Procedure Matters

1. Procedure Matters Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy http://www.terpconnect.umd.edu/~pietro

2. Most of the dots are yellow 15 dots: 9 yellow 6 blue

3. ‘Most of the dots are yellow’ #{DOT &YELLOW} > #{DOT}/2 More than half of the dots are yellow (9 > 15/2) #{DOT &YELLOW} > #{DOT & YELLOW} The yellow dots outnumberthe nonyellow dots (9 > 6) #{DOT &YELLOW} > #{DOT} – #{DOT &YELLOW} The number of yellow dots exceeds the number of dots minusthe number of yellow dots (9 > 15 – 9)

4. ‘Most of the dots are yellow’ MOST[DOT(x), YELLOW(x)] #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x)}/2 More than half of the dots are yellow (9 > 15/2) #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x) & YELLOW(x)} The yellow dots outnumberthe nonyellow dots (9 > 6) #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x)} – #{x:DOT(x) &YELLOW(x)} The number of yellow dots exceeds the number of dots minusthe number of yellow dots (9 > 15 – 9)

5. Most of the dots are yellow 15 dots: 9 yellow 6 blue

6. Hume’s Principle #{x:T(x)} = #{x:H(x)} iff {x:T(x)} OneToOne {x:H(x)} ____________________________________________ #{x:T(x)} > #{x:H(x)} iff {x:T(x)} OneToOnePlus {x:H(x)} αOneToOnePlusβiff for some α*, α* is a proper subset of α, and α*OneToOneβ (and it’s not the case thatβOneToOneα)

7. ‘Most of the dots are yellow’ MOST[DOT(x), YELLOW(x)] #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x)}/2 #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x) & YELLOW(x)} #{x:DOT(x) &YELLOW(x)} > #{x:DOT(x)} – #{x:DOT(x) &YELLOW(x)} OneToOnePlus[{x:DOT(x) &YELLOW(x)}, {x:DOT(x) &YELLOW(x)}]

8. ‘Most of the dots are yellow’ MOST[D, Y] OneToOnePlus[{D& Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y} ???Most of the paint is yellow???

9. Many Conceptions of Human Languages • complexes of “dispositions to verbal behavior” • strings of an elicited (or nonelicited) corpus • a procedure that generates an independently specified corpus • something a radical interpreter ascribes to a speaker • “Something which assigns meanings to certain strings of types of sounds or marks. It could therefore be a function, a set of ordered pairs of strings and meanings.”

10. Many Conceptions of Human Languages • a biologically implementable procedure that generates expressions, which may be characterizable only in terms of the procedure that generates them

11. Many Conceptions of Human Languages • complexes of “dispositions to verbal behavior” • strings of an elicited (or nonelicited) corpus • strings of (perhaps written) words in some corpus • a procedure that generates an independently specified corpus • something a radical interpreter ascribes to a speaker • “a set of ordered pairs of strings and meanings” • a biologically implementable procedure that generates expressions, which may be characterizable only in terms of the procedure that generates them

12. Many Conceptions of Human Languages • complexes of “dispositions to verbal behavior” • strings of an elicited (or nonelicited) corpus • strings of (perhaps written) words in some corpus • a procedure that generates an independently specified corpus • something a radical interpreter ascribes to a speaker • “a set of ordered pairs of strings and meanings” • a biologically implementable procedure that generates expressions, which may be characterizable only in terms of the procedure that generates them (E-Languages) (I-Languages)

13. ‘I’ Before ‘E’ Alonzo Church (of Church-Turing fame) function-in-intension vs. function-in-extension --a procedure that pairs inputs with outputs in a certain way --a set of ordered pairs (with no <x,y> and <x, z> where y≠z)

14. I-Language/E-Language function in Intensionimplementable procedure that pairs inputs with outputs function in Extension set of input-output pairs |x – 1| +√(x2 – 2x + 1) {…(-2, 3), (-1, 2), (0, 1), (1, 0), (2, 1), …} λx . |x – 1| = λx . +√(x2 – 2x + 1) λx . |x – 1| ≠ λx . +√(x2 – 2x + 1) Extension[λx . |x – 1|] = Extension[λx . +√(x2 – 2x + 1)]

15. I-Language/E-Language function in Intensionimplementable procedure that pairs inputs with outputs function in Extension set of input-output pairs With regard to languages, we can... (1) focus on phrasal composition, and worry later about words assume meanings for ‘brown’ and ‘cow’, and ask what ‘brown cow’ means (there are lots of conjunction operations out there)

16. I-Language/E-Language function in Intensionimplementable procedure that pairs inputs with outputs function in Extension set of input-output pairs With regard to languages, we can... (1) focus on phrasal composition, and worry later about words (2) focus on words, and worry later about phrasal composition assume a composition rule for ADJECTIVE^NOUN, and ask what ‘cow’ (or ‘brown’) means

17. I-Language/E-Language function in Intensionimplementable procedure that pairs inputs with outputs function in Extension set of input-output pairs With regard to languages, we can... (1) focus on phrasal composition, and worry later about words (2) focus on words, and worry later about phrasal composition assume a composition rule for QUANTIFIER^NOUN, and ask what the quantifier ‘most’ means

18. Tim Hunter A W le el xl iw so o d Darko Odic J e f f L i d z Justin Halberda

19. Most of the dots are yellow

20. ‘Most of the dots are yellow’ MOST[D, Y] OneToOnePlus[{D& Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y}

21. Some Relevant Facts • many animals are good cardinality-estimaters, by dint of a much studied system (see Dehaene, Gallistel/Gelman, etc.) • appeal to subtraction operations is not crazy (Gallistel/King) • but...infants can do one-to-one comparison (see Wynn) • and Frege’s versions of the axioms for arithmetic can be derived (within a consistent fragment of Frege’s logic) from definitions and Hume’s (one-to-one correspondence) Principle • Lots of references in… The Meaning of 'Most’. Mind and Language (2009). Interface Transparency and the Psychosemantics of ‘most’. Natural Language Semantics (2011).

22. ‘Most of the dots are yellow’ MOST[D, Y] OneToOnePlus[{D& Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y}

23. Are most of the dots yellow? What conditions make the question easy/hard to answer? That might provide clues about how we understand the question (given decent accounts of what information is available to us in those conditions).

24. a model of the “Approximate Number System” (key feature: ratio-dependence of discriminability) distinguishing 8 dots from 4 (or 16 from 8) is easier than distinguishing 10 dots from 8 (or 20 from 10)

25. a model of the “Approximate Number System” (key feature: ratio-dependence of discriminability) correlatively, as the number of dots rises, “acuity” for estimating of cardinality decreases--but still in a ratio-dependent way, with wider “normal spreads” centered on right answers

33. 4:5 (blue:yellow) “scattered pairs”

34. 1:2 (blue:yellow) “scattered pairs”

35. 4:5 (blue:yellow) “scattered pairs”

36. 9:10 (blue:yellow) “scattered pairs”

37. 4:5 (blue:yellow) “column pairs sorted”

38. 4:5 (blue:yellow) “column pairs mixed”

39. 5:4 (blue:yellow) “column pairs mixed”

40. scattered random scattered pairs 4:5 (blue:yellow) column pairs mixed column pairs sorted

41. Basic Design • 12 naive adults, 360 trials for each participant • 5-17 dots of each color on each trial • trials varied by ratio (from 1:2 to 9:10) and type • each “dot scene” displayed for 200ms • target sentence: Are most of the dots yellow? • answer ‘yes’ or ‘no’ by pressing buttons on a keyboard • correct answer randomized • controls for area (pixels) vs. number, yadayada…

42. better performance on easier ratios: p < .001 10 : 20 10 : 10 10 : 15

43. fits for Sorted-Columns trials to an independent model for detecting the longer of two line segments fits for trials (apart from Sorted-Columns) to a standard psychophysical model for predicting ANS-driven performance

44. performance on Scattered Pairs and Mixed Columns was no better than on Scattered Random; looks like ANS was used to answer the question, except in the Sorted Columns trials

45. scattered pairs scattered random 4:5 (blue:yellow) column pairs mixed column pairs sorted

46. Follow-Up Study Could it be that speakers use ‘most’ to access a 1-To-1-Plus concept, but our task made it too hard to use a 1-To-1-Plus verification strategy?

47. 4:5 (blue:yellow) “scattered pairs” What color are the loners?

48. better performance on components of a 1-to-1-plus task 10 : 20 10 : 10 10 : 15

49. We are NOT saying... • that speakers always/usually verify sentences of the form ‘Most of the Ds are Ys’ by computing #{D & Y} > #{D} – #{D & Y} • that if there are some tasks in which speakers do notverify ‘Most of the Ds are Ys’ by using a one-to-one correspondence strategy, then ‘Most’ is not understood in terms of a one-to-one correspondence

50. But we are (tentatively) assuming that... if speakers understand sentences of the form ‘Most of the Ds are Ys’ as claims of the form #{D & Y} > #{D} – #{D & Y} then other things equal, speakers will use this “logical form” as a verification strategy if they can easily do so Compare: ‘Bert arrived and Ernie left’