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

Semantic Typology and Composition

Paul M. Pietroski, University of MarylandDept. of Linguistics, Dept. of Philosophy

Human children naturally acquire languagesthat 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 languagesthat 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



• what types of meaning do human linguistic expressions exhibit?

(1) Fido (5) every cat (2) chase (6) chase every cat(3) every (7) Fido chase every cat(4) cat (8) Fido chased every cat.

5

Human Languages



• 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>

FidoFido chased every cat.

6

<t>

FELIX<e> IS-A-CAT(_)

<e, t>

FIDO<e> CHASED(_, _)<e, <e, t>> FELIX<e>

<t> <e, t>

since <e> and <t> are types,meanings of type <e, t> can be “abstracted”from the meanings of sentences (type <t>)

that contain a name (type <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>

FIDO<e> CHASED(_, _)<e, <e, t>> FELIX<e>

<t>

<et, t> RAN(_)<e, t> EVERY(_, _)<et, <et, t>> CAT(_)<e, t>

<t>

…and so on

“composition”(reverse abstraction)

via Function Application

<β>/ \

<α> <α,β>

<e, t>

8

Human Languages





(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>

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), includingsixteen 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), including45,056 of <2, 3> 45,056 of <3, 2> <e, <e, <e, <et>>1,982,464 of <3, 3>

at Level 5, more than 5 x 1012

10

Human Languages





(iii) if <α> and <β> are types, then so is <α, β>

Languages

Human Languages

That’s a lot of types

11

Human Languages




• another answer (i) a type <M>, for monadic predicates (ii) a type <D>, for dyadic predicates

(iii) all complex expressions are of type <M>

Languages

Human Languages

horse, brown, run on, from, cause

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> (way too many)– a basic type <e>, for entity denoters of (unattested)– a basic type <t>, for truth-value denoters (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


• say a little more about the suggested alternative to a “Fregean” semantic typology


• if time permits, brief discussion of quantification (otherwise, conversations about quantification later)

14

We can invent languages that haveboundlessly 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





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





Monad + Monad Monad Dyad + Monad Monad

ON(_, _) + HORSE(_)

[ON(_, _)^HORSE(_)]

for each entity:


|________| |_______

(thing that is) on a horse






ON(_, _) + HORSE(_)

[ON(_, _)^HORSE(_)]

for each entity:


(thing that is) on a horse# thing that a horse is on

19






for each entity:


for each entity:

[Δ(_, _)^Ψ(_)] applies to it if and only ifit bears Δ to somethingthat Ψ(_) applies to

20

FAST(_)^BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)] is like

is likeFAST(e) & BROWN(e) & HORSE(e) 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


✔ say a little more about the suggested alternative to a “Fregean” semantic typology


• if time permits, brief discussion of quantification (otherwise, conversations about quantification later)

25

<e> <t>


0. <e> <t>(2)

1. <e, e> <e, t> <t, e> <t, t>(4) of <0, 0>


<e, et> and <et, t>

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


and <et, <et, t>>


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> | | | 3 2 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>| | |4 2 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>


0. <e> <t> (2)

1. <e, e> <e, t> <t, e> <t, t>(4)

2. <e, <e, e>> …(32), including

<e, et> and <et, t>

3. <e, <e, <e, e>>> …(1408), including

<e, <e, et>> and

<<e, et>, <e, et>>

4. <e, <e, <e, <e, e>>>> … (2,089,472), including

<e, <e, <e, et>>> and

<<e, et>, <<e, et>, t>

30

a linguist sold a car to a friend for a dollar

Can Human Lexical Items have “Level Four Meanings”?

(sold a friend a car for a dollar)

whatever the order of arguments, the concept SOLD, which differs from GAVE,is plausibly (at least) tetradic 31

a linguist sold a car a friend a dollar


x y z w

(she sold this him that)

So why not…

λy. λz . λw. λx . x sold y to z for w

32


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>


0. <e> <t> (2)

1. <e, e> <e, t> <t, e> <t, t>(4)

2.(32), including

<e, et> and <et, t>

3.(1408), including

<e, <e, et>> and

<<e, et>, <e, et>>

4. (2,089,472), including

<e, <e, <e, et>>> and

<<e, et>, <<e, et>, t>

and <et, <et, <et, t>>>

maybe some kind of “resource limitation”keeps us from going beyond Level Three

34

Can Human Lexical Items have Level Three Meanings?

<e, t>

FIDO<e> CHASED(_, _)<e, <e, t>> GARFIELD<e>

<t>

<e, t>

ROMEO<e> GAVE(_, _)<e, <e, <e, t>> GARFIELD<e>

<t>

<e, et>

JULIET<e>35

Romeo gave it to Juliet

Romeo kicked the rock to Juliet Romeo kicked Juliet the rock

but double-object constructions do not show that verbs can have Level Three Meanings

a thief jimmied a lock with a knife

(x) (y) (z) he jimmied it that

a thief jimmied a lock a knife

Why not instead…

‘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>>>?

(x) (y) (z)

a rock betweens a lock a knife

Why not…

‘betweens’ λz. λy . λx . x is between y and z

Still, one might think thatmany verbs do have Level Three Meanings…

<et>

FIDO<e> BARK(_, _)<e, et>

<e, et>

FIDO<e> CHASE(_, _)<e, <e, et>> GARFIELD<e>

<et>

<t>

-ED(_)<et, t>

40

Can Human Lexical Items have Level Three Meanings?

<e, et>

CHASE(_, _)<e, <e, et>> GARFIELD<e>

<e, 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?

INTO-A-BARN<et>

<<e, et>, <e, et>> <et, … >

THE-SENATOR<e> FROM-TEXAS<et>

41

Garfield was chased <e,<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>>

<e, et>

Kratzer and others “sever” agent-variables from verb meanings:

‘chase’ λy. λe . e is a chase of y

<e, et>

Garfield was chased <e, et>>

<et>

<e>

<et>

CHASE(_, _)<e, et>> GARFIELD<e>

<et>

INTO-A-BARN<et>

<e, et> <et, <e, et>>

<et>

FIDO<e>

“active voice head” Level Three

But if the posited verb meaning is below Level Three, do we really need the covert Level Three element?

43

<et>

CHASE(_, _)<e, et>> GARFIELD<e>

<et>

INTO-A-BARN<et>

<e, et>AGENT

<et>

FIDO<e>

<et>

44

<M>

CHASE(_, _)<D> GARFIELD<e>

<M>

INTO-A-BARN<M>

<D>AGENT

<M>

FIDO<e>

<M>

Are names really expressions of type <e>?

45

<M>

CHASE(_, _)<D> GARFIELD<M>

<M>

INTO-A-BARN<M>

<D>AGENT

<M>

FIDO<M

>

<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


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


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> / \

planet <et>

<et> / \ that 2<et> <et>

/woman/…

49

Do Human i-Languages have expressions of type <t>?

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

e . e is (tenselessly) a John-see-Mary event

Why think tensed phrases denote truth values?

Why think the tense morpheme is of type <et, t> E .

e[Past(e) & E(e)] as opposed to <et> e . Past(e)

S NP aux VP

50


T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


Why think the tense morpheme is of type <et, t> E .

e[Past(e) & E(e)]

a quantifier… |

…that is also a conjunctive

adjunct to V? 51


T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


e . Past(e)

e . Past(e) & e is a John-see-Mary event

?? / \ ?

52

Longer Story1

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.


T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

John-see-Mary(_)

Past(_)^John-see-Mary(_)

Pol(P) / \ +

+[Past(_)^John-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

....… applies to e if and only if there was an event of John seeing Mary

53

‘such that John saw Mary’ is sentential; but it is a sentential predicate, not a truth-value denoter


T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

John-see-Mary(_)

Past(_)^John-see-Mary(_)

Pol(P) / \ −

−[Past(_)^John-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

….… applies to e if and only if there was no event of John seeing Mary

54


✔ 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> (way too many)– a basic type <e>, for entity denoters of (unattested)– a basic type <t>, for truth-value denoters (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>


0. <e> <t>(2)

1. <e, e> <e, t> <t, e> <t, t>(4) of <0, 0>


<e, et> and <et, t>

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


and <et, <et, t>>


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 ife is

both brown and a horse

+[BROWN(_)^HORSE(_)] applies to e if and only if

something is both brown and a horse

61



• a monadic predicate -M that applies to e if and only if M applies to nothing

BROWN(_)^HORSE(_) applies to e if and only ife is

both brown and a horse

−[BROWN(_)^HORSE(_)] applies to e if and only if

nothing is both brown and a horse

62



• 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

−[BROWN(_)^HORSE(_)] applies to e if and only if (e is such

that) there is no brown horse

cp. ~x[Brown(x) & Horse(x)]

cp. x[Brown(x) & Horse(x)]

63

<t> / \ <et, t> <et> / \ ran<et, <et, t>> <et> every cat

<t> / \ <e> <???> Fido / \ <e, et> <et, t>

chased / \

every cat <t> / \ <et, t> <et> Fido / \

/ \ <<et, t>, et> <et, t>

chased / \

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.

<et> / \

which <t> / \

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.

<t> / \ <et, t> / \<et, <et, t>> <et> every cat

<et> / \

1 <t> / \

Fido / \

chased __

Heim&Kratzer

<M> / \

which <M+> / \

+ <M>

/ \

Fido / \

chased __

<t> / \ <et, t> / \<et, <et, t>> <et> every cat

<et> / \

1 <t> / \

Fido / \

chased __

<M> / \ <M> / \ <D <M> every cat

<+M> / \

+ <M> / \

Fido / \

chased __ 1

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

But as a refinable first approximation, suppose that…

• ‘every’ is a plural Dyad:EVERY(_, _) applies to <the Xs, the Ys> if and only ifthe 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

Technical Details Remain (see Conjoining Meanings)

67

Fido chase t1

every cat

<M>

<+M>

+

past

<M>


applies to entity e if and only if (e is such that) Fido chased whatever

whatever A assigns to t1

<M> so relative to any

assignment that assignse to the variable,

the polarized predicate applies to e if and only if

Fido chased e

<M>

but relative to each assignment A of values to variables…

Fido chase t1

every cat

<M>

<+M>

+

past

<M>


and it’s not (too) hard to formulate Tarski-style composition principles according to which…

<M>

<M>

applies to the Xs if and only if (i) the Xs include the cats, and (ii) Fido chased the Xs

(i.e., for each of the Xs, the polarized predicate

applies to that entity when that entity is the value of the variable)

Fido chase t1

every cat

<M>

<+M>

+

past

<M>


and it’s not (too) hard to formulate Tarski-style composition principles according to which…

<M>

<M>

applies to the Xs if and only if (i) the Xs include the cats, and (ii) Fido chased the Xs

<+M>

+

applies to e if and only if Fido chased the cats

i before e

• function in intensiona 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-Languagea 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’ connotes: intensional/procedural; implementable, and hence constrained;

internal/individual, rather than external/public

• e-languagea language in any other sense—e.g.,

a suitable set of meaning-pronunciation pairs

73

i before e



Human i-Languages

Finitei-Languages

Gruesome i-Languages

Less Permissive i-Languages

More Permissive i-Languages

| unbounded “sane” i-Languages |

74

<et> / \ <et, t> <<et,t>,<et>> / \ ran every cat

<et> / \ <et>today

Level 3 <et> / \ <et, t> <<et, t>, et> / \ every dog

<<et, t>, <<et, t>, et> <et, t>

chase / \

every cat

<et> <et> today

Level 4

and without quantifier raising, severing external quantifiers still requires high types

<et> / \ <et, t> <<et,t>,<et>> / \ ran every cat

<et> / \ <et>today

<et> / \ <et, t> <<et, t>, et> / \ every dog

<<et, t>, <et>> <et, t> chase

/ \

every cat

<et> <et> today

<et> / \ <??>

see

<<et>, <<et, t>, et>> <et> Level 4

V(P) / \ D(P) V(P) That / \

D(P) V(P) John / \

V D(P) see Mary


That is a John-see-Mary event

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?

if T is (semantically) the verb’s 3rd argument, then why not…

77

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

e[PAST(e) & John-see-Mary(e)]

<<e, t>, t> E . e[PAST(e) & E(e)]

“God likes Fregean Semantics” theory of tense

| (e < RefTime) & (RefTime = SpeechTime)

e . John-see-Mary(e)

78

…

i-LanguagesFregean 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> <t><et> <t, t><e, et> <t, <tt> <et, t><e, e, et> <t, <t, tt>> <<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> / \

<M> <M>Mary saw John / \

<D> <M> before John saw Mary

[Before(_, _)^PastJohnSeeMary(_)]

[PastMarySeeJohn(_) & …

And

And

81

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

Documents

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