Montague Grammar
EECS 595 - Fall 2004
Amy Kao
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Montague Grammar
• Maps syntactic structure with semantic structure
• Uses formal language to describe natural language
(1970) Universal Grammar
Theory of formal syntax and semantics applied to formal & natural languages
(1970) English as a formal language
Theory of English as a form of formal language
(1973) The Proper Treatment of Quantification in Ordinary English
Application of Universal Grammar theories of a fragment of English
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Relevant Contributors
• Richard Montague (1930-1971)
• Student of Tarski
• Created Philosophies and Theories of
Montague Grammar
• Taught at UCLA
• Barbara Hall Partee
• Student of Chomsky
• Wrote interpretations that made Montague’s
work more understandable
• Teaches at U. Mass Amherst
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Syntactic Categories
Ca tegory
t
A bb rev ia ti on
(p ri m iti ve )
e
(p ri m iti ve )
t/ e
IV
P T Q Na m e
T ru th -va lue expr e ss ion ; or
dec larati ve sen te nce
E nt it y exp res si on; o r
ind iv idua l exp ression
In tran siti ve ve rb phra se
t/ IV
IV /T
IV /IV
T
TV
IAV
T er m
T ran siti ve ve rb phra se
IV- m od if y ing adv e rb
t// e
t/t
IAV /T
IV /t
CN
None
None
None
IV /IV
None
Co mm on noun ph ras e
S en tence -m od if ying adve rb
IAV -m a king p repos iti on
S en tence -tak ing verb
ph rase
IV- tak ing ve rb ph rase
Near e st lin g ui sti c e qui va lent
S en tence
(noun phra se)
tr ans iti ve verb , tr an siti ve ve rb and
it s ob jec t, o r o the r ve rb phra ses
Noun phra se
T ran siti ve ve rb
V P- adve rb and pr e pos iti ona l
ph rase s c on tain ing in a nd abou t.
Noun or NOM
S en tence -m od if ying adve rb
L oca ti ve, etc., pr e pos iti on
V wh ich tak e s th at -C OM P
V wh ich tak e s in fi n iti ve C OM P
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Category Definitions/Generation
• Categories are of form: X/Y
– semantics of Y into the truth value of X
• Abbreviations for first 5 categories
–
–
–
–
–
IV = t/e
T = t/IV
TV = IV/T
IAV = IV/IV
CN = t//e
• Infinite number of possible categories
– May use as many slashes as needed for new categories
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Example Expressions
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5
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Ca tegory
B IV
BT
B TV
B IAV
B CN
B t/t
B IAV /T
B IV /t
B IV /IV
Ba si c E xp res si on
{run, wa lk, t alk, ris e, chang e}
{John , M ary, B ill , nine ty, he0, he1 , he2 , … }
{fi nd los e , ea t, l ove, da te, be , seek , con c eive }
{rap idly, slow ly, vo lun tarily, a lle ged ly}
{m an , wo m an , pa rk, fi sh , pen , un ico rn, p ri ce, t em pe rature}
{nec e ss a ril y}
{in, abou t}
{be li eve t ha t, as sert tha t}
{tr y to , w ish to}
P art ee 1973
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Example Rule F3 For BTV
F3(, ) =
If first word of  is a TV:

if  is not a variable
 himi
if  is hei
If  is 1 2 where 1 is a TV/T:
1  2 if  is not a variable
 himi 2
if  is hei
F3(shave, a fish) = shave a fish
F3(seek, he1) = seek him1
F3(read a large book, Mary) = read Mary a large book
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Syntactic Rules
Mary loves him, I, F1(love him, Mary)
love him, IV, F3(love, he)
love, TV
Mary, T
he, T
If   X/Y and   Y then Fi(,)  X
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Extensions and Intensions
Extension: Semantic interpretation
Intension: Function generating an extension
Extension Problem
C lassm ate o f John
F or me r [Cl as sm ate o f John ]
=
°
G radua te studen t at U M ic h
F or me r [G radua te stud e nt a t U M ic h]
Ex ten sion Cat e gory
T ru th-va lue of a sen tence
A pp ro p riat e Intens io n F un cti on
F ro m i nd ices t o tr uth-v al ues
T hing na m ed by a na m e
F ro m i nd ices t o thing s
S et o f obj e cts a c om m on noun o r intr an siti ve ve rb
ph rase app li es to
F ro m i nd ices t o sets
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Intensional Logic (IL)
• IL = Intensions and Types
• Syntactic Category Rule = Semantic Rule = Type
• ^X = intension of X
– Example: if J = John, ^J = function returning individual named
John
• If   X/Y and   Y and ,  translates into ’’, then
Fi(,) translates into ’(^’).
• Semantic Primitive t = truth values
• Semantic type is function of model view
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R u le
t
e
If a an d b are ty pes , < a, b> i s a ty pe
If a is a ty pe, <s, a> is a t ype
S em antic Ru le
D t is {0 ,1 } .
D e is A.
Da
D <a,b> is D b
W xT
D <s,a > is S a = S a
S em antic S et
all tr uth-va lue s
all en titie s
all fu nct io ns fr om D a to D b for D a
all fu nct io ns fr om w or ld- tim e p air s
to D a
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Truth Definitions
Co n ce pt
T r uth D e finit io n
B a se
X sa tis fies F
(no cond ition s n e eded )
R e cur si ve
X sa tis fies “ F and G ”
if f X sati sfi es F and X sati sfi es G
Con sequenc e
X i s a con sequenc e o f sen tence s in clas s K
if f X is tr ue in eve ry m ode l whe re e very s e n tence in K is tr ue
E quiv al ence
X and Y are log ical ly equ iva len t
if X i s a con sequenc e o f Y and Y is a cons e quence of X
T ru th
X i s log ical ly t ru e
if for all m ode ls , X is tr ue
Con tr ad icti on
C lass K is c ont rad icto ry
if no mod el ex is ts w her e a ll sen tenc e s in K ar e tr ue
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Model Theoretic Semantics
• Semantics based on truth conditions
– Tarski’s Model Theory (1954)
• IL is based on truth Conditions
• 3 Levels of Symbols
– Logical Constants: =, , etc
– Variables: As in traditional math
– Non-logical Constants: , , relation symbols, function
symbols, and constant individual symbols
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General Quantification &
Compositionality
• Compositionality: phrase’s meaning derived
from meaning of constituents & syntactic
structure
• General Quantification: allows for syntax
and semantic structure to be equivalent
S en te nc e
John t alks
E very studen t t alks
F ir st O rd er Log ic
T alks (j )
 x [St uden t( x) -> T alks (x) ]
G en era l Q ua nti fica ti o n
John ’ (^t alk’ )
E very m an ’ (^talk ’)
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Controversies
• Formal Logic
– Opposition: differing views of what semantics is
• Chomskyan: semantics is branch of psychology
• Semantists: semantics different from knowledge of semantics
– Defense: Montague’s “English as a Formal Language”
– Formal Semantics now mainstream
• Model Theory
– Opposition: Prefer more concrete expressions
– Defense: Understands sentences in terms of human mental models
• Truth Conditions
– Opposition: Brings in too many irrelevant factors
– Defense: Humans judge sentences based on context
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Developments Influenced by
Montague Grammar
• Head-Driven Phrase Structure Grammar (HPSG)
– Influenced by syntactic categories
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•
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File Change Semantics(FCS)
Discourse Representation Semantics (DRS)
Situation Semantics
Extended Categorical Grammar
Generalized Phrase Structure Grammar (GPSG)
Lexical Semantics
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Montague Grammar - University of Michigan