Semantic Typology for Human - PowerPoint Presentation

Slide1

Semantic Typology for Human i-Languages

Paul M. Pietroski, University of Maryland

Dept. of Linguistics, Dept. of PhilosophySlide2

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings” with unboundedly many

pronunciations

how many types

of meanings?

one answer, via Frege

on ideal languages:

<e> entity-

denoters

<

t> truth-evaluable sentences

possible

languages

Human Languages

if <

α

> and <

β

> are types,

then so is <

α

,

β

> Slide3

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings” with unboundedly many

pronunciations

how many types

of meanings?

one answer, via Frege on

ideal languages:

<e> entity-

denoters

<

t> truth-evaluable sentences

possible

languages

Human Languages

<

t

>

Sadie<e> is-a-horse(_)<e, t> Slide4

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings” with unboundedly many

pronunciations

how many types

of meanings?

one answer, via Frege on

ideal languages:

<e> entity-

denoters

<

t> truth-evaluable sentences

possible

languages

Human Languages

<

e

,

t

> Sadie<e> saw(_, _)<e,

<e,

t

>>

<

t

>

Sophie<e>Slide5

Barbara Partee (2006), “Do We Need Two Basic Types?”“…

Carstairs-McCarthy argues that the apparently universal distinction in human languages between sentences and noun phrases cannot be assumed to be inevitable….His work suggests…that there is also no conceptual necessity for the distinction between basic types e

and t….If I am asked why we take e

and t as the two basic semantic types, I am ready to acknowledge that it is in part because of tradition, and in part because doing so has worked well.…”

Some of Partee’s Suggested Ingredients for an Alternative

eventish

semantics for VPs: barked 

Bark(e,

e’) & Past(e)

chase 

Chase(e,

e’, e

’’)

Heim/Kamp for indefinite NPs: a dog

 Dog(e)entity/event neutrality, and

maybe predicate/sentence neutrality (cp. Tarski)Slide6

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings” with unboundedly many

pronunciations

how many types

of meanings (basic

or not)?

one answer, via Frege

on ideal languages:

<e

> entity-

denoters

<t> truth-evaluable sentences

possible

languages

Human Languages

if <

α

> and <

β

> are types, then so is <α, β> Slide7

<e> <t

> if <α

> and <β>, then <

α, β

>0. <

e> <t> (2)

1. <e,

e> <e,

t> <t, e

> <t, t

> (4) of <0, 0>2. eight of <0, 1> eight of <1, 0

> (32), including sixteen of <1, 1>

<e,

et> and <et, t> 3. 64 of <0, 2> 64 of <2, 0> (1408),

128 of <1, 2> 128 of <2, 1> including <e, <e

, et>> 1024 of <2, 2> and <et, <et, t>>

4. 2816 of <0, 3> 2816 of <3, 0>

5632 of <1, 3> 5632 of <1, 3

>

(2,089,472), including

45,056 of <2, 3> 45,056 of <3, 2> <

e, <e, <e, <et>> 1,982,464 of <3, 3> that’s a lot of typesSlide8

possible languages

Fregean

Languages with expression types: <

e>, <t

>, and if <

α> and <β

> are types, so is <α

, β>

Human

i

-Languages

Level-

n

Fregean

Languages with expression types: <

e>, <t>, and

the nonbasic types up to Level-

n

Pseudo-

Fregean

Languages with expression types: <e>, <t>, and a few of the nonbasic typesSlide9

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings”

with unboundedly many pronunciations

how many types

of meanings (basic or not)?

another answer:

<M> monadic predicates

<D> dyadic predicates

possible

languages

Human Languages

Horse(_)

On(_, _)

<

e

,

t

>

<

e, <e, t>> Slide10

We can imagine/invent a language that has…(1) finitely many

atomic monadic predicates: M

1(_) … M

k(_

)

(2) finitely many

atomic dyadic predicates:

D1

(_,

_) …

Dj(

_, _

)

(3) boundlessly many

complex monadic predicatesMonad +

Monad  Monad

Dyad + Monad



Monad

BROWN(

_)

+ HORSE(_)  BROWN(_)^HORSE(_) FAST(_)

+ BROWN(_

)

^

HORSE(

_

)



FAST(

_)^

BROWN(_)^HORSE(_)Slide11

We can imagine/invent a language that has…(1) finitely many atomic monadic

predicates: M1(

_) … Mk

(_)

(2)

finitely many atomic dyadic

predicates: D

1(

_, _

) … Dj

(_,

_)

(3) boundlessly many

complex monadic

predicatesMonad + Monad

 Monad Dyad

+ Monad  Monad

Φ(

_

)^Ψ

(

_) applies to e iff Φ(_) applies to e

and

Ψ

(

_

)

applies to

e

ON(_, _) + HORSE(_)  [

ON(_, _

)

^

HORSE(

_

)

]Slide12

We can imagine/invent a language that has…(1) finitely many atomic monadic

predicates: M1(

_) … Mk

(_)

(2)

finitely many atomic dyadic

predicates: D

1(

_, _

) … Dj

(_,

_)

(3) boundlessly many

complex monadic

predicatesMonad + Monad

 Monad Dyad

+ Monad  Monad

Φ(

_

)^Ψ

(

_) applies to e iff Δ(_, _)^Φ(_)

applies to e

iff

Φ

(

_

)

applies to

e and e bears Δ(_, _) to something

Ψ(_)

applies to

e

that

Φ

(

_

)

applies to

BROWN(

_

)

^

HORSE(

_

)



[

ON(

_

,

_

)

^

HORSE(

_

)

]Slide13

Monad + Monad  Monad

Dyad +

Monad  Monad

Φ(

_)^Ψ(

_)

applies to e

iff

Δ(

_, _)

^Φ

(_)

applies to

e

iff

Φ(

_) applies to e

and

e

bears

Δ(_, _) to something Ψ(_) applies to e that Φ(

_) applies to

FAST(

_

)

^

BROWN(

_

)

^

HORSE(_) [ON(_, _)^HORSE(_)]

FAST(

e

)

&

BROWN(

e

)

&

HORSE(

e

)

e’[

ON(

e

,

e

’

)

&

HORSE(

e

’

)

]

v[

BETWEEN(

x

,

y

,

z

)

&

SOLD(

z

, w, v, x)] Triad & Tetrad  Pentad v[Pentad(…v…)]  Tetrad

but ‘

&

’ and ‘

v

[…

v

…]

’ permit

a lot

more than ‘

^

’ and ‘



’Slide14

Monad + Monad  Monad

Dyad +

Monad  Monad

Φ(

_)^Ψ(

_)

applies to e

iff

Δ(

_, _)

^Φ

(_)

applies to

e

iff

Φ(

_) applies to e

and

e

bears

Δ(_, _) to something Ψ(_) applies to e that Φ(

_) applies to

FAST(

_

)

^

BROWN(

_

)

^

HORSE(_) [ON(_, _)^HORSE(_)]

FAST(

e

)

&

BROWN(

e

)

&

HORSE(

e

)

e’[

ON(

e

,

e

’

)

&

HORSE(

e

’

)

]



[

AGENT(

_

,

_

)

^

HORSE(

_

)

]^

FAST(

_

)^RUN(_) e’[AGENT(e, e’) & HORSE(e’)] & FAST(e) & RUN(e) [

AGENT(

_

,

_

)

^

HORSE(

_

)

^

FAST(

_

)

]^

RUN(

_

)

e’[

AGENT(

e

,

e

’

)

&

HORSE(

e

’

)

&

FAST(

e

’

)

]

&

RUN(

e

)Slide15

Monad + Monad  Monad

Dyad +

Monad  Monad

Φ(

_)^Ψ(

_)

applies to e

iff

Δ(

_, _)

^Φ

(_)

applies to

e

iff

Φ(

_) applies to e

and

e

bears

Δ(_, _) to something Ψ(_) applies to e that Φ(

_) applies to

PAST(

_

)

^

SEE(

_

)^

[THEME(_, _)^HORSE(_)]

PAST(e)

&

SEE(

e

)

&

e’[

THEME(

e

,

e

’

)

&

HORSE(

e

’

)

]

PAST(

_

)

^

SEE(

_

)

^



[

THEME(

_

,

_

)

^

RUN(

_

)^[AGENT(_, _)^HORSE(_)]] PAST(e) & SEE(e) & e’[THEME(e, e’) &

RUN(

e’) & e’’[AGENT(e’, e’’)^HORSE(e’’)]]

/ \saw / \ a horse

/ \

saw / \

/ \ run

a horseSlide16

How many types of meanings in human languages?

How do the meanings combine?

<

t> <

β> <M> <M>

/ \ / \ / \ / \<

e>

<e,

t> <

α,

β>

<α

>

<M>

<M> <D> <M> <e

, <e, t>> <

e> <<e,

t

>,

t

> <

e

, t> … … BROWN(_)^HORSE(_) [ON(_, _)

^HORSE(_

)

]

<

e

, t> / \

<e, t> <e, t> brown horse

possible languages

Human Languages

function application

and

(type-shifting or)

a rule for

adjunctionSlide17

Human Languages

acquirable by normal human children

given ordinary courses of experience

pair unboundedly many “meanings” with unboundedly many

pronunciations

in accord with language-specific constraints

Bingley was ready _ to please _.

Georgiana was eager _ to please _.

Darcy was easy _ to please _.

possible

languages

Human Languages

ambiguous

unambiguous

unambiguous

(the other way)Slide18

Human Languages

acquirable by normal human children given ordinary courses of experience

pair unboundedly many “meanings”

with unboundedly many pronunciations

in accord with language-specific

constraints

hiker lost kept walking circles

possible

languages

Human LanguagesSlide19

Human Languages

acquirable by normal human children given ordinary courses of experience

pair unboundedly many “meanings”

with unboundedly many pronunciations

in accord with language-specific constraints

The hiker who was lost kept walking in circles.

The hiker who lost was kept walking in circles.

Was the hiker who lost kept walking in circles?

(meaning 2)

possible

languages

Human LanguagesSlide20

Human Languages

acquirable by children unbounded but constrained

and so presumably

i-Languages

in Chomsky’s (Church-inspired) sense 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

)

possible

languages

Human LanguagesSlide21

Human Languages

acquirable by children unbounded but constrained

and so presumably

i-Languages in Chomsky’s (Church-inspired) sense

function-in-intension vs.

function-in-extension |

x – 1| +√(x

2 – 2x + 1)

{…, (-2, 3), (-1, 2), (0, 1), (1, 0), (2, 1), …}

λ

x .

|

x – 1|

≠ λx

. +√(x

2 – 2x + 1)

λ

x

.

|

x

– 1| = λx . +√(x2 – 2x + 1) Extension[λ

x . |x

– 1|

] =

Extension

[

λ

x

.

+√(x2

– 2x + 1)]possible languages

Human LanguagesSlide22

Human Languages

acquirable by children unbounded but constrained

biologically implemented procedures that pair

“meanings” with sounds/gestures in a human way

how many

types of meanings?

one hypothesis, via Frege on

ideal languages:

<e> entity-denoters

<t> truth-evaluable sentences

if <α> and <

β> are types, then so is <α, β

>

possible

languages

Human LanguagesSlide23

Human Languages

natural generative procedures

how many types of meanings?

one hypothesis, via Frege on

ideal languages: <

e> <

t> if <α> and <

β>, then <α,

β>

THREE CONCERNS: available evidence suggests that… the proposed generative principle

overgenerates (massively) Human Languages don’t generate expressions of type <

e> Human Languages don’t generate expressions of type <

t

>

possible

languages

Human LanguagesSlide24

<e> <t

> if <α

> and <β>, then <

α, β

>one worry…

overgeneration

0. <e> <t> (2)

1. <e,

e> <e,

t> <t,

e> <t, t

> (4)2. eight of <0, 1> eight of <1, 0>

sixteen of <1, 1>

(32)3. 64 of <0, 2> 64 of <2, 0>

128 of <1, 2> 128 of <2, 1> 1024 of <2, 2> (1408)4. 2816 of <0, 3> 2816 of <3, 0>

5632 of <1, 3> 5632 of <1, 3> 45,056 of <2, 3> 45,056 of <3, 2> (2,089,471)

1,982,464 of <3, 3>

possible

languages

Human LanguagesSlide25

Human Languages

natural generative procedures

how many types of meanings?

one hypothesis, via Frege on

ideal languages: <

e> <

t> if <α> and <

β>, then <α,

β>one worry…

overgeneration

<e, <

e, <e, <e

, <

e, t>>>>>

λv. λ

w. λ

z . λy.

λ

x

.

GRONK(x

,

y, z, w, v) <<<e, t>, <<e, t> , <e, t>>> λZ . λY. λX . GLONK(X, Y, Z)

<<e, t>, <t

,

e

>>

<<

e, t

>, e> ???

<e, e> ??? possible languages

Human Languages



x[X(x

)

v

Y(x

)

v

Z(x

)]



x[X(x

)] &



x[Y(x

) &

Z(x

)]Slide26

Human Languages

natural generative procedures

how many types of meanings?

one hypothesis, via Frege on

ideal languages: <

e> <

t> if <α> and <

β>, then <α,

β>one worry…

overgeneration

<e, <

e, <e, <e

, <

e, t>>>>>

λw. λ

z . λ

y. λx .

λ

e

.

SELL(e

,

x, y, z, w) <e, <e, <e, <e, t>>>> λz . λy

. λx .

λ

e

.

GIVE(e

,

x

, y, z)

<e

, <e, <e, t>>> λy. λx . λe.

KICK(e, x, y)

possible

languages

Human Languages

claim

: verbs don’t provide evidence for “

supradyadic

” types Slide27

a linguist sold a car to a friend for a dollar

(

x

) (

y

) (

z

) (

w

)Slide28

a linguist sold a friend a car for a dollar

(

x

) (

z

) (

y

) (

w

)Slide29

a linguist sold a friend a car a dollar

‘sold’



λ

z

.

λ

y

.

λ

w

.

λ

x

.

x

sold

y

to

z

for

w (x) (z) (y) (w)Why not just…

λz.

λ

y

.

λ

w

.

λ

x .

λe . e was a selling by x of y to z for w

a

triple

-object construction

she her this that Slide30

(

x

) (

y

) (

z

)

a thief jimmied a lock with a knife Slide31

(

x

) (

y

) (

z

)

a thief jimmied a lock a knife

Why not instead…

‘jimmied’



λ

z

.

λy

. λx .

x

jimmied

y

with

zSlide32

(

x

) (

y

) (

z

)

a rock betweens a lock a knife

Why not…

‘betweens’



λ

z

.

λy

. λx .

x

is

between

y

and zSlide33

(

x

) kicked (

y

)

gave

a linguist tossed a coinSlide34

(

x

) kicked (

y

) (

z

)

gave

a linguist tossed a coin to a friend Slide35

a linguist tossed a friend a coin

‘tossed’



λ

z

.

λ

y

.

λ

x

.

x

tossed

y to z‘kicked’

 λz

.

λ

y

.

λ

x . x kicked y to z (x) kicked (z) (y) Why not…

λ

z

.

λ

y

.

λ

x .

λe . e was a kicking by x of y to zSlide36

(

y

) kicked (

x

)

a coin was tossed (by a linguist)

‘tossed’



λ

y

.

λ

e

. e was a tossing of

y‘kicked’ 

λ

y

.

λ

e

. e was a kicking of yif Kratzer andothers are on the right track…representing an Agent is as optional as representing a RecipientSlide37

(

x

) kicked (

y

)

a linguist tossed a coin

if

Kratzer

and

others are

on the right track…

‘kicked’



λ

y. λe

. e was a kicking of y

[

AGENT(

_

,

_

)

^LINGUIST(_)]^PAST(_)^[KICK(_, _)^COIN(_)

]

or

… ^

PAST(

_

)

^

KICK(

_)^[

PATIENT(_, _)^COIN(_)]Slide38

Chris hailed from Boston

Why …

and not…

Chris

frommed

Boston

λ

y

.

λ

x

.

λ

e

.

e

was a hailing by

x

from

ySlide39

Chris hailed from Boston

Chris was taller than Sam

Why …

and not…

Chris

frommed

Boston

Chris

talled

Sam

outheighted

Slide40

Human Languages

natural generative procedures

how many types of meanings?

one hypothesis, via Frege on

ideal languages: <

e> <

t> if <α> and <

β>, then <α,

β>

THREE CONCERNS: it seems that…

✔ the proposed generative principle overgenerates (massively)

Human Languages don’t generate expressions of type <e>

Human Languages don’t generate expressions of type <

t>

possible

languages

Human LanguagesSlide41

Which expressions are of type <e>?

NAMES

® ©

--|-- --|--

‖Robin<

e>‖= / \ ‖

Cruso<e>

‖= / \‖Robin<<

e, t>,

t>‖=

P . P(®) ‖

Cruso

<<e,

t>, t>‖= P . P(©)

‖Robin<e,

t>‖= x . x =

®

‖

Cruso

<

e

, t>‖= x . x = © x . Robin(x) x . Cruso(x)‖[D1<e, t> Robin<e,

t>]<e,

t

>

‖ =



x

. Indexes(1,

x) & Called(x, ‘Robin’)Slide42

Proper Nouns

even English tells against the idea that

lexical proper nouns are

i-language expressions of type <

e>

Every Tyler I saw was a philosopher

Every philosopher I saw was a Tyler

That Tyler stayed late, and so did this one

There were three

Tylers

at the party, and

Tylers

are clever

The

Tylers

are coming to dinner (That nice) Professor Tyler Burge

was sitting next to John Jacob Jingleheimer

Schmidt

proper

nouns

seem to be of type <M>, even if they are

related to singular concepts of type <e>Slide43

D(P) D(P) N(P) / \ / \ / \ D N D N N C(P)

every tiger most tigers tiger(s) that I saw the Tyler some

Tylers Tyler(s)

this those

?

/ \Tyler Burge<

e> <e

>

/ \ that <

e, t

> / \ nice ?

/ \

Professor Burge

<

e>

<e, t

> / \ Prof. <e, t

>

<

e

,

t

> / \ Tyler Burge <e, t> <e, t>Slide44

Which expressions are of type <e>?

V(P)

/ \

D(P) V(P)

/ \

/ \

D N

V D(P)

that woman tossed / \

D N this coin

if the nouns are of type <

e

,

t>then presumably, the indexed determiners are not of type <

e>;if ‘that’ and ‘this’ are of type <

e, t>then presumably, the indices are

not

of type <

e

>;

1

2  [AGENT(_, _)

^THAT(_)

^

1

(_)^

WOMAN(

_

)

]^…Slide45

Which expressions are of type <e>?

V(P)

/ \

/ V(P)

/

/ \

D(P)

V D(P)

she tossed \

it

if ‘the pronouns ‘she’ and ‘it’ are of type <e

, t>

then presumably, the indices are

not of type <

e> ‖she

<e, t>‖=

x . x is female

‖1

<

e

,

t>‖= x . Indexes(1, x)12  [

AGENT(_, _

)

^

FEMALE(

_

)

^

1

(_)]^… Slide46

Human Languages

natural generative procedures

how many types of meanings?

one hypothesis, via Frege on

ideal languages: <

e> <

t> if <α> and <

β>, then <α,

β>

THREE CONCERNS: it seems that…

✔ the proposed generative principle overgenerates (massively)

✔ Human Languages don’t generate expressions of type <

e

> Human Languages don’t generate expressions of type <

t>

possible

languages

Human LanguagesSlide47

Which expressions are of type <t>?

SENTENCES S  NP aux VP

NP aux VP

D + N 

D(P) V + D(P)  V(P)

every tree / \ saw / \ / \

D N D N V D(P)

every tree every tree saw / \

D N

every tree

? + ??

 S / \

? ??

S Slide48

Which expressions are of type <t>?

SENTENCES S = TP

T(P)

/ \

T V(P)

past / \

D(P) V(P)

John / \ V D(P)

see Mary



e

. T  e is (tenselessly

) an event of John seeing Mary

y . 

x

.



e

. T

 e is (tenselessly) an event of x seeing y Slide49

Which expressions are of type <t>?

SENTENCES S = TP

T(P) / \

T V(P)

past / \

D(P) V(P)

John / \

V D(P) see Mary





e . T  e is (

tenselessly) an event of John seeing Mary

 Why think

TPs

are of type <

t

> instead of <

e

, t> ?Is this expression of type <<e, t>, t> or type <e

, t> ?Slide50

Which expressions are of type <t>?

SENTENCES S = TP

T(P) / \

T V(P)

past / \

D(P) V(P)

John / \

V D(P) see Mary





e . T  e is (

tenselessly) an event of John seeing Mary

 Why think

TPs

are of type <

t

> instead of <

e

, t> ?<<e, t>, t> E . T  

e[e is in the past & E(e) = T]

<

e

,

t

>



e

. T

 e is in the pastSlide51

Which expressions are of type <t>?

SENTENCES S = TP+

?

/ \

 T(P) / \

T V(P)

past / \

D(P)

V(P) John / \

V D(P) see Mary



e . T 

e is (tenselessly) an event of John seeing Mary





e

. T

 e is in the past & …<e,

t> 

e

. T



e

is in the pastSlide52

Which expressions are of type <t>?

?

/ \

 T(P) / \

T V(P)

past / \

D(P)

V(P) John / \

V D(P) see Mary



e . T

 e is (tenselessly

) an event of John seeing Mary





e

. T

 e is in the past & …<e, t>

e . T



e

is in the past

??

neg

Slide53

Which expressions are of type <t>?

SENTENCES S = TP

T(P) / \

T V(P)

past / \

D(P) V(P)

John / \

V D(P) see Mary





e . T  e is (

tenselessly) an event of John seeing Mary



T





e[e

is in the past & …] <<e, t>, t> E . T 

e[e is in the past &

E(e

) = T]

vv

is T a quantificational argument of V

and

a conjunctive adjunct? Slide54

Which expressions are of type <t>?

SENTENCES S = TP (or TP+)

V(P) / \

D(P) V(P)

That / \

D(P) V(P) John / \

V D(P)

see Mary





e . T

 e is (tenselessly

) an event of John seeing Mary

 T 

That is (

tenselessly

) an event of J seeing M

Tense may be needed (in matrix clauses). But does it do

two semantic

jobs: adding time information via the ‘e’-variable, like the adjunct ‘yesterday’; and closing the ‘e’ variable, as if tense is the 3rd argument of a verb that can’t take a 3

rd argument?

if T is (semantically) the verb’s 3

rd

argument, then why not…Slide55

Which expressions are of type <t>?

SENTENCES S = TP

T(P)

/ \ T

V(P)

past / \

D(P) V(P) John / \

V D(P)

see Mary





e . T

 e is (tenselessly

) an event of John seeing Mary

 T





e[

PAST(e

) & …] <<e, t>, t> E . T 

e[PAST(e

)

&

E(e

) = T]

“God likes

Fregean

Semantics” theory of tense

(e < RefTime) & (

RefTime = SpeechTime

) Slide56

Which expressions are of type <t>?

Maybe None:

T(P) / \

T V(P)

past

/ \ D(P)

V(P) John / \

V D(P)

see Mary



PastSeeingOfMaryByJohn

(_)

a monadic predicate M can be “polarized” into a predicate

+M that applies to e iff

M

applies to

something

or a predicate -M that applies to e iff M applies to nothing+Polarized |

 _ is such that 

[

PastSeeingOfMaryByJohn

(_)]Slide57

But what about

Quantification

?

<

t>

/ \

<et, t

> <et> / \ ran

<et, <et, t

>> <et> every cow

Not at all clear

that the “external argument” of ‘every cow’ is—

or even can be—an expression of type <et>Slide58

<?> / \ Fido <?>

/ \ chased

<et, t>

/ \ every cow

<

e

t>

/ \ <e

t>

<e

t> / \ today

<e

> <

e,

et> Bessie ran

<?>/ \

<et> today

<

e

t>

/ \

<?>

<et> / \ today <et, t> <e, et> / \ ran every cow Slide59

<et> / \ <?>

<et>

/ \ today <et,

t> <

e, e

t> / \ ran

every cow

<

et>

/ \ <e

t>

<e

t>

/ \ today

<e> <e

, et> (i)t

1 ran every cow

<

t

>

<

t

>

<et>



1

<et,

t

>Slide60

<t> / \ <et,

t> <et>

/ \ / \every cow which ran

<

et>

/ \ <

et>

<e

t> / \ today

<e

> <e

, e

t>

(i)t1

ran every cow

<

t

>

<

t

>

<et>



1

<et,

t

>

Why not…

very

syncategorematicSlide61

<?>/ \

<et>

today

<?> / \

Fido

<?>

/ \ chased

<et, t>

/ \

every cow Slide62

<et>/ \ <

et> today

<

et>

/ \

Fido <

e,

et>

/ \ chased

<et, t>

/ \

every cow

<<et,

t

>, <<

e,

et>>

We can discuss the difficulties for this kind of view in Q&A.

But my point is not that an <et,

t

> analysis of quantifiers

cannot be

preserved. My point is that there is no argument here for the standard typology, especially given doubts about <e>.<<1, 0>, <<

0, 1

>>

<

2

, 2>

<3>Slide63

Human Quantification: Still PuzzlingBut maybe… ‘every’ is a

plural Dyad:

EVERY(_,

_) applies to <the Xs, the

Ys> iff

the Xs

include the Ys

‘every cow’ is a

plural Monad

: 

[EVERY(

_,

_

)

^THE-COWS(_)

] applies to the Xs

iff the Xs include the cows

‘every cow ran’ is a

plural

Monad

:



[EVERY(_, _)^THE-COWS(_)]^RAN(_)

applies to the Xs iff the Xs include the cows & the Xs ran

Slide64

Human Quantification: Still PuzzlingBut maybe… ‘most’ is a

plural Dyad:

MOST(_,

_) applies to <the Xs, the

Ys> iff

the

Ys that are Xs outnumber the

Ys that are not Xs

‘most

cows’ is a plural

Monad:

[

MOST (

_

, _)^

THE-COWS(_)

] applies to the Xs iff the

cows that are Xs

outnumber

the cows that are not Xs

‘most cows ran’ is a

plural

Monad: [MOST(_, _)^THE-COWS(_)

]^RAN(_) applies to the Xs

iff

the

cows that are Xs

outnumber

the other cows

Slide65

Human Quantification: Still PuzzlingBut maybe… ‘every’ is a

plural Dyad:

EVERY(_,

_) applies to <the Xs, the

Ys> iff

the Xs

include the Ys

‘every cow’ is a

plural Monad

: 

[EVERY(

_,

_

)

^THE-COWS(_)

] applies to the Xs

iff the Xs include the cows

‘every cow ran’ is a

plural

Monad

:



[EVERY(_, _)^THE-COWS(_)]^RAN(_)

applies to the Xs iff the Xs include the cows & the Xs ran

Slide66

<M> / \ t1

ran every cow

<M>

<M>

polarity

<M>

for any assignment A of values to variables…

applies to

e

iff

there was

an event of A1 running

applies to

e

iff

e

was

an event of A1 runningSlide67

<M> / \ t1

ran every cow

<M>

<M>

polarity

<M>

for any assignment A of values to variables…

applies to

e

iff

A1 ran

so if

e

is the value of 1 (but A is otherwise the same),

then the “polarized” predicate applies to

e

iff

e

ran Slide68

<M> / \ t1

ran every cow

<M>

<M>

polarity

<M>

for any assignment A of values to variables…

applies to

e

iff

A1 ran

and we can define a “

Tarski

Relation” such that

TARSKI(e

,

Polarity[

t

1

ran], 1)

iff

e

ran Slide69

<M> / \ t1

ran every cow

<M>

<M>

polarity

<M>

for any assignment A of values to variables…

applies to

e

iff

A1 ran

TARSKI(e

,

Polarity[

t

1

ran], 1)

A*:A*≈

1

A

{Satisfies

(A*,

Polarity[

t

1

ran]

) & (

e

= A*[

1

])}Slide70

<M> / \ t1

ran every cow

<M>

<M>

polarity

<M>

for any assignment A of values to variables…

applies to

e

iff

e

ran

we can define a “

Tarski

Relation” such that

TARSKI(e

,

Polarity[

t

1

ran], 1)

iff

e

ran

syncategorematic

,

but honestly soSlide71

Human Quantification: Still PuzzlingBut maybe… ‘every’ is a

plural Dyad:

EVERY(_,

_) applies to <the Xs, the

Ys> iff

the Xs

include the Ys

‘every cow’ is a

plural Monad

: 

[EVERY(

_,

_

)

^THE-COWS(_)

] applies to the Xs

iff the Xs include the cows

‘every cow ran’ is a

plural

Monad

:



[EVERY(_, _)^THE-COWS(_)]^RAN(_)

applies to the Xs iff the Xs include the cows & the Xs ran

Slide72

Lots of Further Issues

Quantification Homework (including

conservativity)

Mary saw John,

and John did

n’t

see Mary

lexicalization of singular and relational concepts

lexical

inflexibilities

*Chris sneezed the baby

*Chris put the letter

Gleitman-esque

acquisition of verbs

Your Objection HereSlide73

Human Languages

acquirable by normal human children

given ordinary courses of experience

generatively pair meanings with gestures

in accord with human constraints

possible

languages

Fregean

Languages with expression types:

<

e

>, <

t>, and if <α> and <

β> are types, so is <α

, β>

Human

i

-LanguagesSlide74

possible languages

Fregean

Languages with expression types: <

e>, <t

>, and if <

α> and <β

> are types, so is <α

, β>

Human

i

-Languages

Level-

n

Fregean

Languages with expression types: <

e>, <t>, and

the nonbasic types up to Level-

n

Pseudo-

Fregean

Languages with expression types: <e>, <t>, and a few of the nonbasic typesSlide75

Semantic Typology for Human I-Languages

THANKS!Slide76

Human Quantification: Still PuzzlingBut maybe...DET(

_, _)

applies to <the Xs, the Ys> only if the Xs are among the

Ys

EVERY(_

, _)

applies to <the Xs, the Ys> only if the Xs

are the Ys

MOST(

_,

_) applies to <the Xs, the

Ys> only if #(X) > #(Y) − #(X)Slide77

Monad + Monad  Monad

Dyad +

Monad  Monad

Φ(_

)^Ψ(_

) applies to

e

iff

Δ(_, _

)^

Φ(

_) applies to

e

iff

Φ(_

) applies to e

and

e

bears

Δ

(_, _) to something Ψ(_) applies to e that Φ(_)

applies to

FAST(

_

)

^

BROWN(

_

)

^HORSE(_

) ON(_, _)^HORSE(_) FAST(e)

& BROWN(e

)

&

HORSE(

e

)

e[

ON(

e

’

,

e

)

^

HORSE(

e

)

]

HORSE(

e

)

s[

EXEMPLIFIES(

e

,

s

)

&

HORSEY(

s

)

]

&

i[

AT(

s

, i) & NOW(i)] Slide78

Which expressions are of type <t>?

Maybe None

<t

> / \

<

t>

<t

, t

>Mary saw John / \

<t, <

t

,

t>> <

t> and John saw Mary

Slide79

<M>

/ \ <M>

<M>

Mary saw John / \

<D>

<M>

before John saw Mary



[Before

(_, _)^JohnSeeMary(_)

]



Past(_)^MarySeeJohn(_) & …Slide80

<M> / \

<M>

<M>Mary saw John / \

<e

, M>

<M>

before John saw Mary

and





f[Before(e, f

) & f

is an event of John seeing Mary]



e

was an event of Mary seeing John & …