4:5 (blue:yellow) “scattered random”

8. 4:5 (blue:yellow) “scattered random”

9. 1:2 (blue:yellow) “scattered random”

10. 4:5 (blue:yellow) “scattered pairs”

11. 9:10 (blue:yellow) “scattered random”

12. 4:5 (blue:yellow) “sorted columns”

13. 4:5 (blue:yellow) “mixed columns”

14. 5:4 (blue:yellow) “mixed columns”

15. 4:5 (blue:yellow)

16. Basic Design • 12 naive adults, 360 trials for each participant • 5-17 dots of each color on each trial • trials varied by ratio (from 1:2 to 9:10) and type • each “dot scene” displayed for 200ms • target sentence: Are most of the dots yellow? • participants answered ‘yes’ or ‘no’ by pressing buttons on a keyboard. • correct answer randomized, relevant controls for area (pixels) vs. number, yada yada…

17. better performance on easier ratios: p < .001

18. fits for Sorted-Columns trials to an independent model for detecting the longer of two line segments fits for trials (apart from Sorted-Columns) to a standard psychophysical model for predicting ANS-driven performance

20. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)} #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} {x:Dot(x) & Blue(x)} 1-To-1-Plus {x:Dot(x) & ~Blue(x)} Could it be that speakers use ‘most’ to access a complex 1-To-1-Plusconcept…but our task made it too hard to use a 1-To-1-Plusverification strategy?

21. better performance on components of a 1-to-1-plus task

22. Side Point Worth Noting…

23. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)} #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • if there are only two colors to worry about, blue and red, the non-blues can be identified reds

25. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)} #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • if there are only 2 colors to worry about, blue and red, the non-blues can be identified reds • the visual system can (and will) “select” the dots, the blue dots, and the red dots; so the ANS can estimate these three cardinalities • but adding more colors will make it harder (and with 5 colors, impossible) for the visual system to make enough “selections” for the ANS to operate on

28. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)} #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • adding alternative colors will make it harder (and eventually impossible) for the visual system to make enough “selections” for the ANS to operate on • so given the first proposal (with negation), verification should get harder as the number of colors increases • but the second proposal (with subtraction) predicts relative indifference to the number of alternative colors

29. better performance on easier ratios: p < .001

30. no effect of number of colors

31. fit to psychophysical model of ANS-driven performance

33. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • we need to think about the form-dependence of a priori knowledge, since given the proposed analysis… • speakers of English know a priori that: if most of the dots are blue, then the number blue dots exceeds the result of subtracting that number from the number of dots • this is so, even if speakers cannot put it this way • and speakers could fail to know that most of the dots are blue, even if they knew that: there are 8 blue dots, and 7 yellow dots, and 8 is more than 7

34. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • determiner/adjectival flexibility (for another day) I saw the most dots • mass/count flexibility Most of the dots are blue Most of the goo is blue

37. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • mass/count flexibility Most of the dots are brown Most of the goo is brown • are mass nouns disguised count nouns? #{x:GooUnits(x) & BlueUnits(x)} > #{x:GooUnits(x)} − #{x:GooUnits(x) & BlueUnits(x)}

39. discriminability is BETTER for ‘goo’ (than for ‘dots’)

40. ‘Most’ as a Case Study ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} • mass/count flexibility Most of the dots are brown Most of the goo is brown • I wouldn’t bet that mass nouns disguised count nouns #{x:GooUnits(x) & BlueUnits(x)} > #{x:GooUnits(x)} − #{x:GooUnits(x) & BlueUnits(x)} # • work remains

42. ‘Most of the dots are blue’ #{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)} If this hypothesis about the form of the assembled thought is on the right track, it provides some insight into how quantificational expressions interface with the (presumably older) cognitive systems that make it possible to have thoughts with quantificational content. If ‘most’ fetches a complex concept specified in terms of conjunction, cardinalities and subtraction, perhaps that is because these concepts are basic, so far as prelinguistic cognition is concerned—or at least more natural than other concepts that are equivalent for purposes of logic/math. In which case, the modern study of perception can help revive an ancient research program: study the “logical” vocabulary and its relation to meaning/analyticity/verification, to gain insights about the forms of human judgment. The trick is to pursue this program, and work out its implications for human knowledge, without spoiling semantics by confusing it with epistemology.

43. Tim Hunter Darko Odic Jeff Lidz Justin Halberda

44. an I-language in Chomsky’s sense: the expression-generator generates semantic instructions; and executing these instructions yields concepts that can be used in thought

45. Most Fs are Gs (two registers) #{x:F(x) & G(x)} > #{x:F(x) & ~G(x)}

46. Most Fs are Gs (no registers): {x:F(x) & G(x)} 1-To-1-Plus {x:F(x) & ~G(x)}