Transcript Semantic Typology and Composition Paul M. Pietroski, University of Maryland Dept. of Linguistics, Dept.

Semantic Typology and Composition
Paul M. Pietroski, University of Maryland
Dept. of Linguistics, Dept. of Philosophy
Human children naturally acquire languages
that somehow generate boundlessly many expressions
that connect “meanings” with “pronunciations”
in accord with certain constraints.
Bingley is eager to please.
(a) Bingley is eager to please relevant parties.
#(b) Bingley is eager to be pleased by relevant parties.
Bingley is easy to please.
#(a) Bingley can easily please relevant parties.
(b) Bingley can easily be pleased by relevant parties.
2
Human children naturally acquire languages
that somehow generate boundlessly many expressions
that connect “meanings” with “pronunciations”
in accord with certain constraints.
The senator called the donor from Texas.
(a) The senator called the donor, and
the donor was from Texas.
(b) The senator called the donor, and
the call was from Texas.
#(c) The senator called the donor, and
the senator was from Texas.
3
Human Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what are these meanings?
4
Human Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what types of meaning do human linguistic expressions exhibit?
(1)
(2)
(3)
(4)
Fido
chase
every
cat
(5)
(6)
(7)
(8)
every cat
chase every cat
Fido chase every cat
Fido chased every cat.
5
Human Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what are the Human Meaning Types?
• standard answer, via Frege’s conception of ideal languages
Fido
(i) a basic type <e>, for entity denoters
(ii) a basic type <t>, for truth-value denoters Fido chased every cat.
(iii) if <α> and <β> are types, then so is <α, β>
6
since <e> and <t> are types,
meanings of type <e, t> can be “abstracted”
from the meanings of sentences (type <t>)
FELIX<e> IS-A-CAT(_)
that contain a name (type <e>)
<e, t>
<t>
<t>
<e, t>
FIDO<e>
CHASED(_, _)<e, <e, t>>
FELIX<e>
since <e> and <e, t> are types,
meanings of type <e, <e, t>> can be “abstracted”
from the meanings of sentences that contain two names
7
<t>
FELIX<e> IS-A-CAT(_)
<e, t>
“composition”
(reverse abstraction)
via
Function Application
<β>
/ \
<α> <α,β>
<t>
<e, t>
FIDO<e>
CHASED(_, _)<e, <e, t>>
FELIX<e>
<t>
<et, t>
EVERY(_, _)<et, <et, t>>
RAN(_)<e, t>
…and so on
CAT(_)<e, t>
8
Human Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what are the Human Meaning Types?
• standard answer, via Frege’s conception of ideal languages
(i) a basic type <e>, for entity denoters
(ii) a basic type <t>, for truth-value denoters
(iii) if <α> and <β> are types, then so is <α, β> That’s a lot of types
9
<e>
<t>
if <α> and <β>, then <α, β>
at Level 5,
more than 5 x 1012
0. <e>
(2)
<t>
1. <e, e> <e, t> <t, e> <t, t>
(4) of <0, 0>
2. eight of <0, 1>
sixteen of <1, 1>
eight of <1, 0>
(32), including
<e, et> and <et, t>
3. 64 of <0, 2>
128 of <1, 2>
1024 of <2, 2>
64 of <2, 0>
128 of <2, 1>
(1408),
including <e, <e, et>>
and <et, <et, t>>
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>
1,982,464 of <3, 3>
(2,089,472), including
<e, <e, <e, <et>>
10
Languages
Human Languages
Human
Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what are the Human Meaning Types?
• standard answer, via Frege’s conception of ideal languages
(i) a basic type <e>, for entity denoters
(ii) a basic type <t>, for truth-value denoters
(iii) if <α> and <β> are types, then so is <α, β> That’s a lot of types
11
Languages
Human Languages
Human
Languages
• acquirable by normal children
given ordinary courses of experience
• somehow generate unboundedly many expressions that
connect “meanings” with “pronunciations”
• what are the Human Meaning Types?
• another answer
horse, brown, run
(i) a type <M>, for monadic predicates
(ii) a type <D>, for dyadic predicates
on, from, cause
(iii) all complex expressions are of type <M>
12
Outline for Rest of the Talk
• say a little more about the suggested alternative to a
“Fregean” semantic typology
• then focus on why we need an alternative
– if <α> and <β> are types, then so is <α, β>
– a basic type <e>, for entity denoters of
– a basic type <t>, for truth-value denoters
(way too many)
(unattested)
(unattested)
• if time permits, brief discussion of quantification
– serious difficulties for the idea that ‘every cat’ is of type <et, t>
and ‘every’ is of type <et, <et, t>>
– alternatives available if we don’t insist that <e> and <t> are basic
13
Outline for Rest of the Talk
• say a little more about the suggested alternative to a
“Fregean” semantic typology
• then focus on why we need an alternative
– if <α> and <β> are types, then so is <α, β>
– a basic type <e>, for entity denoters of
– a basic type <t>, for truth-value denoters
(way too many)
(unattested)
(unattested)
• if time permits, brief discussion of quantification
(otherwise, conversations about quantification later)
14
We can invent languages that have
boundlessly many expressions that exhibit…
• no semantic typology (see Tarski)
• finitely many semantic types (2, 3, …, a million, …)
• endlessly many semantic types (see Frege and Church)
But where are human languages located
along this dimension of potential variation?
In addressing the empirical questions, it helps to have
some “typologically spare” languages as possible models.
15
We can imagine a language whose expressions are limited to…
(1) finitely many atomic monadic predicates: M1(_) … Mk(_)
(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)
(3) boundlessly many complex monadic predicates
Monad + Monad  Monad
BROWN(_) + HORSE(_)  BROWN(_)^HORSE(_)
FAST(_) + BROWN(_)^HORSE(_)  FAST(_)^BROWN(_)^HORSE(_)
16
We can imagine a language whose expressions are limited to…
(1) finitely many atomic monadic predicates: M1(_) … Mk(_)
(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)
(3) boundlessly many complex monadic predicates
Monad + Monad  Monad
for each entity:
Φ(_)^Ψ(_) applies to it
if and only if
Φ(_) applies to it, and
Ψ(_) applies to it
17
We can imagine a language whose expressions are limited to…
(1) finitely many atomic monadic predicates: M1(_) … Mk(_)
(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)
(3) boundlessly many complex monadic predicates
Monad + Monad  Monad
for each entity:
Φ(_)^Ψ(_) applies to it
if and only if
_)^HORSE(_)]
Φ(_) applies to it, and
Ψ(_) applies to it
Dyad + Monad  Monad
ON(_, _) + HORSE(_)

[ON(_,
|_______|________|
(thing that is) on a horse
We can imagine a language whose expressions are limited to…
(1) finitely many atomic monadic predicates: M1(_) … Mk(_)
(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)
(3) boundlessly many complex monadic predicates
Monad + Monad  Monad
for each entity:
Φ(_)^Ψ(_) applies to it
if and only if
_)^HORSE(_)]
Φ(_) applies to it, and
Ψ(_) applies to it
Dyad + Monad  Monad
ON(_, _) + HORSE(_)

[ON(_,
(thing that is) on a horse
# thing that a horse is on
19
We can imagine a language whose expressions are limited to…
(1) finitely many atomic monadic predicates: M1(_) … Mk(_)
(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)
(3) boundlessly many complex monadic predicates
Monad + Monad  Monad
for each entity:
Φ(_)^Ψ(_) applies to it
if and only if
Φ(_) applies to it, and
Ψ(_) applies to it
Dyad + Monad  Monad
for each entity:
[Δ(_, _)^Ψ(_)] applies to it
if and only if
it bears Δ to something
that Ψ(_) applies to
20
FAST(_)^BROWN(_)^HORSE(_)
is like
FAST(e) & BROWN(e) & HORSE(e)
[ON(_, _)^HORSE(_)]
is like
e[ON(e’, e) & HORSE(e)]
But ‘&’ and ‘e[…e…]’ allow for much more.
FAST(e) & BROWN(e’) & HORSE(e’’)
e[ON(e, e’) & HORSE(e)]
e[BETWEEN(e’, e, e’’) & SOLD(e’’’, e’’’’, e’’’’’, e)]
And human languages are not this permissive.
21
[AGENT(_, _)^HORSE(_)]^EAT(_)^FAST(_)
is like
e[AGENT(e’, e) & HORSE(e)] & EAT(e’) & FAST(e’)
[AGENT(_, _)^FAST(_)^HORSE(_)]^EAT(_)
is like
e[AGENT(e’, e) & FAST(e) & HORSE(e)] & EAT(e’)]
We don’t need variables to capture the meanings of
‘horse eat fast’ and ‘fast horse eat’.
22
SEE(_)^[THEME(_, _)^HORSE(_)]
is like
SEE(e’) & e[THEME(e’, e) & HORSE(e)]
SEE(_)^[THEME(_, _)^[AGENT(_, _)^HORSE(_)]^EAT(_)]
is like
SEE(e’’) & e’[THEME(e’’, e’) & e[AGENT(e’, e)^HORSE(e)] & EAT(e’)]
We don’t need variables to capture the meanings of
‘see a horse’ and ‘see a horse eat’.
23
What are the Human Meaning Types?
--two basic types, <e> and <t>
--endlessly many derived types
of the form <α, β>
--a monadic type <M>
--a dyadic type <D>, for finitely
many atomic expressions
-- <α> can combine with
<α, β> to form <β>
-- <M> + <M>  <M>
<M> + <D>  <M>
24
Outline for Rest of the Talk
✔ say a little more about the suggested alternative to a
“Fregean” semantic typology
• then focus on why we need an alternative
– if <α> and <β> are types, then so is <α, β>
– a basic type <e>, for entity denoters of
– a basic type <t>, for truth-value denoters
(way too many)
(unattested)
(unattested)
• if time permits, brief discussion of quantification
(otherwise, conversations about quantification later)
25
<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>
sixteen of <1, 1>
eight of <1, 0>
(32), including
<e, et> and <et, t>
3. 64 of <0, 2>
128 of <1, 2>
1024 of <2, 2>
64 of <2, 0>
128 of <2, 1>
(1408),
including <e, <e, et>>
and <et, <et, t>>
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>
1,982,464 of <3, 3>
(2,089,472), including
<e, <e, <e, <et>>
26
But even below Level Five…
λy.λx.Predecessor(x, y)
Level Two function of type <e, et>
λy.λx.Precedes(x, y)
another function of the same type
λR.TRANSITIVE[R]
Level Three function of type <<e, et>, t>
||
|
32
0
λy.λx.Precedes(x, y) = ANCESTRAL[λy.λx.Predecessor(x, y)]
λR.ANCESTRAL[R]
Level Three function of type
<<e, et>, <e, et>>
27
But even below Level Five…
λy.λx.Predecessor(x, y)
Level Two function of type <e, et>
λy.λx.Precedes(x, y) = ANCESTRAL[λy.λx.Predecessor(x, y)]
λR.ANCESTRAL[R]
Level Three function of type
<<e, et>, <e, et>>
ANCESTRAL-OF[λy.λx.Precedes(x, y), λy.λx.Predecessor(x, y)]
λR.λR’.ANCESTRAL-OF[R’, R]
Level Four function of type
<<e, et>, <<e, et>, t>
||
|
42
3
28
Frege had to invent a language
that supported abstraction on relations
The plate outweighs the knife.
The plate is something which
outweighs the knife.
The knife is something which the plate outweighs
.
*Outweighs is somerelat which the plate
the knife.
Three precedes four.
Three is something that precedes four.
λx.Precedes(x, 4)
Four is something that three precedes.
λx.Precedes(3, x)
*Precedes is somerelat that three four.
λR.R(3, 4)
29
<e>
<t>
if <α> and <β>, then <α, β>
0. <e>
<t>
1. <e, e> <e, t> <t, e> <t, t>
2. <e, <e, e>> …
3. <e, <e, <e, e>>> …
4. <e, <e, <e, <e, e>>>> …
(2)
(4)
(32), including
<e, et> and <et, t>
(1408), including
<e, <e, et>> and
<<e, et>, <e, et>>
(2,089,472), including
<e, <e, <e, et>>> and
<<e, et>, <<e, et>, t>
30
Can Human Lexical Items have “Level Four Meanings”?
a linguist sold
(sold
a car
a
to a
friend for a dollar
friend a car
for a dollar)
whatever the order of arguments,
the concept SOLD, which differs from GAVE,
is plausibly (at least) tetradic
31
Can Human Lexical Items have “Level Four Meanings”?
So why not…
a linguist sold
x
(she
sold
a car
y
this
a
friend
z
him
a dollar
w
that)

λy. λz . λw. λx . x sold y to z for w
32
Can Human Lexical Items have “Level Four Meanings”?
λZ . λY. λX . GLONK(X, Y, Z)
x[X(x) v Y(x) v Z(x)]
x[X(x) & Y(x)] & x[Y(x) & Z(x)]
Glonk
cat
friendly
dog
33
<e>
<t>
if <α> and <β>, then <α, β>
0. <e>
<t>
1. <e, e> <e, t> <t, e> <t, t>
2.
(4)
(32), including
<e, et> and <et, t>
3.
4.
(2)
(1408), including
<e, <e, et>> and
<<e, et>, <e, et>>
maybe some kind of
“resource limitation”
keeps us from going
beyond Level Three
(2,089,472), including
<e, <e, <e, et>>> and
<<e, et>, <<e, et>, t>
and <et, <et, <et, t>>>
34
Can Human Lexical Items have Level Three Meanings?
<t>
<e, t>
FIDO<e>
CHASED(_, _)<e, <e, t>>
GARFIELD<e>
<t>
<e, et>
<e, t>
ROMEO<e> GAVE(_, _)<e, <e, <e, t>> GARFIELD<e>
JULIET<e>
35
but double-object constructions do not show
that verbs can have Level Three Meanings
Romeo
Romeo
Romeo
gave
it
kicked
the rock
kicked Juliet the rock
to
Juliet
to
Juliet
a
thief
jimmied a
lock with a
knife
Why not instead…
a
thief
jimmied a
(x)
he
jimmied
lock
a
knife
(y)
(z)
it
that
‘jimmied’  λz. λy . λx . x jimmied y with z
The concept JIMMIED is plausibly (at least) triadic.
So why isn’t the verb of type <e, <e, <et>>>?
Why not…
a
rock
(x)
betweens a
lock
(y)
a knife
(z)
‘betweens’  λz. λy . λx . x is between y and z
Still, one might think that
many verbs do have Level Three Meanings…
<t>
-ED(_)<et, t> <et>
FIDO<e>
BARK(_, _)<e, et>
<et>
<e, et>
FIDO<e>
CHASE(_, _)<e, <e, et>>
GARFIELD<e>
40
Can Human Lexical Items have Level Three Meanings?
<e, et>
<<e, et>, <e, et>>
<e, et>
<et, … >
CHASE(_, _)<e, <e, et>>
INTO-A-BARN<et>
GARFIELD<e>
THE-SENATOR<e> FROM-TEXAS<et>
Saying that expressions of type <e, et> can be modified by
expressions of type <et> is like positing a covert Level 4 element.
And why does the modifier skip over the thing chased,
applying instead to the chase?
41
<e, et>
Garfield was
chased
<e,<e, et>>
<et>
<e, et>
Garfield was
<e>
chased
<e, et>>
if the meaning of ‘chase’
is at Level Three,
then a “passivizer” would
also be at Level Four:
<<e,<e, et>, <e, et>>
Kratzer and others
“sever” agent-variables
from verb meanings:
‘chase’ 
λy. λe . e is a chase of y
<et>
FIDO<e>
<e, et>
<et, <e, et>>
“active voice head”
Level Three
CHASE(_, _)<e, et>>
<et>
<et>
GARFIELD<e>
INTO-A-BARN<et>
But if the posited verb meaning is below Level Three,
do we really need the covert Level Three element?
43
<et>
<et>
FIDO<e> <e, et>
AGENT
<et>
<et>
CHASE(_, _)<e, et>>
GARFIELD<e>
INTO-A-BARN<et>
44
<M>
<M>
FIDO<e> <D>
AGENT
<M>
<M>
CHASE(_, _)<D>
GARFIELD<e>
INTO-A-BARN<M>
Are names really expressions of type <e>?
45
<M>
<M>
FIDO<M <D>
AGENT
<M>
>
<M>
CHASE(_, _)<D>
GARFIELD<M>
INTO-A-BARN<M>
46
Do Human i-Languages have expressions of type <e>?
Initially tempting hypothesis: proper names are of type <e>
Robin<e>
®
 --|-/ \
But alternatives have been considered.
Robin<et>  x . x = ®
x . Robinizes(x)
Robin<<e, t>, t>  P.P(®)
P.the(x):Robinizes(x).P(x)
[D1<e, t> Robin<e, t>]<e, t>  x . Indexes(1, x) & Called(x, ‘Robin’)
47
Do Human i-Languages have expressions of type <e>?
Initially tempting hypothesis: proper names are of type <e>
Robin<e>
®
 --|-/ \
But alternatives have been considered for good reasons.
• Neptune is a planet, but Vulcan isn’t.
• Every Tyler at the party was tall, and every philosopher was a Tyler.
That Tyler stayed late, and so did this one.
There were three Tylers at the party, and Tylers are clever.
• We sat next to (that nice) Professor Tyler Burge.
48
Do Human i-Languages have expressions of type <e>?
Another tempting hypothesis: deictic pronouns are of type <e>
®
that/she/him<e>  --|-/ \
But we need alternatives.
That planet is bright.
This1 trumps that2.
<t>
/ \
That planet
<e> <et>
<et>
/ \
This 1
<et> <et>
<et>
/ \
<et> planet/woman/…
/ \ <et>
that 2
<et> <et>
49
Do Human i-Languages have expressions of type <t>?
S  NP aux VP
T(P) Why think tensed phrases denote truth values?
/ \
T
V(P)  e . e is (tenselessly) a John-see-Mary event
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
Why think the tense morpheme
is of type <et, t>
as opposed to <et>
E . e[Past(e) & E(e)]
e . Past(e)
50
Do Human i-Languages have expressions of type <t>?
T(P)
/ \
T
V(P)  e . e is (tenselessly) a John-see-Mary event
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
Why think the tense morpheme
is of type <et, t>
a quantifier…
|
E . e[Past(e) & E(e)]
…that is also a
conjunctive adjunct to V?
51
Do Human i-Languages have expressions of type <t>?
??  Longer Story1
/ \
T(P)  e . Past(e) & e is a John-see-Mary event
/ \
T
V(P)  e . e is (tenselessly) a John-see-Mary event
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
?
e . Past(e)
1. Everybody needs some version of the following Tarskian idea:
a predicate that is satisfied by some but not all things can be
“polarized” into a predicate that is satisfied by everything or nothing.
52
Do Human i-Languages have expressions of type <t>?
Pol(P)  +[Past(_)^John-see-Mary(_)] ....… applies to e
/ \
if and only if
+ T(P)  Past(_)^John-see-Mary(_) there was an event
/ \
of John seeing Mary
T
V(P)  John-see-Mary(_)
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
‘such that John saw Mary’
is sentential; but
it is a sentential predicate,
not a truth-value denoter
suppose that a monadic predicate M can be “polarized” into a
monadic predicate +M that applies to e iff M applies to something
53
Do Human i-Languages have expressions of type <t>?
Pol(P)  −[Past(_)^John-see-Mary(_)] ….… applies to e
/ \
if and only if
− T(P)  Past(_)^John-see-Mary(_) there was no event
/ \
of John seeing Mary
T
V(P)  John-see-Mary(_)
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
suppose that a monadic predicate M can be “polarized” into a
monadic predicate +M that applies to e iff M applies to something
or a monadic predicate -M that applies to e iff M applies to nothing
54
Outline for Rest of the Talk
✔ say a little more about the suggested alternative to a
“Fregean” semantic typology
✔ then focus on why we need an alternative
– if <α> and <β> are types, then so is <α, β>
– a basic type <e>, for entity denoters of
– a basic type <t>, for truth-value denoters
(way too many)
(unattested)
(unattested)
• if time permits, brief discussion of quantification
– serious difficulties for the idea that ‘every cat’ is of type <et, t>
and ‘every’ is of type <et, <et, t>
– alternatives available if we don’t insist that <e> and <t> are basic
55
…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….
--Barbara Partee, “Do We Need Two Basic Types?”
Next Question: Why take <e> or <t> as basic types?
• little or no independent support for the claim that Human Languages
(as opposed to certain languages of thought) generate denoters
• treating names and deictic pronouns as predicates isn’t hard
• Tarski showed us how to treat sentences as predicates, and
• the grammatical status of sentences is unclear
56
<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>
sixteen of <1, 1>
eight of <1, 0>
(32), including
<e, et> and <et, t>
3. 64 of <0, 2>
128 of <1, 2>
1024 of <2, 2>
64 of <2, 0>
128 of <2, 1>
(1408),
including <e, <e, et>>
and <et, <et, t>>
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>
1,982,464 of <3, 3>
(2,089,472), including
<e, <e, <e, <et>>
57
1. <M>
2. <D>
58
Semantic Typology for Human I-Languages
THANKS!
Extra Slides Follow
60
Suppose that a monadic predicate M can be “polarized” into
• a monadic predicate +M that applies to e
if and only if M applies to something
BROWN(_)^HORSE(_)
applies to e if and only if
e is both brown and a horse
+[BROWN(_)^HORSE(_)]
applies to e if and only if
something is both brown and a horse
61
Suppose that a monadic predicate M can be “polarized” into
• a monadic predicate +M that applies to e
if and only if M applies to something
• a monadic predicate -M that applies to e
if and only if M applies to nothing
BROWN(_)^HORSE(_)
applies to e if and only if
e is both brown and a horse
−[BROWN(_)^HORSE(_)]
applies to e if and only if
nothing is both brown and a horse
62
Suppose that a monadic predicate M can be “polarized” into
• a monadic predicate +M that applies to e
if and only if M applies to something
• a monadic predicate -M that applies to e
if and only if M applies to nothing
+[BROWN(_)^HORSE(_)]
applies to e if and only if
(e is such that) there is a brown horse
cp. x[Brown(x) & Horse(x)]
−[BROWN(_)^HORSE(_)]
applies to e if and only if
(e is such that) there is no brown horse
cp. ~x[Brown(x) & Horse(x)]
63
<t>
/
\
<et, t> <et>
/ \
ran
<et, <et, t>> <et>
every
cat
We can try assigning a
higher type to ‘chased’.
But then (I claim):
adjuncts like ‘into the barn’
and passive constructions
will lead us to posit
Level Four meanings.
<t>
/ \
<e> <???>
Fido / \
<e, et> <et, t>
chased / \
every cat
<t>
/ \
<et, t> <et>
Fido / \
/
\
<<et, t>, et> <et, t>
chased
/ \
every cat
Heim&Kratzer
<t>
/ \
<et, t>
<et>
/ \
/ \
<et, <et, t>> <et> 1
<t>
/ \
every
cat
Fido / \
chased __
But then how do we explain
the unambiguity of strings like
‘every cat which Fido chased’
#Every cat is one which Fido chased.
<et>
/ \
which <t>
/ \
Fido / \
chased __
<M>
/
\
<M>
<+M>
/ \
/ \
<D <M> + <M>
/ \
every cat
Fido / \
chased __
1
Maybe the external argument of a
raised quantifier like ‘every cat’
is a polarized predicate,
while a relative clause is a more
ordinary (“unsentential”) predicate
that can apply to some things
without applying to all things
Heim&Kratzer
<t>
/ \
<et, t>
<et>
/ \
/ \
<et, <et, t>> <et> 1
<t>
/ \
every
cat
Fido / \
chased __
<M>
/ \
which <M+>
/ \
+ <M>
/ \
Fido / \
chased __
1
Technical Details Remain (see Conjoining Meanings)
But as a refinable first approximation, suppose that…
• ‘every’ is a plural Dyad:
EVERY(_, _) applies to <the Xs, the Ys> if and only if
the Xs include the Ys
•
‘every cat’ is a plural Monad:
[EVERY(_, _)^THE-CATS(_)]
applies to the Xs if and only if the Xs include the cats
• ‘Fido chased every cat’ is a plural Monad:
[EVERY(_, _)^THE-CATS(_)]^FIDO-CHASED-THEM(_)
applies to the Xs if and only if
the Xs include the cats, and Fido chased the Xs
67
Technical Details Remain (see Conjoining Meanings)
but relative to each assignment A of values to variables…
<M>
+
<M>
every
cat
applies to entity e
<+M> if and only if (e is such that)
Fido chased whatever
whatever A assigns to t1
<M>
so relative to any
<M> assignment that assigns
past
e to the variable,
Fido chase t1 the polarized predicate
applies to e if and only if
Fido chased e
Technical Details Remain (see Conjoining Meanings)
and it’s not (too) hard to formulate
Tarski-style composition principles according to which…
<M>  applies to the Xs if and only if
(i) the Xs include the cats, and
<+M> (ii) Fido chased the Xs
+
<M>
every
cat
(i.e., for each of the Xs,
<M> the polarized predicate
applies to that entity
<M> when that entity is the
past
value of the variable)
Fido chase t1
Technical Details Remain (see Conjoining Meanings)
and it’s not (too) hard to formulate
Tarski-style composition principles according to which…
<+M> applies to e if and only if
Fido chased the cats
+
<M>  applies to the Xs if and only if
(i) the Xs include the cats, and
<+M> (ii) Fido chased the Xs
+
<M>
every
cat
<M>
past
<M>
Fido chase t1
i before e
• function in intension
a procedure that pairs inputs with outputs in a certain way
• function in extension
a set of ordered pairs (with no <x, y> and <x, z> where y ≠ z)
|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)]
71
i before e
• i-Language
a procedure that connects meanings with pronunciations
in a certain way, thereby respecting certain constraints
Bingley is eager to please.
(a) Bingley is eager to please relevant parties.
#(b) Bingley is eager to be pleased by relevant parties.
Bingley is easy to please.
#(a) Bingley can easily please relevant parties.
(b) Bingley can easily be pleased by relevant parties.
72
i before e
• i-Language
a procedure that connects meanings with pronunciations
in a certain way, thereby respecting certain constraints
‘i’ connotes: intensional/procedural;
implementable, and hence constrained;
internal/individual, rather than external/public
• e-language
a language in any other sense—e.g.,
a suitable set of meaning-pronunciation pairs
73
i before e
• i-Language
a procedure that connects meanings with pronunciations
in a certain way, thereby respecting certain constraints
Finite
i-Languages
Gruesome
i-Languages
Human
i-Languages
Less Permissive
i-Languages
More Permissive
i-Languages
| unbounded “sane” i-Languages
|
74
<et>
/
\
<et>
<et>
today
/
\
<et, t> <<et,t>,<et>>
/ \
ran
every cat
Level 3
<et>
<et>
/
\
<et, t> <<et, t>, et>
/ \
every dog
and without quantifier raising,
severing external quantifiers
still requires high types
<et>
today
<<et, t>, <<et, t>, et> <et, t>
chase
/ \
every cat
Level 4
<et>
/
\
<et>
<et>
today
/
\
<et, t> <<et,t>,<et>>
/ \
ran
every cat
/
<??>
see
<et>
\
<et>
<et>
/
\
<et, t> <<et, t>, et>
/ \
every dog
<<et>, <<et, t>, et>>
Level 4
<<et, t>, <et>>
chase
<et>
today
<et>
<et, t>
/ \
every cat
if T is (semantically) the verb’s 3rd argument, then why not…
V(P)  That is a John-see-Mary
/ \ event
D(P)
V(P)  e . e is (tenselessly) a John-see-Mary event
That
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
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 3rd argument?
77
T(P)  e[PAST(e) & John-see-Mary(e)]
/ \
T
V(P)  e . John-see-Mary(e)
past
/ \
D(P) V(P)
John / \
V
D(P)
see Mary
<<e, t>, t>
E . e[PAST(e) & E(e)]
|
(e < RefTime) & (RefTime = SpeechTime)
“God likes Fregean Semantics” theory of tense
78
i-Languages
Fregean i-Languages:
expressions of the types:
<e>, <t>, and if
<α> and <β> are types,
so is <α,
β>
Human
i-Languages
Level-n Fregean i-Languages:
…
expressions of the types:
<e>, <t>, and the nonbasic types
up to Level n
Psuedo-Fregean Languages:
expressions of the types:
<e>, <t>, and some nonbasic
types
79
A Pseudo-Fregean Language Might Have a Dozen Types
<e>
<et>
<e, et>
<e, e, et>
<t>
<t, t>
<t, <tt>
<t, <t, tt>>
<et, t>
<<e, et>, t>
<<et>, <et, t>>
<<e, et>, <et, t>>
80
<t>
/
\
<t>
<t, t>
Mary saw John /
\
<t, <t, t>> <t>
and
John saw Mary
<M>  [PastMarySeeJohn(_) & …
/
\
<M>
<M>  [Before(_, _)^PastJohnSeeMary(_)]
Mary saw John
/ \
And
<D> <M>
before John saw Mary
And
81