Fine, K. 2013, Truth-Maker Semantics for Intuitionistic Logic
Beyond Truth Conditions: The semantics of ‘most’
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Beyond Truth Conditions: The semantics of ‘most’ Tim Hunter UMD Ling. Justin Halberda JHU Psych. Jeff Lidz UMD Ling. Paul Pietroski UMD Ling./Ph
description
Beyond Truth Conditions: The semantics of ‘most’. Tim HunterUMD Ling. Justin HalberdaJHU Psych. Jeff LidzUMD Ling. Paul PietroskiUMD Ling./Phil. What are meanings?. The language faculty pairs sounds with meanings Maybe meanings are truth conditions - PowerPoint PPT Presentation
Transcript of Beyond Truth Conditions: The semantics of ‘most’
Beyond Truth Conditions:The semantics of ‘most’
Tim Hunter UMD Ling.Justin Halberda JHU Psych.Jeff Lidz UMD Ling.Paul Pietroski 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 3-4 year olds’ comprehension of ‘most’
Easiest ratio: 1:9 Hardest ratio: 6:7
‘most’ without cardinalities
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Non-Counters (n=14)
Full-Counters (n=35)
One-to-one Correspondence
|A| > |B|iff
A [OneToOne(A, B) and A A]
AB
A
One-to-one Correspondence|DOTS YELLOW| > |DOTS – YELLOW|
iffA [OneToOne(A, (DOTS – YELLOW)) and A (DOTS
YELLOW)]
DOTS YELLOW DOTS – YELLOW
One-to-one Correspondence|DOTS YELLOW| > |DOTS – YELLOW|
iffA [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 1997Feigenson, Spelke & Dehaene 2004Whalen, Gallistel & Gelman 1999
Analog Magnitude System
• Discriminability of two numbers depends only on their ratio
• Noise in the representations increases with the number represented
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– predicts sensitivity to ratio– predicts no sensitivity to pairing of dots
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Experiment 1
• Test different ratios, looking for signs of analog magnitude ratio-dependence
Experiment 1
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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
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Scattered Pairs
Experiment 1
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Experiment 1
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Experiment 1
• Success rate does depend on ratio
• Success rate does not depend on the arrangement’s amenability to pairing
• Results support Hypothesis 2: analog magnitude system
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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 1988Wolfe 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 well-understood– truth-conditional semantics– psychology of number– constraints on vision
[email protected]://www.ling.umd.edu/~timh/
‘most’
tmost pmost
Cardinality OneToOne+ Approximate
count 1-to-1+ count 1-to-1+
Level 1Computation(truth conditions)
Level 1.5Families ofAlgorithms
(understanding)
#HP HP
ANSbANSa
a. ANS Gaussian numerosity identificationb. ANS Gaussian GreaterThan operation via subtraction
Word
Further Distinctions
(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
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Avera
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SE (
Data
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Column Pairs Sorted
Control studies
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Original Scattered Pairs DataFind Pairs ControlFind Loners Control
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