Procedure Matters Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy...

Procedure Matters

Paul M. PietroskiUniversity of Maryland

Dept. of Linguistics, Dept. of Philosophyhttp://www.terpconnect.umd.edu/~pietro

Most of the dots are yellow

15 dots:9 yellow6 blue

‘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 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)

‘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 outnumber the 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 minus the number of yellow dots

(9 > 15 – 9)

Most of the dots are yellow

15 dots:9 yellow6 blue

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)}

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)

‘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)}]

‘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???

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.”

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

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

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(I-Languages)

(E-Languages)

‘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)

I-Language/E-Language

function in Intension implementable 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)]

I-Language/E-Language

function in Intension implementable 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)

I-Language/E-Language

function in Intension implementable 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

I-Language/E-Language

function in Intension implementable 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

TimHunter

DarkoOdic

J e f f

L i d z

Justin Halberda

A Wl ee lx li ws o o d

Most of the dots are yellow

‘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}

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).

‘Most of the dots are yellow’

MOST[D, Y]

OneToOnePlus[{D & Y},{D & Y}]

#{D & Y} > #{D & Y}

#{D & Y} > #{D} – #{D & Y}

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

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)

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

4:5 (blue:yellow)“scattered pairs”

1:2 (blue:yellow)“scattered pairs”

4:5 (blue:yellow)“scattered pairs”

9:10 (blue:yellow)“scattered pairs”

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

4:5 (blue:yellow)“column pairs mixed”

5:4 (blue:yellow)“column pairs mixed”

4:5 (blue:yellow)scattered random

column pairs mixed

scattered pairs

column pairs sorted

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, yada yada…

50

60

70

80

90

100

1 1.5 2Ratio (Weber Ratio)

Perc

en

t C

orr

ect

Scattered Random

Scattered Pairs

Column Pairs Mixed

Column Pairs Sorted

better performance on easier ratios: p < .001

10 : 1010 : 15

10 : 20

fits for trials (apart from Sorted-Columns) to a standard psychophysical model for predicting ANS-driven performance

fits for Sorted-Columns trials to an independent model for detecting the longer of two line segments

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

4:5 (blue:yellow)scattered random

column pairs mixed

scattered pairs

column pairs sorted

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?

4:5 (blue:yellow)“scattered pairs”

What color are the loners?

better performance on components of a 1-to-1-plus task

10 : 1510 : 10 10 : 20

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 not verify

‘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

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’

fits for trials (apart from Sorted-Columns) to a standard psychophysical model for predicting ANS-driven performance

fits for Sorted-Columns trials to an independent model for detecting the longer of two line segments

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

Side Point Worth Noting…50% plus a tad

‘Most of the dots are yellow’

MOST[D, Y]

OneToOnePlus[{D & Y},{D & Y}]

#{D & Y} > #{D & Y}

#{D & Y} > #{D} – #{D & Y}

‘Most of the dots are blue’

#{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)}

#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

• if there are only two colors to worry about, say blue and red, then the non-blues can be identified with the reds

‘Most of the dots are blue’

#{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)}#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

if there are only two colors to worry about, say blue and red, then the non-blues can be identified with the reds

• the visual system can (and will) “select” the dots, the blue dots, and the red dots;

so the ANS can estimate these three cardinalities

but adding more colors will make it harder (and with 5 colors, impossible) for the visual system to make enough “selections” for the ANS to operate on

‘Most’ as a Case Study

‘Most of the dots are blue’

#{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)}#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

• adding alternative colors will make it harder (and eventually impossible) for the visual system to make enough “selections” for the ANS to operate on

• so given the first proposal (with negation), verification should get harder as the number of colors increases

• but the second proposal (with subtraction) predicts relative indifference to the number of alternative colors

better performance on easier ratios: p < .001

no effect of number of colors

fit to psychophysical model of ANS-driven performance

‘Most’ as a Case Study

‘Most of the dots are blue’

#{x:Dot(x) & Blue(x)} > #{x:Dot(x) & ~Blue(x)}#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

• adding alternative colors will make it harder (and eventually impossible) for the visual system to make enough “selections” for the ANS to operate on

• so given the first proposal (with negation), verification should get harder as the number of colors increases

• but the second proposal (with subtraction) predicts relative indifference to the number of alternative colors

‘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???

‘Most’ as a Case Study

‘Most of the dots are blue’#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

• mass/count flexibilityMost of the dots (blobs) are brown

Most of the goo (blob) is brown

• are mass nouns (somehow) disguised count nouns? #{x:GooUnits(x) & BlueUnits(x)} > #{x:GooUnits(x)} − #{x:GooUnits(x) & BlueUnits(x)}

discriminability is BETTER for ‘goo’ (than for ‘dots’) w = .18r2 = .97

w = .27r2 = .97

Are more of the blobs blue or yellow? If more the blobs are blue, press ‘F’. If more of the blobs are yellow, press ‘J’.

Is more of the blob blue or yellow? If more the blob is blue, press ‘F’. If more of the blob is yellow, press ‘J’.

Ratio (Bigger Quantity/ Smaller Quantity)

1.0 1.2 1.4 1.6 1.8 2.0 2.2

% C

orre

ct

50

55

60

65

70

75

80

85

90

95

100

Mass DataMass ModelCount DataCount Model

w = .20r2 = .99

w = .29r2 = .98

Performance is better (on the same stimuli) when the question is posed with a mass noun

‘Most’ as a Case Study

‘Most of the dots are blue’#{x:Dot(x) & Blue(x)} > #{x:Dot(x)} − #{x:Dot(x) & Blue(x)}

• mass/count flexibilityMost of the dots (blobs) are brown

Most of the goo (blob) is brown

• are mass nouns disguised count nouns? #{x:GooUnits(x) & BlueUnits(x)} > #{x:GooUnits(x)} − #{x:GooUnits(x) & BlueUnits(x)}

SEEMS NOT

Procedure Matters

...Psychophysics, on the other hand, is related more directly to the level of algorithm and representation. Different algorithms tend to fail in radically different ways as they are pushed to the limits of their performance or are deprived of critical information.

As we shall see, primarily psychophysical evidence proved to Poggio and myself that our first stereo-matching algorithm was not the one used by the brain, and the best evidence that our second algorithm (Marr and Poggio, 1976) is roughly the one used also comes from psychophysics. Of course, the underlying computational theory remained the same in both cases, only the algorithms were different.Psychophysics can also help to determine the nature of a representation...

THANKS

GLONK

GLONKS GLONK

GLONKS GRUP GLONK

FLIB GRUP FLIB GRONK FLIB GRONK

FLORT GRONK

But often, the world isn’t this helpful

GLONKS GRUPS GLONKLED FLIB GRUPE FLIB GRONK FLIB GRONK

FLORT GRONK

DRIV WONK HORTLE BING

GLONKS GRUPS GLONKLED

FLIB GRUPE FLIB GRONK

FLIB GRONK FLORT GRONK

DRIV WONK HORTLE BING

GLONKTRIANGLETRILATERALDETACHED DIAMOND-HALFetc.

I-Language/E-Language

function in Intension implementable procedure

that pairs inputs with outputs

function in Extension set of input-output pairs

In drawing this distinction, 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 ‘glonk’) means

Church (1941) on Lambdas1: 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 caluclus 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.”

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 ...”

Later on, in replies to objections...

• “We may define a class of objects called grammars... A grammar uniquely determines the language it generates. But a language does not uniquely determine the grammar that generates it...”

Lewis, “Languages and Language”

“I know of no promising way to make objective sense of the assertion that a grammar Γ is used by a population P, whereas another grammar Γ’, which generates the same language as Γ, is not. I have tried to say how there are facts about P which objectively select the languages used by P. I am not sure there are facts about P which objectively select privileged grammars for those languages...a convention of truthfulness and trust in Γ will also be a convention of truthfulness and trust in Γ’ whenever Γ and Γ’ generate the same language.”

“I think it makes sense to say that languages might be used by populations even if there were no internally represented grammars. I can tentatively agree that £ is used by P if and only if everyone in P possesses an internal representation of a grammar for £, if that is offered as a scientific hypothesis. But I cannot accept it as any sort of analysis of “£ is used by P”, since the analysandum clearly could be true although the analysans was false.”

‘I’ Before ‘E’

• Church: 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)

• Chomsky: I-language vs. E-language

--an implementable procedure that generates expressions: π-λ DS-SS-PF DS-SS-PF-

LF PHON-SEM

(a) ‘generate’ as in ‘These axioms generate the natural numbers’

(b) procedure...a LEXICON plus a COMBINATORICS

(c) open question how such procedures are used in events of comprehension/production/thinking/judging-

acceptability