1 / 46

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

london
Download Presentation

Beyond Truth Conditions: The semantics of ‘most’

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related