.org/10.1007/s10849-014-9197-9

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