PLIN3004/PLING218 Advanced Semantic Theory
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Assignment 7
More Quantificational Determiners
In class we analysed the meanings of no, every and some as follows:
(1)
Function Talk
a. veverywa,M “ rλg P Dxe,ty .rλ f P Dxe,ty . 1 iff for every x P De such that gpxq “ 1, f pxq “ 1ss
b. vnowa,M “ rλg P Dxe,ty .rλ f P Dxe,ty . 1 iff for no x P De such that gpxq “ 1, f pxq “ 1ss
c. vsomewa,M “ rλg P Dxe,ty .rλ f P Dxe,ty . 1 iff for some x P De such that gpxq “ 1, f pxq “ 1ss
We also discussed how these meanings can be viewed as relations between two sets of individuals.
(2)
Set Talk
a. veverywa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff setpgq Ď setp f qss
b. vnowa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff setp f q X setpgq “ Hss
c. vsomewa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff setp f q X setpgq ‰ Hss
In this exercise, you will analyse the quantificational determiners in the following sentences in
similar ways.
(3)
a.
b.
c.
d.
Between five and ten students were asleep.
Some but not all professors came.
An even number of cats purred.
Fewer boys than girls smiled.
Throughout this exercise we ignore the number marking on the noun, so we assume vlinguistwa, M
= vlinguistswa, M , for example. Also, we don’t analyze the composition of complex phrases like
between five and ten and an even number of, essentially treating them as single determiners.
Here are some useful set theoretic notions (see also Lecture 1):
• A X B is the intersection of A and B, defined as t x | x P A and x P B u.
• A Y B is the union of A and B, defined as t x | x P A or x P B u.
• A ´ B is defined as t x | x P A and x R B u.
• |A| is the cardinality of A, i.e. the number of (distinct) members of A. E.g. | t z, 3, John u | “ 3.
Question: Write the denotations of the determiners in the following sentences both in terms
functions like (1) (function talk) and in terms of sets like (2) (set talk).
(a) Between five and ten
• Function talk: vbetween five and tenwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
• Set talk: vbetween five and tenwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
(b) Some but not all
• Function talk: vsome but not allwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
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• Set talk: vsome but not allwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
(c) An even number of
• Function talk: van even number ofwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
• Set talk: van even number ofwa, M “ rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???ss
(d) Assume that fewer ... than ... has the following structure (ignoring the word order):
S
VP
DP
NP
D
NP
smiled
girls
fewer...than... boys
Notice that there are three type xe, ty functions, namely, vboyswa, M , vgirlswa, M and
vsmiledwa, M , rather than two. So fewer...than... must be of type xet, xet, xet, tyyy.
What should vfewer...than...wa, M look like?
• Function talk: vfewer...than...wa, M “ rλh P Dxe,t y . rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???sss
• Set talk: vfewer...than...wa, M “ rλh P Dxe,t y . rλg P Dxe,t y .rλ f P Dxe,t y . 1 iff ???sss
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Proper Names as Generalized Quantifiers
In class, we analyzed proper names as denoting individuals (of type e) and quantificational DPs
like every linguist and no linguist as denoting functions of type xet, ty. This means that there
are two kinds of DPs.
However, there is a way to assign a single type to all DPs. Recall that we concluded that
quantificational DPs cannot be of type e, but there’s nothing that prevents us from assigning
proper names type-xet, ty functions as their denotations.
For the sake of this exercise, let’s assume that there are two lexical entries Johne and Johnxet ,t y
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8a, M
and that vJohne wa, M = John. Of course we want vJohne smiledwa, M = Johnxet ,t y smiled
,
which is 1 if John smiled in M, and 0 otherwise.
What is the type xet, ty meaning for John? Hint: It should map any function f of type xe, ty to
1 iff f maps John to 1.
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8a, M
Johnxet ,t y
“ rλ f P Dxe,t y . ???s
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Optional: ‘Only John’
The phrase only John is not of type e, essentially for the same reasons that quantificational DPs
are not, and can be analyzed as a Generalized Quantifier (i.e. a function of type xet, ty) We
assume the following structure:
2
S
smiled
DP
only John
We also assume that this sentence is true with respect to any model M (and any assignment
function a) iff John smiled in M and nobody else smiled in M.
(a) What is the generalised quantifier denotation of only John? You can write the denotation
in function talk or set talk.
vonly Johnwa, M “ rλ f P Dxe,t y . ???s
Hint: This is a function of type xet, ty that maps any function f of type xe, ty to 1 iff f
maps John to 1 and everybody else to 0.
(b) Assuming that vJohnwa, M is of type e, annotate the above true with semantic types.
(c) What is the denotation of only? Hint: only maps an individual (e.g. John) to a Generalized
Quantifier like your answer to (a) above.
vonlywa, M “ rλ x P De . ???s
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