Semantic composition

Chris Potts, Ling 130a/230a: Introduction to semantics and pragmatics, Winter 2020

Jan 28

1 Overview

• This handout describes our theory of semantic composition. It’s a bare-bones theory, but still

powerful.

• The most important conceptual move is to interpret lexical items as functions. Meanings

might not literally be functions, but, following Lewis’s advice, functions at least do what mean-

ings do.

• The grammar is defined by a set of rules for interpreting syntactic structures. Some of these

rules come from the optional Heim & Kratzer reading, while others are new. My notation is

slightly different (to more strongly suggest a computational perspective), but the underlying

ideas are the same as in H&K. (I’ve set up these correspondences largely to try to build some

bridges into this advanced optional reading.)

• We need a bunch of rules in order to respect the syntactic structures. However, all the rules

just do function application.

• So, in essence, if you know the lexical meanings of your language and you can put them

together according to the syntactic rules, then the only other concept you need to be a full-

fledged interpreter is function application.

2 Basic semantic objects

2.1 Truth values

I use T for truth and F for falsity. (Heim & Kratzer use 1 and 0, respectively.)

2.2 Universe

U =⇢

, , ,

�

3 Characteristic sets and characteristic functions

Characteristic function of a set For a universe of entities U and any set A✓ U , the characteristic

function for A is the function that maps every member of A to T and all members of (U � A) to F.

Characteristic set of a function If f is a function into truth values, the characteristic set of f is

{x | f (x) = T}

Set Function

⇢, ,

�

26666666666664

7! T

7! T

7! F

7! T

37777777777775

⇢,

�

26666666666664

7!

7!

7!

7!

37777777777775

;

26666666666664

7!

7!

7!

7!

37777777777775

Set Function

26666666666664

7! T

7! F

7! F

7! F

37777777777775

26666666666664

7! T

7! T

7! T

7! T

37777777777775

26666666666664

7! F

7! F

7! F

7! F

37777777777775

Emas

amassed

Maggie

T

F UF

T

F

F

FF

O t GET

sizes 3IE

4 Notation for describing functions

4.1 Functions

26666666666664

7! T

7! T

7! T

7! F

37777777777775

CHILD(x)

1 if x 2ß

, ,

™

2 return T3 else return F

�x

0B@T if x 2ß

, ,

™, else F

1CA

4.2 Function application

0B@�x

0B@T if x 2ß

, ,

™, else F

1CA

1CAÅ ã

=

T if 2ß

, ,

™, else F =

T

5 Semantic lexicon

5.1 PNs

JMaggieK= JLisaK= JBartK= JHomerK=

5.2 Ns

(1) JSimpsonK= �x

0B@T if x 2ß

, , ,

™, else F

1CA

(2) JchildK= �x

0B@T if x 2ß

, ,

™, else F

1CA

type'ifunctionfronentitiattrutludes

CHILD

i

To

zeuioyfch.ldDCes.frouztahfowonta.es

1

the.EE4 Tifxe elseFµremind sore.se

Characteristicfunctionoftheses X

Type: entities

Type: functions from entities to truth values

(3) JstudentK= �x

0B@T if x 2ß

,

™, else F

1CA

(4) JparentK= �x

0B@T if x 2ß ™

, else F

1CA

5.3 Intransitive Vs

(5) JskateboardsK= �x

0B@T if x 2ß

,

™, else F

1CA

(6) JstudiesK= �x

0B@T if x 2ß ™

, else F

1CA

(7) JintrospectsK= �x

0B@T if x 2ß

,

™, else F

1CA

(8) JspeaksK=

5.4 Transitive Vs

(9) JteaseK= �y

0BBBBBBBBBBB@

�x

0BBBBBBBBBBB@

T if hx , yi 2

8>>>>>>>>>><>>>>>>>>>>:

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑

9>>>>>>>>>>=>>>>>>>>>>;

, else F

1CCCCCCCCCCCA

1CCCCCCCCCCCA

Isametypeandmeaning

framework

as Ns

t1x Tif EU B elseF

I

iA teased Labar

tease PNE µypekassiaieasess37D hs.DGb

directors't E a

Type: functions from entities to truth values

Type: functions from entities into functions from entities to truth values

(10) JadmiresK= �y

0BBBBB@�x

0BBBBB@T if hx , yi 2

8>>>>><>>>>>:

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑

9>>>>>=>>>>>;

, else F

1CCCCCA

1CCCCCA

(11) JlovesK=

5.5 Adjectives

(12) JfemaleK= � f

0B@�x

0B@T if x 2ß

,

™and f (x) = T, else F

1CA

1CA

(13) JmaleK= � f

0B@�x

0B@T if x 2ß

,

™and f (x) = T, else F

1CA

1CA

(14) JhungryK= � f

0B@�x

0B@T if x 2ß

,

™and f (x) = T, else F

1CA

1CA

(15) JallegedK= � f (�x (T if someone claimed that f (x) = T, else F))

(16) JSpringfieldianK=

Setd ordered

fqfIIIIIE.es GxGitLx.DEPienes eluFD

r tfemaleskateboarder

l

framework 7f Xx 3gtie afootie function

GlendeDCashdeboarderD

XfGxTis xE Ehsaan.gsauefkdT.elseFDEshdeboa.eeiD

XxCTifxCElissaErassied3anaffshdeboarderDGd I elseF

WTifxE asadAnansie andGYGifyefabartkdhonerH.ieseF x elseFD

Gif xEElisaGrossed aneTifxEEbartD.Ehonerd3eeseFelseF

cacti xcLEhsaa.serao.ie are C aboard.GhorerB elseF

Type: functions from functions from entities to truth values into functions from entities to truth values

Intersective: T if the argument is in the first set and true of the characteristic function of the second

5.6 Negation

To be completed:

(17) � f

✓�x

✓ ◆◆

5.7 Quantificational determiners

(18) JeveryK= � f

�g

ÄT if {x | f (x) = T} ✓ {x | g(x) = T} , else F

äã

(19) JsomeK= � f

�g

ÄT if {x | f (x) = T}\ {x | g(x) = T} 6= ;, else F

äã

(20) JnoK= � f

�g

ÄT if {x | f (x) = T}\ {x | g(x) = T}= ;, else F

äã

(21) JmostK=

(22) JthreeK=

(23) JfewK=

Etta LEIF Bartnotshieboards

amodelforhowto

approach

thatquestion

onthe

Tiff Felse XfGxEndalf assignment

µgJA B

X pi ToutputoutputjnputddIFH.IE

tfGgGif t I IIcharset I 2 etat

charselff x fCx T

XfXg Tif IchorsetCArcharsetts lZ3elseFD

Type: functions from functions from entities to truth values into (functions from entities to truth values into truth values)

6 Semantic grammar

(NB) Given a syntactic structure X

Y

, JXK= JYK

(S) Given a syntactic structure S

PN VP

, JSK= JVPK(JPNK)

(A) Given a syntactic structure NPj

AP NPi

, JNPjK= JAPK(JNP

iK)

(N) Given a syntactic structure VPj

not VPi

, JVPjK= JnotK(JVP

iK)

(With apologies to the morphosyntax of English for providing no do support.)

(TV) Given a syntactic structure VP

V PN

, JVPK= JVK(JPNK)

(Q1) Given a syntactic structure QP

D NP

, JQPK= JDK(JNPK)

(Q2) Given a syntactic structure S

QP VP

, JSK= JQPK(JVPK)

metavandices

overthesyntacticcategories

PNlPNBlade

where CHD is

A wellformed the

functorsure't

reverse Days

nee is

i am j justhelpus distinguish thetwoidenticalcategorylabels

Quantifier

b

TDeterminer

7 Illustrations

(24) S

PN

Bart

VP

V

skateboards

�x

0BBB@T if x 2

ß,

™, else F

1CCCA

✓ ◆=

T if 2ß

,

™, else F

�x

0BBB@T if x 2

ß,

™, else F

1CCCA

JskateboardsK

JskateboardsK

(25) NP

AP

A

female

NP

N

student

� f

0BBBBB@�x

0BBBBB@T if x 2

ß,

™and f (x) = T, else F

1CCCCCA

1CCCCCA

0BBBBB@�x

0BBBBB@T if x 2

ß,

™, else F

1CCCCCA

1CCCCCA

� f

0BBBBB@�x

0BBBBB@T if x 2

ß,

™and f (x) = T, else F

1CCCCCA

1CCCCCA

JfemaleK

�x

0BBBBB@T if x 2

ß,

™, else F

1CCCCCA

JstudentK

S

NBis

ANB

NB

NB NB Thismeaningiscorrectbut

wouldnotsatisfythe requirement

of thehomeworkthatallsubstitutions beperformed

(26) S

PN

Maggie

VP

V

teases

PN

Bart

T if h , i 2

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

, else F

�x

0BBBBBBBBBBBBBBBBB@

T if hx , i 2

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

, else F

1CCCCCCCCCCCCCCCCCA

�y

0BBBBBBBBBBBBBBBBB@

�x

0BBBBBBBBBBBBBBBBB@

T if hx , yi 2

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑,

≠,

∑

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

, else F

1CCCCCCCCCCCCCCCCCA

1CCCCCCCCCCCCCCCCCA

JteaseK

S

µ

(27) S

QP

D

every

NP

N

child

VP

V

skateboards

T if

�x | JchildK(x) = T

✓�

x | JskateboardsK(x) = T

, else F

�g

ÄT if

�x | JchildK(x) = T

✓ {x | g(x) = T} , else F

ä

� f

�g

ÄT if {x | f (x) = T} ✓ {x | g(x) = T} , else F

äã

JeveryK

JchildK

JchildK

JchildK

JskateboardsK

JskateboardsK

JskateboardsK

QP2

QPl NB

NB NB NB

NB

(28) S

PN

Homer

VP

V

teases

PN

Lisa

(29) S

PN

Homer

VP

not VP

V

teases

PN

Lisa

(30) QP

D

a

NP

Simpson

S

N4Elisac Laid

akaesb3deF

GfGzL ftp.DJGYG Lg.EusuDEKaDakoesb3elseFDGzEE3EJGgGit4ausakkasiateamb3elur.DK

Gcc Giftisabella.ir.aleausB.elser

(31) S

QP

D

no

NP

N

parent

VP

V

skateboards

(32) S

QP

D

every

NP

AP

A

female

NP

N

child

VP

V

studies

(33) S

QP

D

a

NP

AP

A

hungry

NP

N

child

VP

not VP

V

studies