1. Tim
Hunter
A W
l e
e l
x l
i w
s o o d
Darko
Odic J e f L i d z
Justin Halberda

2. a language: something that connects signals with interpretations a Slang: a language of a special sort (i) connects boundlessly many pronunciations— e.g., the sounds of spoken English, or signs of ASL— with boundlessly many meanings (whatever they are) (ii) acquirable by children, given ordinary experience What sort of thing is a Slang? What are the meanings that Slangs connect with pronunciations?

3. Lewis, “Languages and Language”
What is a language? 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.
What is language? A social phenomenon which is part of the natural history of human beings; a sphere of human action…
What could a meaning of a sentence be? Something which, when combined with factual information about the world—or factual information about any possible world—yields a truth value. It could therefore be a function from worlds to truth-values.
General conception of language as sets…Slangs as special cases
Meanings as sets that are independent of Slangs
SLIDES OPTIONAL

4. A Different Picture
What are Slangs? Child-acquirable languages that generate pronunciation-meaning pairs in certain (limited) ways.
Languages are “things” of various sorts.
Human Language is a hodgepodge: varied uses of Slangs.
What could a sentence meaning be? Something that determines a thought, perhaps by interacting with human cognitive capacities.
A sentence meaning might be a thought, and represent a truth condition. Or maybe a sentence meaning is more like an instruction (or blueprint) for how to build a thought of a certain sort.
Specific conception of Slangs as generative procedures
SLIDES OPTIONAL

5. Language: something that connects signals with interpretations Slang: a language of a special sort (i) connects boundlessly many pronunciations— e.g., the sounds of spoken English, or signs of ASL— with boundlessly many meanings (ii) acquirable by children, given ordinary experience ______________________________________________________ a Slang is an I-Language in Chomsky’s sense: a procedure that generates pronunciation-meaning (π-μ) pairs, as opposed to a mere set of such pairs

6. function in intension function in extension (computational procedure) (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)]

7. Church (1941) on Lambdas
1: a function is a “rule of correspondence” 2: underdetermined when “two functions shall be considered the same” 2-3: functions in extension, functions in intension In the calculus of L-conversion and the calculus of restricted λ-K-conversion, as developed below, it is possible, if desired, to interpret the expressions of the calculus as denoting functions in extension. However, in the calculus of λ-δ-conversion, where the notion of identity of functions is introduced into the system by the symbol δ, it is necessary, in order to preserve the finitary character of the transformation rules, so to formulate these rules that an interpretation by functions in extension becomes impossible. The expressions which appear in the calculus of λ-δ-conversion are interpretable as denoting functions in intension of an appropriate kind. 3: “The notion of difference in meaning between two rules of correspondence is a vague one, but in terms of some system of notation, it can be made exact in various ways.”

8. focus on focus on languages as intensions: languages as extensions: procedures that generate sets of pronunciation-meaning pairs pronunciation-meaning pairs In principle, distinct procedures might generate the same pronunciation-meaning pairs Language-1 ≠ Language-2 Extension[Language-1] = Extension[Language-2]

9. (1) Bingley is eager to please (a) Bingley is eager to be one who pleases. #(b) Bingley is eager to be one who is pleased. (2) a boy saw a man with a telescope (a) A boy saw a man who had a telescope. (b) A boy saw a man by using a telescope. #(c) A boy saw a man and had a telescope. Children acquire procedures that connect boundlessly many pronunciations with boundlessly many meanings in constrained ways

10. Language: something that connects signals with interpretations Slang: a language of a special sort (i) connects boundlessly many pronunciations— e.g., the sounds of spoken English, or signs of ASL— with boundlessly many meanings (ii) acquirable by children, given ordinary experience ______________________________________________________ a Slang is an I-Language in Chomsky’s sense: a procedure that generates pronunciation-meaning (π-μ) pairs, as opposed to a mere set of such pairs

11. Slang: a child-acquirable procedure that generates boundlessly many pronunciation-meaning (π-μ) pairs, What are these meanings? What are the “human interpretations” that Slangs connect pronunciations with? Are they more like functions-in-extension (sets), more like functions-in-intension (procedures), or neither?

12. Three traditional ideas:
What are Meanings?
Three traditional ideas:
concepts (mental representations of some sort),
with thoughts as special cases of concepts
extensions of ideal concepts,
with truth conditions as special cases of extensions
ideal concepts: concepts that have extensions
extensions include sets of
cats, dogs, prime numbers, …
ordered pairs <w, s> where s is a set of things at w
ordered pairs <C, F> where C is a context and F is a set of <w, s>-pairs

13. Three traditional ideas:
What are Meanings?
Three traditional ideas:
concepts (mental representations of some sort),
with thoughts as special cases of concepts
extensions of ideal concepts,
with truth conditions as special cases of extensions
instructions of some kind
for how to “use” pronunciations
for how to build concepts of a special sort

14. What are Meanings? And do they compose? Three traditional ideas:
concepts (mental representations of some sort), ✔
with thoughts as special cases of concepts
extensions of ideal concepts, ✖
with truth conditions as special cases of extensions
instructions of some kind ✓?
for how to “use” pronunciations
for how to build concepts of a special sort

15. Are most of the dots yellow?
6 blue
1 to 2 SUPER EASY
How is the sentence understood?
What question is getting asked?

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

17. Most of the dots are yellow
MOST[DOT, YELLOW]
1 to 2 SUPER EASY
#{DOT & YELLOW} > #{DOT}/2
#{DOT & YELLOW} > #{DOT & YELLOW}
#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

18. Hume’s Principle #{Triangle} = #{Heart} iff
{Triangle} OneToOne {Heart}
____________________________________________
#{Triangle} > #{Heart}
{Triangle} OneToOnePlus {Heart}
α OneToOnePlus β iff for some α*,
α* is a proper subset of α, and α* OneToOne β
(and it’s not the case that β OneToOne α)

19. Most of the dots are yellow
MOST[DOT, YELLOW]
OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW]
1 to 2 SUPER EASY
#{DOT & YELLOW} > #{DOT}/2
#{DOT & YELLOW} > #{DOT & YELLOW}
#{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}

20. Most of the dots are yellow
What conditions
make the
sentence
easy/hard
to evaluate?
That might
provide
clues
about
how the
sentence is understood
1 to 2 SUPER EASY
(given independent accounts of the information available
to human beings in those conditions).

21. ‘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}
Number Representations
These analyses are provably equivalent (for finite cases)
and not crazy

22. ‘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}
Number Representations

23. 4:5 (blue:yellow) scattered random scattered pairs column pairs mixed
column pairs sorted

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

25. 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 Sorted Columns

26. but even better performance on the components of a
1-to-1-plus task if the question is not posed with ‘most’
10 : 20
10 : 10
10 : 15

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

28. better performance on easier ratios: p < .001

29. no effect of number of colors

30. fit to psychophysical model of ANS-driven performance

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

33. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be as primitive
as ‘dot’ and ‘yellow’ seem to be?
Some of the yellow dogs barked  Some of the dogs barked
Some of the dogs barked loudly  Some of the dogs barked
None of the yellow dogs barked  None of the dogs barked
None of the dogs barked loudly  None of the dogs barked

34. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be as primitive
as ‘dot’ and ‘yellow’ seem to be?
All of the yellow dogs barked  All of the dogs barked
All of the dogs barked loudly  All of the dogs barked
Most of the yellow dogs barked -- Most of the dogs barked
Most of the dogs barked loudly  Most of the dogs barked

35. ‘Most of the dots are yellow’
MOST[D, Y]
Why analyse at all? Why not take ‘Most’ to be as primitive
as ‘dot’ and ‘yellow’ seem to be?
Most of the dogs barked  More than half of the dogs barked
Most of the dogs barked  More dogs barked than didn’t
Most of the yellow dogs barked -- Most of the dogs barked
Most of the dogs barked loudly  Most of the dogs barked

36. Most of the dots are yellow
What conditions
make the
sentence
easy/hard
to evaluate?
That might
provide
clues
about
how the
sentence is understood
1 to 2 SUPER EASY
(given independent accounts of the information available
to human beings in those conditions).

37. There are some yellow dots, and there are some blue dots.
1 to 2 SUPER EASY
Many strategies that you might use to evaluate
the conjunctive sentence in a particular situation,
depending on the situation and relevant knowledge.

38. There are some yellow dots, and there are some blue dots.
1 to 2 SUPER EASY
Many strategies that you might use to evaluate
the conjunctive sentence in a particular situation.
But the meaning of ‘and’ invites a “default strategy.”

39. Question: α & β? Background Knowledge: (1) α & β if and only if (β & α) v ~(α & ~α) (2) α & β if Tim nodded (3) α & β only if Tim nodded

40. Question: #{D & Y} > #{D} – #{D & Y}
Question: #{D & Y} > #{D} – #{D & Y}? Background Knowledge: (1) #{D & Y} > #{D} – #{D & Y} if and only if #{D & Y} > #{D & Y} (2) #{D & Y} > #{D} – #{D & Y} if Tim nodded (3) #{D & Y} > #{D} – #{D & Y} only if Tim nodded

41. Meanings are Verification Procedures
Two ways of hearing this claim
(1) What are meanings?
(2) Please tell us some things about meanings.  

42. Meanings are Verification Procedures
Two ways of hearing this claim
(1) What are meanings?
 Meanings are mappings from contexts to contents.
Q: What are contexts?
A: Mappings from indices to domain entities
Q: What are contents?
A: Mappings from possible worlds to truth values.
Q: What are possible worlds?
A: Please be quiet. I'm trying to say what meanings are.

43. Meanings are Verification Procedures
Two ways of hearing this claim
(1) What are meanings?
Meanings are mappings from contexts to contents.
Meanings are concepts.
Q: What are concepts?
A: Composable mental representations.
Q: What are those?
A: Go read all of Jerry Fodor's books.

44. Meanings are Verification Procedures
Two ways of hearing this claim
(1) What are meanings?
Meanings are mappings from contexts to contents.
Meanings are concepts.
Meanings are Verification Procedures
Q: What are Verification Procedures?
A: Rules for connecting words with epistemically special concepts.
Q: What are epistemically special concepts?
A: I’ll tell you later.

45. Meanings are Verification Procedures
Two ways of hearing this claim
(1) What are meanings?
Meanings are mappings from contexts to contents.
Meanings are concepts.
Meanings are Verification Procedures [this often goes very badly]
(2) Please tell us some things about meanings.
Slangs connect meanings with pronunciations.
Meanings compose.
It's often said that meanings are sets
Meanings provide default strategies
for how to evaluate sentences as true/false

46. Most of the dots are blue
What conditions
make the
sentence
easy/hard
to evaluate?
That might
provide
clues
about
how the
sentence is understood
1 to 2 SUPER EASY
(given independent accounts of the information available
to human beings in those conditions).

47. Most of the dots are blue?
MOST[D, B] #{D & B} > #{D} – #{D & B}
framing the question
with ‘most’
has effects that are
expected if the question
is understood in terms of
cardinality subtraction
Prima facie, this requires
a representation of #{D}
and a computation on
this representation.
Does use of ‘most’ reflect
ease/difficulty of representing #{D}?

48. % Of Undergrads Who Choose This Side (N= 48) Asked, “Most” = 58%
left
right
% Of Undergrads Who
Choose This Side (N= 48)
Asked, “Most” = 58%
Asked, “More” = 13%
G1: most of the dots are blue
Back
G2: there are more blue dots than yellow dots
2nd time: click for data

49. A) More of the dots are grey. B) Most of the dots are grey.
Which sentence would you choose to describe this picture?
A) More of the dots are grey.
B) Most of the dots are grey.
65% choose “more”

50. 74% choose “most” A) More of the dots are grey.
Which sentence would you choose to describe this picture?
A) More of the dots are grey.
74% choose “most” 
B) Most of the dots are grey.

51. Not just about recognition
80 participants asked to “draw” on an iPad More/Most of the dots are blue

52. Not just about recognition
Typical for “More of the dots are blue”

53. Not just about recognition
Typical for “Most of the dots are blue”

54. Centroid distance (adults)
More Most
Halberda, Pietroski, Hunter, Odic, Wellwood, & Lidz. (2012). More & most: spatial vision affects word understanding on an iPad. Vision Science Society annual meeting.

55. 4-8 yr olds (n=92)
More
Most

56. I Cognize, ergo I am prone to Framing Effects
A bat and a ball cost $1.10
The bat costs a dollar more than the ball
How much does the ball cost?
NOT ten cents…a dollar is not a dollar more than ten cents
Adam and Beth drive equal distances in a year.
Adam switches from a 12-mpg to 14-mpg car.
Beth switches from a 30-mpg to 40-mpg car.
Who will save more gas?
Adam: 10,000/12 = ,000/14 = 714 saving of 119 gallons
Beth: 10,000/30 = ,000/40 = 250 saving of 83 gallons

57. Tim
Hunter
A W
l e
e l
x l
i w
s o o d
Darko
Odic J e f L i d z
Justin Halberda