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April 23
Semantics I
Generalized Conjunction
Generalized conjunction in relative clause
(1)
a.
b.
teacher whom Mary likes
x[teacher'(x)  like'(Mary', x)]
What is the correct analysis of conjunction to make the above work out?
As above: ' = PQx[P(x)  Q(x)]
[P  Q]' = x[P(x)  Q(x)] <e, t>
(2)
P
[ Q]' = Px[P(x)  Q(x)] <<e, t>, <e, t,>>
<e, t>

' = QPx[P(x)  Q(x)]
<<e, t>, <<e, t>, <e, t,>>>
VP
Q
<e, t>
Compare:
(3)
(4)
Pavarotti is boring and hates Bond
a.
[boring'(Pavarotti')]  [hate'(Pavarotti', Bond')]
b.
[boring'(Pavarotti')  hate'(Pavarotti', Bond')]
(sentential conjunction)
(VP conjunction)
A man is boring and hates Bond
a.
x[man'(x)  boring'(x)]  y[man'(y)  hate'(y, Bond')] (sentential conjunction)
b.
x[man'(x)  boring'(x)  hate'(x, Bond')]
(VP conjunction)
One further consequence
(5)
John [VP1 [VP2 is nice] and [VP3 likes everybody]]
Where does the quantified NP everybody raise to?
It can’t be raised to S. Extraction out of conjoined structures is subject to restrictions; i.e.
extraction is possible only when an identical constituent is extracted out of the conjuncts across
the board.
(6)
a.
b.
the boy whoi [S [VP1 Mary likes ei] and [VP2 John hates ei]]
*[S everybodyi [S John [VP1 [VP2 is nice] and [VP3 likes ei]]]]
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The quantified NP in (5) must be adjoined to the minimal VP that contains its trace.
(7)
a. John [VP1 [VP2 is nice] and [VP3 everybodyi [VP4 likes ei]]]
b. yxi[like'(y, xi)] ( VP3, the property of liking everybody)
So, having coordinated VPs leads us to extend our rule for interpreting quantified NPs by adding
the category VP as a suitable landing site for raised NPs.
This has nice empirical consequences. Compare:
(8)
a.
b.
A banner was hanging in front of every window
A banner was hanging in front of every window and was ripped off by the wind
How many banners?
(9)
a.
a’.
b.
[S a bannerj [S every windowi [S ej was hanging [in front of ei]]]
(, one banner)
[S every windowi [S a bannerj [S ej was hanging [in front of ei]]]
(, many banners)
[S a bannerj [S ej [VP [VP every windowi [VP was hanging in front of ei]] and [VP was ripped off by the wind]]]]
(, one banner)
The across-the-board constraint on extraction prevents every window from being extracted from
the VP that contains it in (9b), and thus the NP winds up having narrow scope with respect to a
banner.
The present analysis matches our intuition that (8a) has two readings, while (8b) has only one
reading.
Going even further
Conjunctions seem to be cross-categorical:
(10)
a.
b.
c.
d.
e.
John and every student liked the show.
Most or every student came.
John walked in and around the building.
John saw and bought a shirt.
John saw an old and ugly house.
(NP  NP)
(Det  Det)
(Prep  Prep)
(Vt  Vt)
(Adj  Adj)
So the semantic type of conjunction is <α < α, α>>, where α can be of any type.
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Type Shifting Principles (brief introduction, more in Semantics II)
Montague (1973, 1974)
All NPs are of type <<e, t>, t>
(11)
NP
John
every man
the man
a man
translation
P[P(j)]
P[x[man'(x) P(x)]]
P[x[y[man'(y)  y=x]  P(x)]]
P[x[man'(x)  P(x)]]
type
<<e, t>, t>
<<e, t>, t>
<<e, t>, t>
<<e, t>, t>
Partee (1986)
Noun phrase interpretation: referring, predicative, quantificational
(12) NP
e
<e, t>
<<e, t>, t>
“referential”
“predicative”
“quantificational”
(Einstein was smart.)
(He is a real Einstein.)
(Einstein and every other genius)
Partee and Rooth (1983)
It is the “traditional view that proper names and singular pronouns are individual constants and
variables, respectively"
Assign “to all members of a given syntactic category, the ‘highest’ type needed for any of them”
(p.117) with the following principles:
(13)
i.
each basic expression is lexically assigned the simplest type adequate to
capture its meaning
ii. there are general type-lifting rules that provide additional higher-type meanings
for expressions, so that the uniform higher-type meanings Montague posited for
a given syntactic category will be among the available meanings for all
expressions of that category
iii. there is a general processing strategy of trying lowest types first, using higher
types only when they are required in order to combine meanings by available
compositional rules
John is basically of type e, but Montague’s interpretations are available through the general
“lifting” rule. John would have to be “lifted” from j to P[P(j)] to interpret John and every man.
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Linguistic evidence for multiple types of NPs
It is assumed because of Montague’s work that all NPs can be type <<e, t>, t>
• Evidence for type e
Proper names, indefinites, and definites license discourse anaphora. More “quantificational”
NPs do not.
(14) a. John walked in. He looked tired.
The man walked in. He looked tired.
A man walked in. He looked tired.
e
e
e
b. Every man walked in. *He looked tired.
<<e, t>, t>
No man walked he. *He looked tired.
<<e, t>, t>
More than one man walked in. *He looked tired. <<e, t>, t>
• Evidence for type <e, t>
Consider as a test for arguments of type <e, t>
(15) a. Mary considers that an island.
Mary considers that two islands.
Mary considers that many islands.
Mary considers that the prettiest island.
Mary considers that the island.
b. *Mary considers that every island.
*Mary considers that most islands.
*Mary considers that this island.
*Mary considers that Schiermonnikoog.
*Mary considers that Utopia.
<e, t>
<e, t>
<e, t>
<e, t>
<e, t>
<<e, t>, t>
<<e, t>, t>
e
e
e
In sum, some NPs can have different kinds of semantic contributions. Thus, assigning distinct
types accordingly seems to be meaningful, rather than saying that all NPs are <<e, t>, t> at all
times.
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Type shifting operations
(16)
(17) e  <<e, t>, t>
a.
lift ()
e.g. j  P[P(j)]
maps the denotation of j onto the denotation of P[P(j)], P a set of
properties
e.g. j  P[P(j)]
maps a set of properties onto an individual
b.
lower ()
c.
lower(lift(j)) = j
lift(lower(P) = P
(18) e  <e, t>
a.
ident ()
e.g. j  x[x=j]
maps any element onto its singleton set: j  {j}
e.g. x[P(x)]  P
maps any singleton set onto its member: j  {j}
b.
iota ()
c.
iota(ident(j)) = j
ident(iota(P)) = P, when iota(P) is defined (i.e. when there actually is a unique
individual, since iota is simply not defined if no such individual exists)
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(19)
a.
<e, t>  <<e, t>, t>
THE ()
king' = {x such that x is a king}
THE = QP[xy[Q(y)  y=x]  P(x)] (Montague’s definition)
[the king]' = Qx[king'(x) y[king'(y)  y = x]  Q(y)]
THE is a function which maps any set Q onto a generalized quantifier.
If there is a unique king, THE(king') denotes the set of all his properties;
otherwise it denotes an empty set of properties.
b.
BE ()
[the king]' = Qx[king'(x) y[king'(y)  y = x]  Q(y)]
BE = Px[P(y[y=x])] (Montague’s definition)
BE(THE(king'))
=Px[P(y[y=x])](P[xy[king'(y)  y=x]  P(x)])
= x[y[king'(x)  y=x]
(the set of x's that are all kings and are all identical to some
individual y)
“treat this operator not as the meaning of the word be but as a type-shifting
functor that we apply to the generalized quantifier meaning of an NP whenever
we find the NP in an <e, t> position.” (p.124).
• Two readings of the king
The English phrase the king has two semantic interpretations related to two interpretations of the
definite article: iota and THE.
(20) a. iota(king')
<e, t>  e
defined iff there is one and only one king
b. THE(king')
<e, t>  <<e, t>, t>
“If we assume that e is the unmarked type for subject positions, and the preferred type for
arguments of extensional verbs generally, this would help to explain the strong but not absolute
preference for taking existence and uniqueness as presuppositions” (p.124).
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Some examples
(21) John is the king.
a. John'
b. king'
c. (THE(king'))
d. BE(THE(king'))
e. [John is the king]'
(As an answer to the question, Who is John?)
e
<e, t>
<e, t>  <<e, t>, t>
<<e, t>, t>  <e, t>
t (by Functional Application)
(22) John is the king.
a. John'
b. king'
c. (iota(king'))
d. BE(iota(king'))
e. [John is the king]'
(As an answer to the question, Who is the king?)
e
<e, t>
<e, t>  e
e  <e, t>
t (by FA)
(23) Einstein was smart.
a. Einstein'
b. smart'
c. [Einstein was smart]'
e
<e, t>
t (by FA)
(24) He is a real Einstein.
a. he'
b. Einstein'
c. (ident(Einstein'))
d. real'
e. [real Einstein]'
f. A([real Einstein]')
g. BE([a real Einstein]')
h. [He is a real Einstein]'
e
e
e  <e, t>
<<e, t>, <e, t>>
<e, t> (by FA)
<e, t>  <<e, t>, t>
<<e, t>, t>  <e, t>
t (by FA)
(25) Einstein and every woman ran.
a. Einstein'
b. [every woman]'
c. and'
d. lift(Einstein')
e. [Einstein and every woman]'
f. ran'
g. [Einstein and every woman ran]'
e
<<e, t>, t>
<, <, >>
e  <<e, t>, t>
<<e, t>, t> (by FA)
<e, t>
t (by FA)
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References
Montague, Richard (1973) The proper treatment of quantification in ordinary English. Reprinted in
Montague (1974). 247–270.
Montague, Richard (1974) Formal Philosophy: Selected Papers of Richard Montague, edited and with an
introduction by Richmond Thomason. Yale University Press, New Haven.
Partee, Barbara (1986) Noun phrase interpretation and type-shifting principles. In Jeroen Groenendijk,
Dick de Jongh, and Martin Stokhof (eds.), Studies in Discourse Representation Theory and the
Theory of Generalized Quantifiers. Foris Publications, Dordrecht. 115–143.
Partee, Barabara and Matts Rooth (1983) Generalized conjunction and type ambiguity. In R Bäuerle, C
Schwartze, and A von Stechow (eds), Meaning, Use, and Interpretation of Language. Walter de
Gruyter, Berlin. 361–383.