1. Beyond Truth Conditions: The semantics of ‘most’ Tim HunterUMD Ling. Justin HalberdaJHU Psych. Jeff LidzUMD Ling. Paul PietroskiUMD 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’ Easiest ratio: 1:9 Hardest ratio: 6:7

12. ‘most’ without cardinalities

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

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

29. 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. 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+ Level 1 Computation (truth conditions) Level 1.5 Families of Algorithms (understanding) #  HP ANS b ANS a a.ANS Gaussian numerosity identification b.ANS Gaussian GreaterThan operation via subtraction Word Further Distinctions (towards verification)

42. Multiple Sets Enumerated In Parallel Probe Before 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