1. Most, Mass, and maybe More
Meanings and Minds:
Most, Mass, and maybe More
Paul M. Pietroski
Rutgers University

2. Tim
Hunter
A W
l e
e l
x l
i w
s o o d
Darko
Odic J e f L i d z
Justin Halberda

3. Are most of the dots yellow?
6 blue
1 to 2 SUPER EASY
How is the Yes/No question understood?
What question is getting asked?

4. Most of the dots are yellow.
MOST[DOT, YELLOW]
#{DOT & YELLOW} > #{DOT}/2
More than half of the dots are yellow (9 > 15/2)
#{DOT & YELLOW} > #{DOT & YELLOW}
The yellow dots outnumber the nonyellow dots (9 > 6)
#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}
The number of yellow dots exceeds
the number of dots minus the number of yellow dots
(9 > 15 – 9)
(And there is at least one more option to consider.)
15 dots: 9 yellow, 6 blue

5. Hume’s Principle #{Triangle} = #{Heart} iff
{Triangle} OneToOne {Heart}
____________________________________________
#{Triangle} > #{Hearts}
{Triangle} OneToOnePlus {Heart}
α OneToOnePlus β iff for some α*,
α* is a proper subset of α, and α* OneToOne β
(and it’s not the case that β OneToOne α)

6. Most of the dots are yellow.
OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW]
1 to 2 SUPER EASY
#{DOT & YELLOW} > #{DOT}/2
#{DOT & YELLOW} > #{DOT & YELLOW}
#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

7. Most of the dots are yellow.
What conditions
make it
easy/hard
to evaluate
the sentence?
That might
provide
clues
about
how the
sentence is understood
1 to 2 SUPER EASY
(given independent accounts of the information used
by the evaluators in those conditions).

8. ‘Most of the dots are yellow’
MOST[D, Y] OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y}
Number Representations
These analyses are provably equivalent (for finite cases)
and none of them are crazy

9. ‘Most of the dots are yellow’
MOST[D, Y] OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y}
Number Representations
??Most of the paint is yellow??

10. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be primitive,
as ‘dot’ and ‘yellow’ seem to be?
Some of the yellow dogs barked  Some of the dogs barked
Some of the dogs barked loudly  Some of the dogs barked
None of the yellow dogs barked  None of the dogs barked
None of the dogs barked loudly  None of the dogs barked

11. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be primitive,
as ‘dot’ and ‘yellow’ seem to be?
All of the yellow dogs barked  All of the dogs barked
All of the dogs barked loudly  All of the dogs barked
Most of the yellow dogs barked -- Most of the dogs barked
Most of the dogs barked loudly  Most of the dogs barked

12. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be primitive,
as ‘dot’ and ‘yellow’ seem to be?
Most of the dogs barked  More than half of the dogs barked
Most of the dogs barked  More dogs barked than didn’t
Most of the yellow dogs barked -- Most of the dogs barked
Most of the dogs barked loudly  Most of the dogs barked

13. Most of the dots are yellow?
MOST[DOT, YELLOW]
OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW]
1 to 2 SUPER EASY
#{DOT & YELLOW} > #{DOT}/2
#{DOT & YELLOW} > #{DOT & YELLOW}
#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

14. Most of the dots are yellow?
What conditions
make it
easy/hard
to evaluate
the sentence?
That might
provide
clues
about
how the
sentence is understood
1 to 2 SUPER EASY
(given independent accounts of the information used
by the evaluators in those conditions).

15. a model of the “Approximate Number System”
(key feature: ratio-dependence of discriminability)
distinguishing 8 dots from 4 (or 16 from 8)
is easier than
distinguishing 10 dots from 8 (or 20 from 10)

16. a model of the “Approximate Number System”
(key feature: ratio-dependence of discriminability)
correlatively, as the number of dots rises, “acuity” for estimating of cardinality decreases--but still in a ratio-dependent way, with wider “normal spreads” centered on right answers

17. various ways of presenting dots (8 blue, 10 yellow)
scattered random
scattered pairs
various ways of presenting dots (8 blue, 10 yellow)
column pairs mixed
column pairs sorted

18. 4:5 (blue:yellow)
“scattered random”
4 to 5

19. 1:2 (blue:yellow)
“scattered random”
1 to 2 SUPER EASY

20. 4:5 (blue:yellow)
“scattered pairs”

21. 9:10 (blue:yellow)
“scattered pairs”

22. 4:5 (blue:yellow)
“column pairs mixed”
4 to 5

23. 5:4 (blue:yellow)
“column pairs mixed”
9 to 10

24. 4:5 (blue:yellow)
“column pairs sorted”

25. 4:5 (blue:yellow) scattered random scattered pairs column pairs mixed
column pairs sorted

26. 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?
answer ‘yes’ or ‘no’ by pressing buttons on a keyboard
correct answer randomized
controls for area (pixels) vs. number, etc.

27. 4:5 (blue:yellow) scattered random scattered pairs
THIS IS THE ONLY ODDBALL
column pairs mixed
column pairs sorted

28. better performance on easier ratios: p < .001
10 : 10
10 : 20
10 : 15

29. performance on Scattered Pairs and Mixed Columns
was no better than on Scattered Random…
looks like ANS was used to answer the question,
except in Sorted Columns

30. ANS
ANS
scattered random
scattered pairs
Are most of the dots yellow?
ANS
column pairs mixed
column pairs sorted

31. but even better performance on the components of a
1-to-1-plus task if the question is not posed with ‘most’
10 : 20
10 : 10
10 : 15

32. ‘Most of the dots are yellow’
OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y}
Number Representations
Prima facie,
posing a question
with ‘most’
is not a way
of posing a OneToOnePlus question

33. ‘Most of the dots are yellow’
#{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y}
framing the question
with ‘most’ has effects
that are expected
if the question is understood in terms of
cardinality comparison
(and answered via the ANS) but unexpected if the question is understood
in terms of 1-to-1 correspondence
Number Representations

34. Most of the dots are blue.
What conditions
make it
easy/hard
to evaluate
the sentence?
That might
provide clues
about how the
sentence is understood
But there is nothing special about 200ms,
or colors, or dots, or
any particular task.
(given independent accounts of the information used
by the evaluators in those conditions).

35. Most of the dots are blue.
The trick is to create contexts such that…
(i) we know something about the information available to evaluators, and
(ii) we can get evidence about which kinds of information evaluators are apt to use/ignore.
This raises questions about how (in general) meaning is related to understanding and verification. But let’s save that discussion for the Q&A, after we’ve seen more evidence about which kinds of information evalutaors are apt to use/ignore.

36. Most of the dots are blue?

37. Four Trial Types (2-5 colors)

38. ‘Most of the dots are blue’
#{Dot & Blue} > #{Dot & ~Blue}
in scenes with two colors, blue and red,
the non-blues can be identified with the reds
#{Dot & ~Blue} = #{Dot & Red}
the visual system will “select” the dots, the blue dots, and
the red dots; these 3 sets will be estimated for cardinality
but adding colors will make it harder (and with 5 colors,
impossible) to estimate the cardinality of the non-blues
#{Dot & Blue} > #{Dot} − #{Dot & Blue}

39. ‘Most of the dots are blue’
#{Dot & Blue} > #{Dot & ~Blue}
verification should get harder
as the number of colors increases
but in the two-color case,
“acuity” should be relatively good (w ≈ .15) since
no estimate of the total figures in the computation
#{Dot & Blue} > #{Dot} − #{Dot & Blue}
predicts indifference to the number of colors
“acuity” should be less good (w ≈ .3) since
an estimate of the total will figure in the computation
15 dots, 9 blue
9 > 6
15 dots, 9 blue
9 > (15 − 9)

40. 6 9 15 lower acuity #{Dot & Blue} > #{Dot & ~Blue}
#{Dot} − #{Dot & Blue} 6 9 15
lower
acuity

41. ‘Most of the dots are blue’
#{Dot & Blue} > #{Dot & ~Blue}
verification should get harder
as the number of colors increases
but in the two-color case,
“acuity” should be relatively good (w ≈ .15) since
no estimate of the total figures in the computation
#{Dot & Blue} > #{Dot} − #{Dot & Blue}
predicts indifference to the number of colors
“acuity” should be less good (w ≈ .3) since
an estimate of the total will figure in the computation
15 dots, 9 blue
9 > 6
15 dots, 9 blue
9 > (15 − 9)

42. better performance on easier ratios: p < .001
no effect of number of colors

43. fit to psychophysical model of ANS-driven performance

44. ‘Most of the dots are blue’
#{D & B} > #{D & B} #{D & B} > #{D} – #{D & B}
Number Representations
Prima facie,
posing a question
with ‘most’
is not a way
of posing a
cardinality
question
formulated in
terms of negation

45. ‘Most of the dots are blue’
#{D & B} > #{D} – #{D & B}
framing the question
with ‘most’
has effects that are
expected if the question
is understood in terms of
cardinality subtraction
Number Representations
But even if this is right for
cases involving count nouns
(e.g., ‘most of the dots’)
what about mass nouns, as in…….Most of the paint is blue?

48. ‘Most of the dots are blue’
#{Dot & Blue} > #{Dot} − #{Dot & Blue}
mass/count flexibility
Most of the dots (blobs, peas, shoes) are blue
Most of the paint (blob, corn, footwear) is blue
are mass nouns (somehow) disguised count nouns?
#{StuffUnit & BlueUnit} >
#{StuffUnit} − #{StuffUnit & BlueUnit}

51. Look at that goo/stuff/paint…
Is most of it blue?
Is most the paint blue?
Was most of the blob blue?
Look at those spots/blobs…
Are most of them blue?
Are most of the spots blue?
Were most of the blobs blue?

52. First Study…’blob’ vs. ‘blobs’: performance is better,
holding the scene constant,
if the question is posed
with ‘blob’.
w = .20
r2 = .99
w = .29
r2 = .98
Follow-Up Studies:
you can replace ‘most’ with
‘more’, or
‘blob’
with
‘goo’
It seems that performance is better,
holding the scene constant,
if the question is posed
with a mass noun

53. ‘Most of the dots are blue’
#{Dot & Blue} > #{Dot} − #{Dot & Blue}
mass/count flexibility
Most of the dots (blobs, peas, shoes) are blue
Most of the paint (blob, corn, footwear) is blue
are mass nouns (somehow) disguised count nouns?
#{StuffUnit & BlueUnit} >
#{StuffUnit} − #{StuffUnit & BlueUnit}
SEEMS NOT
and that raises
more questions

54. Most of the dots are blue?
#{D & B} > #{D} – #{D & B}
framing the question
with ‘most’
has effects that are
expected if the question
is understood in terms of
cardinality subtraction
Prima facie, this requires
a representation of #{D}
and a computation on
this representation.
Does use of ‘most’ reflect
ease/difficulty of representing #{D},
the (total) number of dots?

55. Which side is the better example of… (1) Most of the dots are blue
left
right
% Of Undergrads Who
Choose This Side (N= 48)
“Most…” = 58%
“More…” = 13%
on both sides,
12 blues and 8 yellows
G1: most of the dots are blue
Back
G2: there are more blue dots than yellow dots
2nd time: click for data
Which side is the better example of…
(1) Most of the dots are blue
(2) More of the dots are blue

56. A) More of the dots are grey. B) Most of the dots are grey.
Which sentence would you choose to describe this picture?
A) More of the dots are grey.
B) Most of the dots are grey.
65% choose “more”

57. 74% choose “most” A) More of the dots are grey.
Which sentence would you choose to describe this picture?
A) More of the dots are grey.
74% choose “most” 
B) Most of the dots are grey.

58. Not just about recognition
80 participants asked to “draw” on an iPad More/Most of the dots are blue

59. Not just about recognition
Typical for “More of the dots are blue”

60. Not just about recognition
Typical for “Most of the dots are blue”

61. Centroid distance (adults)
More Most
Halberda, Pietroski, Hunter, Odic, Wellwood, & Lidz. (2012). More & most: spatial vision affects word understanding on an iPad. Vision Science Society annual meeting.

62. 4-8 yr olds (n=92)
More
Most

63. Ongoing Work with Tyler Knowlton
Each cow is brown Every cow is brown All cows are brown x:Cow(x)[Brown(x)] ~x:Cow(x)[~Brown(x)] {x: Cow(x)}  {x: Brown(x)} {x: Cow(x)} = {x: Cow(x) & Brown(x)}

64. Tim
Hunter
A W
l e
e l
x l
i w
s o o d
Darko
Odic J e f L i d z
Justin Halberda

65. THANKS