Conjunction, Subtraction,

and Comparison

Paul M. PietroskiDept. of Philosophy, Dept. of Linguistics

University of Maryland

TimHunter

DarkoOdic

J e f f

L i d z

Justin Halberda

A Wl ee lx li ws o o d

Outline

• ‘Most’ is often understood as a “number word”...– Most of the dots are yellow– ‘Most’ is not understood in terms of one to one correspondence

• …that expresses a kind of Cardinality Comparison– #( & ) > #()/2– #( & ) > #( & ~)– #( & ) > #() – #( & )

• ‘Most’ is not always understood as a “number word” – Most of the paint is yellow– massy ‘Most’ does not express cardinality comparison

Most of the dots are yellow?

15 dots:9 yellow6 blue

How is the sentence understood? What question is getting asked?

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)

15 dots: 9 yellow, 6 blue

Most of the dots are yellow?

#{DOT & YELLOW} > #{DOT}/2

#{DOT & YELLOW} > #{DOT & YELLOW}#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

MOST[DOT, YELLOW]

Hume’s Principle

#{Triangle} = #{Heart} iff {Triangle} OneToOne {Heart} ____________________________________________#{Triangle} > #{Heart} iff {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)

Most of the dots are yellow

#{DOT & YELLOW} > #{DOT}/2

#{DOT & YELLOW} > #{DOT & YELLOW}#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

MOST[DOT, YELLOW]

OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW]

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}

These analyses are truth-conditionally equivalent and not crazy

Number Representations

Most of the dots are yellow

MOST[D, Y]

#{D & Y} > #{D} – #{D & Y}

??Most of the paint is yellow??

Most of the dots are yellow

MOST[D, Y]

Why analyse at all? Why not take ‘Most’ to be as primitive

as ‘dot’ and ‘yellow’ seem to be?

Some of the yellow dogs barked Some of the dogs barkedSome of the dogs barked loudly Some of the dogs barked

None of the yellow dogs barked None of the dogs barkedNone of the dogs barked loudly None of the dogs barked

Most of the dots are yellow

MOST[D, Y]

Why analyse at all? Why not take ‘Most’ to be as primitive

as ‘dot’ and ‘yellow’ seem to be?

All of the yellow dogs barked All of the dogs barkedAll of the dogs barked loudly All of the dogs barked

Most of the yellow dogs barked -- Most of the dogs barkedMost of the dogs barked loudly Most of the dogs barked

Most of the dots are yellow

MOST[D, Y]

Why analyse at all? Why not take ‘Most’ to be as primitive

as ‘dot’ and ‘yellow’ seem to be?

Most of the dogs barked More than half of the dogs barkedMost of the dogs barked More dogs barked than didn’t

Most of the yellow dogs barked -- Most of the dogs barkedMost of the dogs barked loudly Most of the dogs barked

Most of the dots are yellow?

#{DOT & YELLOW} > #{DOT}/2

#{DOT & YELLOW} > #{DOT & YELLOW}#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

MOST[DOT, YELLOW]

OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW]

Most of the dots are yellow?What conditions make the interrogative easy/hard to answer?That mightprovideclues about how the sentence is understood

(given decent accounts of the information available to human beings in those conditions).

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)

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

Most of the dots are yellow?What conditions make the interrogative easy/hard to answer?That mightprovideclues about how the sentence is understood

(given decent accounts of the information available to human beings in those conditions).

4:5 (blue:yellow)“scattered pairs”

1:2 (blue:yellow)“scattered pairs”

4:5 (blue:yellow)“scattered pairs”

9:10 (blue:yellow)“scattered pairs”

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

4:5 (blue:yellow)“column pairs mixed”

5:4 (blue:yellow)“column pairs mixed”

4:5 (blue:yellow)scattered random

column pairs mixed

scattered pairs

column pairs sorted

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.

4:5 (blue:yellow)scattered random

column pairs mixed

scattered pairs

column pairs sorted

THIS IS THE ONLY ODDBALL

50

60

70

80

90

100

1 1.5 2Ratio (Weber Ratio)

Perc

ent

Corr

ect

Scattered RandomScattered PairsColumn Pairs MixedColumn Pairs Sorted

better performance on easier ratios: p < .001

10 : 1010 : 15

10 : 20

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

but even better performance on the components of a 1-to-1-plus task if the question is not posed with ‘most’

10 : 1510 : 10 10 : 20

4:5 (blue:yellow)scattered random

column pairs mixed

scattered pairs

column pairs sorted

Most of the dots are yellow

MOST[D, Y]

OneToOnePlus[{D & Y},{D & Y}]

#{D & Y} > #{D & Y}

#{D & Y} > #{D} – #{D & Y}

Prima facie,posing the question with ‘most’ is not a way of posing a OneToOnePlus question

Most of the dots are yellow

MOST[D, Y]

#{D & Y} > #{D & Y}

#{D & Y} > #{D} – #{D & Y}

framing the question with ‘most’ has effects that are expected if the questionis understood in terms ofcardinality comparison(and answered via the ANS)

Most of the dots are blue?

What conditions make the interrogative easy/hard to answer?That mightprovideclues about how the sentence is understood

(given decent accounts of the information available to human beings in those conditions).

Most of the dots are blue?

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 obtain a cardinality estimate for each color

• #{Dot & Blue} > #{Dot} − #{Dot & Blue}

Most of the dots are blue

• #{Dot & Blue} > #{Dot & ~Blue}

verification should get harder as the number of colors increases

“acuity” should be relatively good (w ≈ .15)

• #{Dot & Blue} > #{Dot} − #{Dot & Blue}predicts indifference to the number of

alternative colors

“acuity” should be less good (w ≈ .3)

15 dots, 9 blue9 > 6

15 dots, 9 blue15 > (15 − 9)

6 9 15

#{Dot & Blue} > #{Dot & ~Blue}

#{Dot & Blue} > #{Dot} − #{Dot & Blue}

Most of the dots are blue

• #{Dot & Blue} > #{Dot & ~Blue}

verification should get harder as the number of colors increases

“acuity” should be relatively good (w ≈ .15)

• #{Dot & Blue} > #{Dot} − #{Dot & Blue}predicts indifference to the number of

alternative colors

“acuity” should be less good (w ≈ .3)

15 dots, 9 blue9 > 6

15 dots, 9 blue15 > (15 − 9)

better performance on easier ratios: p < .001

no effect of number of colors

fit to psychophysical model of ANS-driven performance

Most of the dots are blue?

MOST[D, B]

#{D & B} > #{D & B}

#{D & B} > #{D} – #{D & B}

Prima facie,posing the question with ‘most’ is not a way of posing a cardinalityquestionformulated in terms of negation

Most of the dots are blue?

MOST[D, B]

#{D & B} > #{D} – #{D & B}

framing the question with ‘most’ has effects that are expected if the questionis understood in terms ofcardinality subtraction

Prima facie, this requires a representation of #{D}and a computation onthis representation.

Does use of ‘most’ reflect ease/difficulty of representing #{D}?

Slide# 48

left right

% Of Undergrads WhoChoose This Side (N= 48)

Asked, “Most” = 58%Asked, “More” = 13%

A) More of the dots are grey.

B) Most of the dots are grey.

Which sentence would you choose to describe this picture?

65% choose “more”

A) More of the dots are grey.74% choose “most”

B) Most of the dots are grey.

Which sentence would you choose to describe this picture?

Not just about recognition• 80 participants asked to “draw” on an iPad

More/Most of the dots are blue

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

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

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.

4-8 yr olds (n=92)

More Most

Most of the dots are blue?

MOST[D, B]

#{D & B} > #{D} – #{D & B}

framing the question with ‘most’ has effects that are expected if the questionis understood in terms ofcardinality subtraction ??Most of the paint is blue??

Most of the dots are blue#{Dot & Blue} > #{Dot} − #{Dot & Blue}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• are mass nouns (somehow) disguised count nouns? #{GooUnit & BlueUnit} > #{GooUnit} − #{GooUnit & BlueUnit}

Ratio (Bigger Quantity/ Smaller Quantity)

1.0 1.2 1.4 1.6 1.8 2.0 2.2

% C

orre

ct

50

55

60

65

70

75

80

85

90

95

100

Mass DataMass ModelCount DataCount Model

w = .20r2 = .99

w = .29r2 = .98

First Study (blob/blobs…others in the works) Performance is better, holding the scene constant, if the question is posed with a mass noun

Most of the dots are blue#{Dot & Blue} > #{Dot} − #{Dot & Blue}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• are mass nouns (somehow) disguised count nouns? #{GooUnit & BlueUnit} > #{GooUnit} − #{GooUnit & BlueUnit}

SEEMS NOT

Most of the dots are blue#{Dot & Blue} > #{Dot} − #{Dot & Blue}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• Most of the goo is blue μ{Goo & Blue} > μ{Goo} − μ{Goo & Blue}

But how do we deal with blue goo, given a Fregean metalanguage, without resorting to goo units? What mapping from objects to truth/falsity corresponds to ‘goo’?

Most of the dots are blue#{Dot(x) & Blue(x)} > #{Dot(x)} − #{Dot(x) & Blue(x)}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• Most of the goo is blue μ{Goo(?) & Blue(?)} > μ{Goo(?)} − μ{Goo(?) & Blue(?)}

Most of the dots are blue#{Dot & Blue} > #{Dot} − #{Dot & Blue}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• singular/plural flexibilityThe child saw a dogThe children saw the dogsThe children counted the dogs

The children counted the dogs

Xy[Xy Dog(y)]

xy[(y x) Dog(y)] Xy[OneOf(y, X) Dog(x)]

there is a set, x, such that there are sm things, the Xs, such that

each thing, y, is such that each thing, y, is such that

ity is an element of itx iff ity is a dog ity is one of themX iff ity is a dog

The logicians counted the sets

Xy[Xy Set(y)]

xy[(y x) Set(y)] Xy[OneOf(y, X) Set(x)]

there is a set, x, such that there are sm things, the Xs, such that

each thing, y, is such that each thing, y, is such that

ity is an element of itx iff ity is a set ity is one of themX iff ity is a set

— a b ba c ca cb cba d da db dba dc dca dcb dcba e ea eb eba ec eca ecb ecbaed eda edb edba edc edca edcb edcba

TWO CONCEPTIONS OF PLURAL VARIABLES

Five entities: a, b, c, d, e

a = 1, b = 10, c = 100, d = 1000, e = 10000

00000 00001 00010 00011 00100 00101 00110 0011101000 01001 01010 01011 01100 01101 01110 0111110000 10001 10010 10011 10100 10101 10110 1011111000 11001 11010 11011 11100 11101 11110 11111

Link: 01011 = 1000⊕10⊕1 a mereological sum with 3

atoms (d, b, a); it can be the value of a

singular variable Boolos: 01011 = 1000+10+1

five answers to a yes/no question: is … a value of an unsingular

variable?

TWO CONCEPTIONS OF PLURAL VARIABLES

Xy[OneOf(y, X) Sheep(x)]there are one or more things, the Xs, such that each thing, y, is such thatity is one of themX iff ity is a sheep

XY:~Plural(Y)[SomeOf(Y, X) Sheep(Y)]there are one or more things, the Xs, such that any one or more things that are not plural, the Ys, are such thattheyy are some of themX iff theyy are sheep

X:Countish(X){Y:Countish(Y)[SomeOf(Y, X) Sheep(Y)]}there be one or more things, the Xs, such that any one or more things, the Ys, be such thattheyy be some of themX iff theyy be sheep

X:~Countish(X){Y:~Countish(Y)[SomeOf(Y, X) Sheep(Y)]}there be some stuff, the X, such that any stuff, the Y, be such thatity be some of itX iff ity be sheep (stuff)

we can allow for assignmentsof value (e.g., sm mutton or sm mud) to variables, and not insist that each assignment assign one or more values to each variable

X:Countish(X){Y:Countish(Y)[SomeOf(Y, X) Sheep(Y)]}there be some (one or more things) that be countish, the X, such that any (one or more things) that be countish, the Y, be such thatit-or-theyy be some of it-or-themX iff it-or-theyy be sheep

X:~Countish(X){Y:~Countish(Y)[SomeOf(Y, X) Sheep(Y)]}there be some (stuff) that be not countish, the X, such that any (stuff) that be not countish, the Y, be such thatit-or-theyy be some of themX iff it-or-theyy be sheep

XY[SomeOf(Y, X) Sheep(Y)]}there be some (stuff-or-thing-or-things), the X, such that any (stuff-or-thing-or-things), the Y, be such thatit-or-theyy be some of themX iff it-or-theyy be sheep

Ratio (Bigger Quantity/ Smaller Quantity)

1.0 1.2 1.4 1.6 1.8 2.0 2.2

% C

orre

ct

50

55

60

65

70

75

80

85

90

95

100

Mass DataMass ModelCount DataCount Model

w = .20r2 = .99

w = .29r2 = .98

First Study (blob/blobs…others in the works) Performance is better, holding the scene constant, if the question is posed with a mass noun

Most of the dots are blue#{Dot & Blue} > #{Dot} − #{Dot & Blue}

• mass/count flexibilityMost of the dots (blobs, peas, shoes) are blue

Most of the paint (blob, corn, footwear) is blue

• Most of the goo is blue μ{Goo & Blue} > μ{Goo} − μ{Goo & Blue}

Maybe we can deal with blue goo, without resorting to goo units, by using suitably neutral variables in our metalanguage (and positing correspondingly neutral mental symbols)

Most of the dots are blue?

MOST[D, B]

#{D & B} > #{D & B}

#{D & B} > #{D} – #{D & B}

Prima facie,this is a cardinality question, but not a questionformulated in terms of negation

TimHunter

DarkoOdic

J e f f

L i d z

Justin Halberda

A Wl ee lx li ws o o d

Conjunction, Subtraction,

and Comparison

Paul M. PietroskiDept. of Philosophy, Dept. of Linguistics

University of Maryland