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Cognitive Processing of Quantifiers

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Quantifiers and Cognition: Logical and Computational Perspectives

Part of the book series: Studies in Linguistics and Philosophy ((SLAP,volume 96))

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Abstract

In this chapter, I set out to study the cognitive task of sentence verification. In particular, I investigate the cognitive capacity to recognize the truth-value of sentences with simple quantifiers (like ‘some’, ‘an even number of’, ‘more than 7’, ‘less than half’). As the exact strategies people use to verify quantifier sentences are mostly uncertain, I study optimal (computationally minimal) algorithms that can handle the tasks, i.e., semantic automata. I overview a number of cognitive science experiments on the processing of natural language quantifiers, which establish the psychological generality of the semantic automata model. The experiments include, behavioral measures of reaction times, accuracy, and working memory involvement, neurocognitive studies, experiments with schizophrenic patients, and linguistic analysis of quantifier distributions in corpora. The empirical data shows that the computational distinctions described in the previous chapter are reflected in human quantifier processing. However, there are many cognitive findings for the explanation of which we need a more fine-grained semantic theory, combining computational, logical, and linguistic insights with cognitive modeling.

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Notes

  1. 1.

    See also Zajenkowski et al. (2013).

  2. 2.

    Recall the discussion of inferential meaning from Sect. 2.4. The ordering here reduces computational complexity of the problem.

  3. 3.

    Compare with Column Pairs Sorted trials in the experiment of (Pietroski et al., 2009).

  4. 4.

    To make this processing step more cognitively plausible, we could introduce a probabilistic sampling to encode the models, but we leave this for future work.

  5. 5.

    All models were implemented in JAGS (Plummer 2003).

  6. 6.

    See also McMillan et al. (2006) for evidence on quantifier comprehension in patients with focal neurodegenerative disease, and Clark and Grossman (2007); Troiani et al. (2009) for a more general discussion.

  7. 7.

    Chapter 4 contains the mathematical details of the correspondence between quantifiers and automata.

  8. 8.

    See Baddeley (1986, 2003), Baddeley and Hitch (1974).

  9. 9.

    See Conway and Engle (1996), Daneman and Green (1986), Just and Carpenter (1992), King and Just (1991).

  10. 10.

    See also Szymanik and Zajenkowski (2011) for a more detailed comparison between proportional and parity quantifiers.

  11. 11.

    Cf. Zajenkowski et al. (2013).

  12. 12.

    For the POS tagging, we relied on the NLTK 3-gram tagger by Bird et al. (2009).

  13. 13.

    Of course, a really good explanation would try to connect computational complexity with Zipf laws via some information-theoretic analysis (cf. Piantadosi et al. 2011).

  14. 14.

    See Barwise and Cooper (1981), Peters and Westerståhl (2006).

  15. 15.

    For an example, see the discussion of collective quantifiers in Chap. 10 and a similar distribution analysis for various readings of reciprocal sentences in Sect. 8.6.2.

  16. 16.

    See Chap. 1.

  17. 17.

    See the next section for more details.

  18. 18.

    Alternatively, if we assume that subjects search through all the cars without quick perceptual identification of the target set, then the estimation of the verification times would depend on the order in which subjects process the elements. As we cannot know this, we would need to consider average cases (even more probable taking into account that the cars were randomly distributed). In such a situation the number of states would not change, but the number of transitions could vary between the minimal case (7 or 8 as in the first analysis) and the maximal case of ‘looking at’ all the cars, i.e., 15. As a result we would get \(11,5t+8s\) for true ‘more than 7’ and false ‘fewer than 8’. In the case of false ‘more than 7’ and true ‘fewer than 8’ the subjects would still need to check all 15 cars. As we will see later, such an alternative analysis is inconsistent with the obtained data.

  19. 19.

    Compare the discussion of the Interface Transparency Thesis in the next section.

  20. 20.

    Note that Lidz et al. (2011) argue that the number of dots (i.e., elements in the restrictor set) and the number of blue dots (i.e., elements in the intersection of the restrictor set and the scope) are computed directly while the number of nonblue dots is computed indirectly by means of subtraction. Therefore, they would have to predict the effects of monotonicity even with proportional quantifiers. However, under the Approximate Number System model they consider this prediction to not necessarily be guaranteed as the Approximate Number System talks about ‘estimating numbers’ rather than counting. Anyway, step-by-step counting was not even possible with the 150 ms time limit. See also the next section.

  21. 21.

    For example, Moxey et al. (2001) note that statements quantified by negative and positive quantifiers can put attentional focus on different subsets of the model and that the focus patterns have their roots in denial rather than in monotonicity (cf. Sanford et al. 1994).

  22. 22.

    One of the claims of Koster-Moeller et al. (2008) is that processing difficulty is affected by the number n mentioned in the sentences instead of the number N that determines the truth-value.

  23. 23.

    Tomaszewicz (2013) additionally presented evidence that participants are prompted to switch between verification procedures by a change in the linguistic input. In addition to ‘most’ she also tested subjects on verification tasks involving a different superlative quantifier in Polish and Bulgarian meaning ‘the biggest group of’.

  24. 24.

    The Weber fraction expresses the smallest numerical difference between two quantities that participants can distinguish. The Weber fraction of \(n_1\) vs \(n_2\) is calculated as \((n_1-n_2)/n_2\).

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  1. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Jakub Szymanik

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Szymanik, J. (2016). Cognitive Processing of Quantifiers. In: Quantifiers and Cognition: Logical and Computational Perspectives. Studies in Linguistics and Philosophy, vol 96. Springer, Cham. https://doi.org/10.1007/978-3-319-28749-2_5

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