[PPT] - The Logical Analysis of Plurals and Mass Terms Godehard Link (1983) PowerPoint Presentation

SLIDE 1

The Logical Analysis of Plurals and Mass Terms

Godehard Link (1983)

Ling 720 Presen tation Jon Ander Mendia De em b er 10, 2013

Overview

Main

n

tribution

f

the pap er: Extend the logi al language presen ted in PTQ to

v

er plural and mass terms. The Plan: 1. The

n

tology

f

plurals and mass terms. 2. The ingredien ts

f

LPM 3. Some appli ations to Mon tague Grammar 4. A brief nal remark

1 The ontology of plurals and mass terms

The Basi Question: what do plural and mass nouns denote? Mon tague pro vided no a oun t

f

plural and mass terms in his system, the domain

f

en tities

nsisted
nly
f

a set

f

singular individuals. 1. Plural T erms (1) a. *The kid met together b. The kids met together ea h

ne
f

the kids met together . The kids are b

ys ⇒

ea h

ne
f

the kids is a b

y

Ho w do w e a oun t for these dieren t fa ts? 2. Mass T erms Plurals and mass nouns b eha v e similarly in some resp e ts. (2) a. If a is w ater and b is w ater, then the sum

f a

and b is w ater. b. If a are horses and b are horses, then the sum

f a

and b are horses. Link dubb ed this prop ert y the umulative r efer en e pr

p

erty

f

plurals and mass terms. Ho w ev er: if t w

expressions a

and b refer to en tities

urring

in spa e and time but ha v e dieren t sets

f

predi ates that an b e true

f

them, then a = b . Example: tak e a ring re en tly made up from some

ld

Egyptian gold. Then the gold the ring is made

f

is

ld,

whereas the ring itself is new. Example: tak e a

mmittee
nstituted

b y all the fa ult y mem b ers under 30. Then, if the

mmittee

w as

nstituted

man y y ears ago it is true that the

mmittee

is

ld,

whereas the fa ult y mem b ers are y

ung.

Note: the um ulativ e referen e prop ert y b eha v es the same: if a and b are rings, the gold in a and b is still

ld,

whereas the rings a and b are new. The distin tiv eness

f

these linguisti expressions p

in

ts

ut

that ev en if the ring and the gold in the ring share the p

rtion
f

matter they are made

f,

they are not the same en tit y . Link's prop

sal:

enri h the

underlying-

set theoreti metalanguage to a oun t for prop erties lik e um ulativ e referen e, but without app ealing to sets. 1

SLIDE 2

The Logical Analysis of Plurals and Mass Terms (Link 1983)

Jon Ander Mendia Idea 1: within the domain

f

en tities E w e an distinguish t w

dieren

t (sub)- domains: the domain A

f

atomi individuals and the domain D

f

stu

r

p

rtions
f

matter. (3) E as generated b y the predi ate P , where P = {a, b, c} : E an b e dened as a ( omplete b

lean)

algebra losed under join: E, ⊕ . A, the set

f

atoms, and D, the set

f

individual p

rtions
f

matter, are

mplete

join- subsemilatti es

f

E generated b y 1-pla e predi ates P . (4) On tology: a. S attered Singular En tities:

x, y ∈ D

(water, gold, air, et .) b. Con rete Singular En tities:

x, y ∈ A \ D

(the ring, the

mmitte

e, the gold in this ring, et .) . Plural En tities:

x, y ∈ E \ A

(the rings, the

mmitte

es, et .) Note: 1. Both mass terms and group nouns are atoms, they are dieren t en tities from the p

rtions
f

matter

r

individuals that they are

mp
sed
f

(re all the example

f

the ring and the

mmitte

e ab

ve

). 1 2. Plural en tities are just sums

f

individuals (and not sets), as

n rete

as the individuals that serv e to dene them and

f

the same logi al t yp e. 3. Substan es are abstra t en tities and annot b e dened in terms

f

their

n rete

manifestations (that's wh y water ∈ D whereas the water in this glass ∈ A ). 4. Joining together an y t w

n rete

individuals (in E \ D ) returns a plural en tit y (in E\A ), whereas joining together t w

p
rtions
f

matter (in D ) returns another p

rtion
f

matter, an atom in D . 2 5. The n ull elemen t is assigned the role

f

dumm y

b

je t to whi h no predi ate applies and whi h an tak e are

f

denotation gaps in the theory (≈

ur
n ept
f

garbage). 1 In the ase

f

group nouns, Link distinguishes to t yp es

f

predi ates: those that are p ermeable to the inner stru ture

f

group nouns and those that are not (variant vs. invariant pr e di ates ): i. V arian t Predi ate: The p ersonnel

mmittee

is

ld
the

professors is

ld

ii. In v arian t Predi ate: The p ersonnel

mmittee

met

⇒

the professors met 2 Graphi ally this is represen ted b y app ealing to t w

dieren

t instan tiations

f

the join

p

erator `⊕' for

n rete

individuals and `+' for p

rtions
f

matter. 2

SLIDE 3

The Logical Analysis of Plurals and Mass Terms (Link 1983)

Jon Ander Mendia 6. Ev en though D ⊆ A ⊆ E , w e annot extend the algebra

f

the domain

f

individuals do wn to the algebra

f

p

rtions
f

matter; they are dieren t stru tures. Idea 2: Elemen ts in E \ D and D are

nne ted

b y a

nstitution

r elation.

h

is the materialization fun tion denoting the

nstitution

relation. It maps an y individual

f E

in to its

rresp
nding

p

rtion
f

matter in D . Example: the departmen t

f

linguisti s has t w

dieren

t

mmittees:

the p ersonnel

mmittee c1

and the urri ulum

mmittee c2

. Both

mmittees

are formed b y all the professors in the departmen t (p1 ⊕ pn ), but they ha v e dieren t hairs, b

ard

mem b ers and se retaries; they are dieren t represen tativ e b

dies,

therefore, c1 = c2 = p1 ⊕pn . (5) a. The p ersonnel

mmittee

met y esterda y

the

urri ulum

mmittee

met y esterda y b. The professors in the departmen t are y

ung
the

urri ulum

mmittee

is y

ung

Ho w ev er, the materialization fun tion maps ev ery individual in to its

rresp
nding

material parts. Th us, if follo ws that: (6)

h(p1 ⊕ pn) = h(c1) = h(c2)

The same holds for the example with the rings: Example: t w

rings, r1

and r2 are made

f

p

rtions
f
ld

Egyptian gold g1 and g2 resp e tiv ely . Then, the rings r1 ⊕ r2 are made

f

the p

rtions
f

matter in g1 + g2 . (7)

a. g1 + g2 = h(r1 ⊕ r2)
b. g1 + g2 = r1 ⊕ r2

g1+g2

is the material fusion

f g1

and g2 , it's still a s attered singular en tit y , whereas

r1 ⊕ r2

is the plural en tit y

mp
sed
f

b

th

rings. Ev en though g1 + g2 and r1 ⊕ r2 ha v e the same materialization, the theory is su h that g1 + g2

nstitutes

but is not equal to r1 ⊕ r2 .

2 The Ingredients of LPM

2.1 Individuals

Plural morphology signals the presen e

f

a pluralization

p

eration `*' whi h generates all the individual sums

f

mem b ers

f

the extension

f

an y 1-pla e predi ate P. That is, E is losed under the join

p

eration: E, ⊕ , where a ⊕ b is the individual- sum (i-sum) a and b . Sums are partially

rdered

through an

rdering

relation ≤i

n E

expressed in the

b

je t language b y the t w

pla e

relation : the i(ndividual)-p art relation.

is

in terpreted in the seman ti s as the Bo

lean

relation ⊔i (where ⊔i denotes the join

p

eration ⊕ ). (8)

a. a b ↔ a ⊕ b = b

i a ≤i b

b. a ≤i b

i a ⊔i b = b . a b = 1 i a = 0 and i a ≤i b (b y (8b)) i a ⊔i b = b (b y def.

f

`⊔i ') i a ⊕ b = b So, the seman ti in terpretation

f

plurals is as follo ws: (9)

a. ∗P :=

[P ℄ the

mplete

joini

latti

e gener ate d by P

b. ∗P := {x ∈ E|∃X ⊆ P & X = ∅
st. x = supiX}

If P is a 1-pla e predi ate and ∗P is the

rresp
nding

plural predi ate, then w e an dene ⊛P , the prop er plural predi ate

f P

. (10)

a. ⊛P := ∗P \ A
b. σxPx := ιx(∗Px ∧ ∀y(∗Py → y x))

the sum

f

P's . σ∗xPx := ιx(⊛Px ∧ ∀y(∗Py → y x)) the pr

p

er sum

f

P's

σ∗xPx

arries the presupp

sition

that there at least t w

P

's; in this ase σ∗xPx and

σxPx

in ide.

(11)

a. σxPx =

supiP, where supi∅ =

b. σ∗xPx = σxPx

if P has > 2 elemen ts,

therwise

2.2 Stuff

D is losed under join making D a

mplete

(but not-ne essarily atomi ) join-semilatti e. Lik e E, D is losed under the join

p

eration `+'

n D

(denoted b y ⊔m ). Just lik e with individuals, there is a

rresp
nding
rdering

relation alled the material part (m- part) relation, denoted b y ⊤ (equiv alen t to in the

un

t domain). It establishes a preorder (it's not an ti-symmetri )

n

p

rtions
f

matter ≤m (akin to ≤i ). If t w

b

je ts are m-parts

f

ea h

ther

then they are materially equiv alen t. 3

SLIDE 4

The Logical Analysis of Plurals and Mass Terms (Link 1983)

Jon Ander Mendia (12)

a. a b a⊤b

b. If a⊤b and b⊤a , then a is material ly e quivalent to b . x ≤m y i x ⊔m y = y Seman ti ally , ⊤ an b e dened in terms

f

the materialization fun tion h : (13)

a⊤b = 1

i a, b = 0 and h(a) ≤m h(b) and a ≤m b if a, b ∈ D Under the assumption that a, b ∈ D seman ti fa t ab

v

e follo ws trivially , giv en that h denotes the iden tit y fun tion

n D

. Building

n

h to

,

w e an pro vide a seman ti in terpretation for the

nstitution

relation, denoted b y `⊲ ' in the

b

je t language and dene the

n ept
f

material fusion: (14)

a. a ⊲ b = 1

i b = 0 and a = h(b)

nstitution

r elation

b. a + b = h(a) ⊔m h(b)

if a = 0 and b = 0 material fusion Th us, if P is a mass term, P is a n um b er/quan tit y

f

p

rtions
f

matter losed under join.

2.3 The relation between individuals and stuff

In terestingly , the material part-whole relation in (12 ) an b e used to

rder

the individuals

f

E materially : the seman ti

un

terpart

f

the

nstitution

relation `⊲ ' is pre isely the semilatti e homomorphism h from E \ 0 to D. (15)

h : E \ {0} → D

is a semilatti e homomorphims st. i.

h ↿ D = idD

(for all x ∈ D, h(x) = x) , and ii.

h(

supB) = suph [B ℄ , for all B ⊆ E \ {0} (16) The Homomorphi Relation:

a. x ≤i y ⇒ h(x) ≤ h(y)

(∀x, y ∈ E,

where x = 0)

b. x ≤m y

i h(x) ≤ h(y)

(∀x, y ∈ E \ {0})

Ho w is h useful for linguisti analysis? Consider that ev ery predi ate P has a mass term

rr

esp

ndent mP

. (17)

a. mP =

suph [P ℄ ( ompare with ∗P in (9a))

= {x ∈ D|x ≤

suph [P ℄}

= {x ∈ D|∃y ∈ ∗P

st. x ≤m h(y)}

= h(

supP ) Note that if P is already a mass term, it follo ws that P ⊆ mP . (18) a. There is apple in the salad

b. ∃x(mPx ∧ Qx)

, where P = is an apple, mP = is apple and Q = is in the salad. What is the relation b et w een the ar ds and the de k

f

ar ds ? W e use the notion

f

material fusion, denoted b y the µ-op erator: (19)

µxPx = ιx(x ⊲ σxPx)

Example: Let P = is a ar d fr

m
ne
f

the de ks

f

ar ds and Q = is a de k

f

ar ds, then σxPx = σxQx . And this is what w e w an t, for if w e ha v e to de ks

f

Bridge ards, then σxPx will

n

tain 104 atoms, whereas σxQx will

n

tain

nly

t w

.

Ho w ev er,

µxPx = µxQx

. Note that the µ-op erator builds des riptions

f
n rete

singular terms

ut
f

p

rtions
f

matter: (20)

a. x ⊲ σxPx = 1

i x ⊲ σxPx = 0 and x = h(x ⊲ σxPx)

b. µxPx = ιx(h(σxPx) ⊲ σxPx)

Similarly , re all the example

f

the rings: The

nstitution

relation denoted b y `⊲ ' relates the gold

r

the ring and the ring ; if r is a ring and g is the gold in r , then g ⊲ r (and same for plurals: g1 + g2 ⊲ r1 ⊕ r2 ).

2.4 Some examples

(21) a. The hild built the raft (Px: x is a hild )

b. ∃y(y = ιxPx ∧ Qy)

(Qx: x built the r aft ) (22) a. The hildren built the raft

b. ∃y(y = σ∗xPx ∧ Qy)

(23) a. T

m

and Jerry arried the piano (t: T

m,

j: Jerry )

b. P(t ⊕ j)

(Px : x arrie d the piano ) (24) a. John and P aul are p

p

stars and George is a p

p

star b.

⊛P(j ⊕ p) ∧ Pg ⇒⊛ P(j ⊕ p ⊕ g)

(Px : x is a p

pstar

) (25) a. W ater is w et

b. ∃x(Px → Qx)

(Px : x is water ) 4

SLIDE 5

The Logical Analysis of Plurals and Mass Terms (Link 1983)

Jon Ander Mendia (26) a. The w ater

f

the Rhine is dirt y

b. Q(µxPx)

(Px : x is R hine water ) (27) a. The gold in the ring is

ld,

but the ring is not

ld
b. Qιx(Px ∧ x ⊲ a) ∧ ¬Qa

(Px : x is gold, Qx : x is

ld,

a: the ring )

3 Application to PTQ

With these ingredien ts, Link builds a mo del M for LPM: an

rdered

pair B, st.

1. B = E, A, D, h

is a tuple where E is the domain

f

individuals in M , A is the set

f

atoms in M , D is the set

f

p

rtions
f

matter in M and h is the materialization fun tion in M . 2. is a rst

rder

assignmen t

f

denotations to the primitiv e expressions

f

LPM. The syn tax and seman ti s are as in PTQ, with the follo wing additions: 1. The ategory CN is split in to MCN (mass NPs), SCN (singular

un

t NPs) and PCN (plural

un

t NPs). 2. There is plural rule, where ζ ∈ PSCN ⇒ ζpl ∈ PP CN . So, he denes an In terpretation: E, A, D, h, I, J, , as dened ab

v

e and in PTQ. Here are some translations for quan tiers: let U b e the translation relation: (28) a. the U

λQλP∃x[Q(x) ∧ P(x) ∧ ∀y[Q(y) → y x]]

b. some, ∅pl U

λQλP∃x[Q(x) ∧ P(x)∧]

Both some and the an apply to singular and plural phrases: (29) a. some hild U

λP∃x[child′(x) ∧ P(x)]

b. (some) hildr en U

λP∃x[⊛child′(x) ∧ P(x)]

. (some) water U

λP∃x[water′(x) ∧ P(x)]

Note: it is

nly

the CN phrase whi h dieren tiates b et w een the appropriate singular and plural readings. As a

nsequen e

some an

m

bine with

njoined

CNs. (30)

a. (ζ

and η) U

λz∃x∃y[ζ′(x) ∧ η′(y) ∧ z = x ⊕ y]

b. b

y

and girl who dated ea h

ther

. λz∃x∃y[boy′(x) ∧ girl′(y) ∧ z = x ⊕ y ∧ dated-ea h-other(z)] This sho ws ho w pluralization has the for e

f

group formation. As a

nsequen e,

w e an

m

bine phrases lik e (ζ and η) with quan tiers lik e some and get the

rre t

in terpretation. (31) some ((b

y

and girl) su h that they met) U

λP∃z[∃x∃y[boy′(x) ∧ girl′(y) ∧ z = x ⊕ y ∧

meet'(z)] ∧ P(z)]

4 Wrapping Up

4.1 One final remark

Link's view

n

plurals is ex lusiv e, that is, plurals don't in lude atoms (⊛P = ∗P\

A

). Ho w ev er, from a linguisti p v., this is problemati : i. Do wn w ard en tailing

texts:

(32) a. No professors are in lass b. Are there professors in lass? ii. Plural NPs with quan tities less than 1: (33) Put 0.5 grams

f

salt in the soup

4.2 Summary

1. The logi

f

plurals and mass nouns share a

mmon

stru ture, a latti e, the dieren e b et w een b

th

latti es b eing that the former is atomi , whereas the latter is not. 2. The star

p

erator allo ws us to treat plural morphology

mp
sitionally

is a w a y su h that the parallels b et w een plurals and mass terms are aptured. 3. Plural en tities and group nouns are equiv alen t in that they are in ter hangeable in some en vironmen t (with invariant predi ates, see fn.1), but this not mak e them

referen

tial. This

n

trasts with redu tionists approa hes where b

th

the ar ds and the de k

f

ar ds denote the same set. Referen es Landman, F., 1991. Stru tures for seman ti s. Klu w er A ademi , Netherlands. Link, G., 1983. The logi al analysis

f

plurals and mass terms. In: Baeuerle, R. et. al. (Eds), Meaning, Use and In terpretation

f

Language. DeGruyter, pp. 303-329. Motague, R., 1974. The prop er treatmen t

f

quan ti ation in

rdinary

English. In: Thoma- son, R.H. (Ed.), F

rmal

philosoph y: Sele ted P ap ers

f

Ri hard Mon tague. Y ale Uni- v ersit y Press, pp. 247-270. P artee, B.H. et. al. 1990. Mathemati al metho ds in linguisti s. Dordre h t, Boston. 5