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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

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Beyond Truth Conditions: The semantics of ‘most’

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  1. Beyond Truth Conditions:The semantics of ‘most’ Tim Hunter UMD Ling. Justin Halberda JHU Psych. Jeff Lidz UMD Ling. Paul Pietroski UMD Ling./Phil.

  2. 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

  3. 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

  4. 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

  5. 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?

  6. 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?

  7. 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

  8. ‘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

  9. ‘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|

  10. ‘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

  11. ‘most’ without cardinalities • Halberda, Taing & Lidz (2008) tested 3-4 year olds’ comprehension of ‘most’ Hardest ratio: 6:7 Easiest ratio: 1:9

  12. ‘most’ without cardinalities

  13. B A A One-to-one Correspondence |A| > |B| iff A [OneToOne(A, B) and A  A]

  14. DOTS – YELLOW DOTS  YELLOW One-to-one Correspondence |DOTS  YELLOW| > |DOTS – YELLOW| iff A [OneToOne(A, (DOTS – YELLOW)) and A  (DOTS  YELLOW)]

  15. 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]

  16. 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

  17. Analog Magnitude System • Discriminability of two numbers depends only on their ratio Noise in the representations increases with the number represented

  18. 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

  19. 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?

  20. 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

  21. 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

  22. Experiment 1 • Test different ratios, looking for signs of analog magnitude ratio-dependence

  23. Experiment 1

  24. Experiment 1 • Test different arrangements of dots, looking for effects of clear pairings

  25. Experiment 1 • Test different arrangements of dots, looking for effects of clear pairings

  26. Experiment 1

  27. Experiment 1

  28. Experiment 1

  29. 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

  30. 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

  31. 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?

  32. 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)

  33. 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

  34. More facts from psychology • Can’t attend to the non-yellow dots directly • Can select on colours; but only two

  35. 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

  36. 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

  37. 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

  38. Experiment 2

  39. 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|

  40. 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 timh@umd.edu http://www.ling.umd.edu/~timh/

  41. ‘most’ tmost pmost Cardinality OneToOne+ Approximate count 1-to-1+ count 1-to-1+ Word Level 1 Computation (truth conditions) Level 1.5 Families of Algorithms (understanding) HP # HP Further Distinctions (towards verification) ANSb ANSa ANS Gaussian numerosity identification ANS Gaussian GreaterThan operation via subtraction

  42. Probe Before Multiple Sets Enumerated In Parallel Halberda, Sires & Feigenson 2006

  43. Multiple Sets Enumerated In Parallel Probe After Halberda, Sires & Feigenson 2006

  44. Divergence from predictions of the model

  45. Column Pairs Sorted

  46. Control studies

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