Quantifiers & Cognition

[ Course description | Lectures |Schedule | Assignments | Literature | Additional Materials ]

National Graduate Course in Philosophy

Stockholm University

November 22-26, 2010

Instructor: Jakub Szymanik

Office hours: Monday, Tuesday and Thursday 10-12

Course description:

Non-human animals are able to count and represent quantities, but reasoning with linguistic expressions of (relative) quantities (known as quantifiers) seems a uniquely human ability. Humans can understand, for example, sentences such as “Most linguists are logicians”, “Less than half of the cognitive neuroscientists are computer scientists”, and “At least 3 of the applicants are psychologists.” Humans can also assess the conditions that make such sentences true or false. While the cognitive bases of counting and quantity representations have been extensively studied (see e.g. Dehaene 1999), the cognitive processing of linguistically expressed quantities is far from being understood. Quantifier expressions occur whenever we describe the world, and communicate about it. Generalized quantifier theory studies the possible meanings and the inferential power of quantifier expressions by logical means. The classical version was developed in the 1980s, at the interface of linguistics, mathematics and philosophy. Until now, advances in "classical" generalized quantifier theory mainly focused on definability questions and their applications to linguistics (see Peters and Westerståhl 2006). However, applications of generalized quantifier theory in cognitive settings have also been developed (Moxey and Sanford 1993; Clark 1976). The lectures will overview the links between generalized quantifier theory and cognitive science. In particular, we will be concerned with extending generalized quantifier theory with computational aspects in order to draw and empirically test psycholinguistic predictions. One major focus will be computational complexity and its interplay with "difficulty" as experienced by subjects asked to verify quantifier sentences. To achieve this we will combine classical generalized quantifier theory (linguistics and mathematics) with newer generalized quantifier theory (computation and cognition).


There will be 5 lectures covering topics on the intersection of generalized quantifier theory and psycholinguistics. We will discuss various cognitive strategies triggered by quantifiers (e.g. precise counting and approximation), computational complexity of various quantifier constructions, cognitive difficulty of quantifier processing, and reasoning with quantifiers. As the lecture will progress I will upload slides containing detailed bibliography.

  1. Quantifiers and cognitive strategies, slides
  2. Quantifiers and approximation, slides
  3. Quantifiers and counting, slides
  4. Quantifiers and monotonicity, slides
  5. Quantifiers and computational complexity, slides


  1. Read about GQs asap.
  2. Read about finite- and push-down automata before Wednesday.
  3. Prepare some ideas for an experiment to present in the class on Wednesday.
  4. Read about computational complexity theory (P vs. NP) before Friday class.
  5. As a final assignment you should write a short (experimental) research proposal before December, 6.


  1. Van Benthem, Towards a Computational Semantics, in: Gärdenfors (ed.) `Generalized Quantifiers', pp. 31-37. PDF
  2. Geurts, Reasoning with quantifiers, Cognition, 86, 2003, pp. 223-251. PDF
  3. Geurts and Van der Slik, Monotonicity and Processing Load, Journal of Semantics, 22, 2005, pp. 97-117. PDF
  4. Gierasimczuk and Szymanik, Branching Quantification vs. Two-way Quantification, Journal of Semantics, 26(4), 2009, pp. 329-366. PDF
  5. Hackl, On the Grammar and Processing of Proportional Quantifiers: Most versus More Than Half, Natural Language Semantics, 17, 2009, pp. 63-98. PDF
  6. Halberda et al., The Development of `Most' Comprehension and Its Potential Dependence on Counting Ability in Preschoolers, Language Learning and Development, 4(2), 2008, pp. 99-121. PDF
  7. Just and Carpenter, Comprehension of negation with quantification, Journal of Verbal Learning and Verbal Behavior, 10(3), 1971, pp. 244-253. PDF
  8. Lidz et al., Interface Transparency and the Psychosemantics of most, Natural Language Semantics, in press. PDF
  9. McMillan et al., Neural Basis for Generalized Quantifier Comprehension, Neuropsychologia, 43, 2005, pp. 1729-1737. PDF
  10. Mostowski, Computational Semantics for Monadic Quantifiers, Journal of Applied Non-Classical Logics, 8, 1998, pp. 107-121. PDF
  11. Pietroski et al., The Meaning of `Most': semantics, numerosity, and psychology, Mind and Language, 24(5), 2009, pp. 554-585. PDF
  12. Szymanik, A Note on some Neuroimaging Study of Natural Language Quantifiers Comprehension, Neuropsychologia, 45 (9), 2007, pp. 2158-2160. PDF
  13. Szymanik, Computational Complexity of Polyadic Lifts of Generalized Quantifiers in Natural Language, Linguistics and Philosophy, in press. PDF
  14. Szymanik and Zajenkowski, Comprehension of Simple Quantifiers. Empirical Evaluation of a Computational Model, Cognitive Science, 34(3), 2010, pp. 521-532. PDF
  15. Szymanik and Zajenkowski, Quantifiers and Working Memory, Lecture Notes in Artificial Intelligence 6042, M. Aloni and K. Schulz (Eds.), Springer, 2010, pp. 456-464. PDF
  16. Szymanik and Zajenkowski,  Computational Approach to Monotonicity in Sentence-picture Verification, under review. PDF

Additional Materials:


  1. Monday, November 22, 13-15, room F347
  2. Tuesday, November 23, 13-15, room F363
  3. Wednesday, November 24, 10-12 and 13-15, room F355
  4. Thursday, November 25, 16-18, room D700 (changed!)
  5. Friday, November 26, 10-12, room F263. A joint meeting with Logic, Language, and Mind Seminar.