i Languages Paul M Pietroski University of Maryland Dept of Linguistics Dept of Philosophy Human Languages acquirable by normal human children given ordinary courses of experience ID: 593603
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Slide1
Semantic Typology for Human iLanguages
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> truthevaluable 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> truthevaluable sentences
possible
languages
Human Languages
<
t
>
Sadie<e> isahorse(_)<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> truthevaluable 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?”“…
CarstairsMcCarthy 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> truthevaluable 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
(typeshifting 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 languagespecific 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 languagespecific
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 languagespecific 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
iLanguages
in Chomsky’s (Churchinspired) sense functioninintension vs.
functioninextension 
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
iLanguages in Chomsky’s (Churchinspired) sense
functioninintension vs.
functioninextension 
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> entitydenoters
<t> truthevaluable 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
ilanguage 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(
_,
_
)
^THECOWS(_)
] applies to the Xs
iff the Xs include the cows
‘every cow ran’ is a
plural
Monad
:
[EVERY(_, _)^THECOWS(_)]^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 (
_
, _)^
THECOWS(_)
] 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(_, _)^THECOWS(_)
]^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(
_,
_
)
^THECOWS(_)
] applies to the Xs
iff the Xs include the cows
‘every cow ran’ is a
plural
Monad
:
[EVERY(_, _)^THECOWS(_)]^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(
_,
_
)
^THECOWS(_)
] applies to the Xs
iff the Xs include the cows
‘every cow ran’ is a
plural
Monad
:
[EVERY(_, _)^THECOWS(_)]^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
Gleitmanesque
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 ILanguages
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 & …