Transcript Document

Beyond Truth Conditions:
The semantics of ‘most’
Tim Hunter
Justin Halberda
Jeff Lidz
Paul Pietroski
UMD Ling.
JHU Psych.
UMD Ling.
UMD Ling./Phil.
What are meanings?
• The language faculty pairs sounds with
meanings
• Maybe meanings are truth conditions
– Various truth-conditionally equivalent expressions
are all equally appropriate
• Maybe meanings are actually something
richer, and make reference to certain kinds of
algorithms and/or representations
– Stating a truth condition doesn’t finish the job
What are meanings?
• If meanings do make reference to certain
kinds of algorithms (and not others), then …
• … we would expect that varying the suitability
of stimuli to algorithms of some type(s) will
affect accuracy …
• … whereas varying the suitability of stimuli to
algorithms of some other type(s) will not
What are meanings?
• Quantifiers like ‘most’ are a good place
to start because relevant background is
well-understood
– truth-conditional semantics
– psychology of number
– constraints on vision
Outline
• What are meanings?
• Possible verification strategies for ‘most’
• Experiment 1
– Does the meaning of ‘most’ involve some
notion of cardinality?
• Experiment 2
– How do constraints from the visual system
interact with this meaning?
Outline
• What are meanings?
• Possible verification strategies for ‘most’
• Experiment 1
– Does the meaning of ‘most’ involve some
notion of cardinality?
• Experiment 2
– How do constraints from the visual system
interact with this meaning?
Verification strategies for
‘most’
• Hackl (2007): meanings may inform
verification strategies
– Hypothesis 1: most(X)(Y) = 1 iff |X  Y| > |X – Y|
– Hypothesis 2: most(X)(Y) = 1 iff |X  Y| > ½|X|
• Participants showed different verification
strategies for ‘most’ and ‘more than half’
– ‘Most of the dots are yellow’
– ‘More than half of the dots are yellow’
• Hackl rejects Hypothesis 2
‘most’ without cardinalities
• There are multiple ways to determine the
truth/falsity of
|X  Y| > |X – Y|
which do not require computing the value of
½ |X|
• There are even ways which don’t involve
computing any cardinalities at all
‘most’ without cardinalities
• There are multiple ways to determine the
truth/falsity of
|X  Y| > |X – Y|
which do not require computing the value of
½ |X|
‘most’ without cardinalities
• There are multiple ways to determine the
truth/falsity of
|X  Y| > |X – Y|
which do not require computing the value of
½ |X|
• Children with no cardinality concepts can
verify ‘most’ statements
‘most’ without cardinalities
• Halberda, Taing & Lidz (2008) tested 34 year olds’ comprehension of ‘most’
Easiest ratio: 1:9
Hardest ratio: 6:7
‘most’ without cardinalities
Percent Correct
100
Non-Counters (n=14)
Full-Counters (n=35)
90
80
70
60
50
1
3
5
7
Ratio (Weber Ratio)
9
One-to-one Correspondence
|A| > |B|
iff
A [OneToOne(A, B) and A  A]
B
A
A
One-to-one Correspondence
|DOTS  YELLOW| > |DOTS – YELLOW|
iff
A [OneToOne(A, (DOTS – YELLOW)) and A  (DOTS  YELLOW)]
DOTS  YELLOW
DOTS – YELLOW
One-to-one Correspondence
|DOTS  YELLOW| > |DOTS – YELLOW|
iff
A [OneToOne(A, (DOTS – YELLOW)) and A  (DOTS  YELLOW)]
iff
OneToOnePlus(DOTS  YELLOW, DOTS – YELLOW)
where:
OneToOnePlus(A,B)

A [OneToOne(A,B) and A  A]
Analog Magnitude System
• In the cases where it’s not possible to count
…
– kids without cardinality concepts
– adults without time to count
• … perhaps we approximate using our analog
magnitude system
– present at birth, no training required
– in rats, pigeons, monkeys, apes
Dehaene 1997
Feigenson, Spelke & Dehaene 2004
Whalen, Gallistel & Gelman 1999
Analog Magnitude System
• Noise in the representations increases with
the number represented
• Discriminability of two numbers depends only
on their ratio
What are meanings?
• If meanings do make reference to certain
kinds of algorithms (and not others), then …
• … we would expect that varying the suitability
of stimuli to algorithms of some type(s) will
affect accuracy …
• … whereas varying the suitability of stimuli to
algorithms of some other type(s) will not
Outline
• What are meanings?
• Possible verification strategies for ‘most’
• Experiment 1
– Does the meaning of ‘most’ involve some
notion of cardinality?
• Experiment 2
– How do constraints from the visual system
interact with this meaning?
Experiment 1
• Display an array of yellow and blue dots
on a screen for 200ms
• Target: ‘Most of the dots are yellow’
• Participants respond ‘true’ or ‘false’
• 12 subjects, 360 trials each
• 9 ratios × 4 trial-types × 10 trials
Experiment 1
• Trials vary in two dimensions
– ratio of yellow to non-yellow dots
– dots’ amenability to pairing procedures
• Hyp. 1: one-to-one correspondence
– predicts no sensitivity to ratio
– predicts sensitivity to pairing of dots
• Hyp. 2: analog magnitude system
100
Percent Correct
– predicts sensitivity to ratio
– predicts no sensitivity to pairing of dots
90
80
70
60
50
1
1.5
Ratio (Weber Ratio)
2
Experiment 1
• Test different ratios, looking for signs of
analog magnitude ratio-dependence
Experiment 1
Percent Correct
100
90
80
70
Scattered Random
60
50
1
1.5
Ratio (Weber Ratio)
2
Experiment 1
• Test different arrangements of dots,
looking for effects of clear pairings
Experiment 1
• Test different arrangements of dots,
looking for effects of clear pairings
Experiment 1
Percent Correct
100
90
80
70
Scattered Random
60
Scattered Pairs
50
1
1.5
Ratio (Weber Ratio)
2
Experiment 1
Percent Correct
100
90
80
70
Scattered Random
60
Scattered Pairs
Column Pairs
50
1
1.5
Ratio (Weber Ratio)
2
Experiment 1
Percent Correct
100
90
80
70
Scattered Random
60
Scattered Pairs
Column Pairs
50
1
1.5
Ratio (Weber Ratio)
2
Experiment 1
• Success rate does
depend on ratio
Percent Correct
100
90
80
70
Scattered Random
60
Scattered Pairs
Column Pairs
50
1
1.5
Ratio (Weber Ratio)
• Success rate does not depend on the
arrangement’s amenability to pairing
• Results support Hypothesis 2: analog
magnitude system
2
What are meanings?
• We shouldn’t conclude that the meaning of
‘most’ requires the use of analog magnitude
representations/algorithms in absolutely
every case
• But there at least seems to be some
asymmetry between this procedure and the
one-to-one alternative
• Not all algorithms for computing the relevant
function have the same status
Outline
• What are meanings?
• Possible verification strategies for ‘most’
• Experiment 1
– Does the meaning of ‘most’ involve some
notion of cardinality?
• Experiment 2
– How do constraints from the visual system
interact with this meaning?
A more detailed question
• How do we actually compute the
numerosities to be compared?
|DOTS  YELLOW| > |DOTS – YELLOW|
• Selection procedure: detect (DOTS –
YELLOW) directly
• Subtraction procedure: detect DOTS, detect
YELLOW, and subtract to get (DOTS –
YELLOW)
More facts from psychology
• You can attend to at most three sets in
parallel
• You automatically attend to the set of all dots
in the display
• You can quickly attend to all dots of a certain
colour (“early visual features”)
• You can’t quickly attend to all dots satisfying a
negation/disjunction of early visual
features
Halberda,
Sires & Feigenson 2007
Triesman & Gormican 1988
Wolfe 1998
More facts from psychology
• Can’t attend to the non-yellow dots directly
• Can select on colours; but only two
Experiment 2
• Same task as Experiment 1
• Trials with 2, 3, 4, 5 colours
• 13 subjects, 400 trials each
• 5 ratios × 4 trial-types × 20 trials
Experiment 2
• Selection procedure: attend (DOTS –
YELLOW) directly
– only works with two colours present
• Subtraction procedure: attend DOTS,
attend YELLOW, and subtract to get (DOTS –
YELLOW)
– works with any number of colours present
• Hyp. 1: Use whatever procedure works best
• Hyp. 2: The meaning of ‘most’ dictates the
use of the subtraction procedure
Experiment 2
• Hypothesis 1: Use whatever procedure works
best
– selection procedure with two colours
– subtraction procedure with three/four/five colours
– better accuracy with two colours
• Hypothesis 2: The meaning of ‘most’ dictates
the use of the subtraction procedure
– performance identical across all numbers of
colours
Experiment 2
Experiment 2
• The curve is the same as in Experiment 1, no
matter how many colours are present
• Even when the non-yellow dots were easy to
attend to, subjects didn’t do so
• The meaning of ‘most’ forced them into a
suboptimal verification procedure;
presumably by requiring a subtraction
|DOTS  YELLOW| > |DOTS – YELLOW|
Conclusions
• Meanings can constrain the range of
procedures speakers can use to verify a
statement
• Quantifiers like ‘most’ are a good place to
start because relevant background is wellunderstood
– truth-conditional semantics
– psychology of number
– constraints on vision
[email protected]
http://www.ling.umd.edu/~timh/
‘most’
Word
tmost
pmost
Cardinality OneToOne+ Approximate
Level 1
Computation
(truth conditions)
Level 1.5
Families of
Algorithms
(understanding)
HP
count 1-to-1+
a.
b.
#
HP
ANSa count
Further
Distinctions
1-to-1+
ANSb
ANS Gaussian numerosity identification
ANS Gaussian GreaterThan operation via subtraction
(towards verification)
Multiple Sets Enumerated In Parallel
Probe Before
Halberda, Sires & Feigenson 2006
Multiple Sets Enumerated In Parallel
Probe After
Halberda, Sires & Feigenson 2006
Divergence from predictions of
the model
Average Signed Percent
Difference ±SE (Data Model)
8
6
4
2
0
-2
-4
-6
1
1.5
Ratio (Weber Ratio)
2
Column Pairs Sorted
Control studies
Percent Correct
100
90
80
70
60
50 +
1
Original Scattered Pairs Data
Find Pairs Control
Find Loners Control
1.5
Ratio (Weber Ratio)
2