Spring 2012 IA165 Combinatory Logic for Computational Semantics by Juyeon Kang
Classwork N°11
due to 11th May 2012
Exercise: “Application of combinators to natural language analysis: Quantification”
1. Universal and existential quantifiers
Please represent the following sentences using the operators of the universal or
existential quantification. Are there ambiguous sentences having more than one reading?
a) All cats are mammals. → Universal quantification
Answer:
b) Some cats were sleeping by the fire. → Existential quantification
c) Some chairs are in the lounge. → Existential quantification
d) Every man loves a woman. → Universal and Existential quantification
Answer: It has two readings.
1. There is one woman who is loved by every man.
2. For each man there is one woman whom he loves.
2.
a) (∀x)[cats’ (x) =>are-mammals’(x)]
b)  (λP.λQ((∀x)[P(x) => Q(x)])(cats)(are-mammals’))
a) x [ man (x)    ∀ → ∃y [ woman (y) & loves (x,y) ] ]
a)  (∃x)[cat’(x) & were­slepping’(x)]
b)  (λP.λQ((∃x)[P(x) & Q(x)])(cat)(were­slepping’))
a)  (∃x)[chairs’(x) & are­in the lounge’(x)]
b)  (λP.λQ((∃x)[P(x) & Q(x)])(chairs)(are­in the lounge’))
Spring 2012 IA165 Combinatory Logic for Computational Semantics by Juyeon Kang
1.
e) A boy kissed every girl. → Universal and Existential quantification
Answer: It has two readings.
1. There is one individual boy who kissed each girl (in a particular domain).
On this analysis, the existential phrase (a boy) takes score over the universal quantifier (every
girl).
2. There are as many boys as there are girls. For every girl, there is at least one boy who kissed
her.
On this analysis, the universal quantifier (every girl) takes score over the existential (a boy).
1.
2.
2. Formal semantic analysis of the quantifiers
Give the analysis of the following sentences which contain the quantifiers using the
combinators. The illative quantifiers Π2 and Σ2 are defined in term of the combinators.
f) Some girl likes Fred
1/ (some(girl))(like­Fred)
2/(Σ2(girl))(like­Fred)
3/ ((B(CB
2
)Φ) & Σ1(girl)) (like­Fred)
4/ ((CB
2
) (Φ  &) Σ1(girl)) (like­Fred)
5/ (B
2
 Σ1 (Φ  &) (girl)) (like­Fred)
[ Σ2
 =def
 (B(CB2
)Φ) & Σ1
 ][ Π2
 =def
 ((B(CB2
) ) => Φ Π1
) ]
a) ∃ x [ boy (x) & ∀y [ girl (y)   kissed (x,y) ] ]→
a) ∃ y [ woman (y) &  x [ man (x)   loves (x,y) ] ]∀ →
a)   y [ girl (y)     ∀ → ∃x [ boy (x) & kissed (x,y) ] ]
Spring 2012 IA165 Combinatory Logic for Computational Semantics by Juyeon Kang
6/ Σ1  ((Φ  &) (girl) (like­Fred))
7/  ((Φ  &) (girl) (like­Fred)) x
8/ & (girl(x)) ((like­Fred) x)
g) Every boy admires a saxophonist (ambiguous)
Answer: It has two readings
1. There is one saxophonist who is admired by every boy.
∃ y [ saxophonist (y) &  x [ boy (x)   admires (x,y) ] ]∀ →
 1/(a saxophonist) (is­admired­by every boy)
2/ Σ2 saxophonist (is­admired­by every boy)
3/ ((B(CB
2
) ) &  Φ Σ2) saxophonist (is­admired­by every boy)
4/ (CB
2
)(   &) Φ Σ2 saxophonist (is­admired­by every boy)
5/ B
2
Σ2 (   &)  Φ saxophonist (is­admired­by every boy)
6/ Σ2 ((   &)  Φ saxophonist (is­admired­by every boy)
7/ ((   &)  Φ saxophonist (is­admired­by every boy) x
8/  &  ( saxophonist x) ((is­admired­by every boy) x)
2. For each boy there is one saxophonist whom he admires.
x [ boy (x)    ∀ → ∃y [ saxophonist (y) & admires (x,y) ] ]
 1/(every boy) (admires a Saxophonist)
2/ Π2 boy (admires a Saxophonist)
3/ ((B(CB
2
) ) => Φ Π1) boy (admires a Saxophonist)
4/ (CB
2
)(   =>) Φ Π1 boy (admires a Saxophonist)
5/ B
2
Π1 (   =>)  Φ boy (admires a Saxophonist)
6/ Π1 ((   =>)  Φ boy (admires a Saxophonist))
7/ ((   =>)  Φ boy (admires a Saxophonist)) x
8/  =>  ( boy x) ((admires a Saxophonist) x)
h) Every man knows Dexter
 1/(every man) (knows Dexter)
Spring 2012 IA165 Combinatory Logic for Computational Semantics by Juyeon Kang
2/ Π2 man (knows Dexter)
3/ ((B(CB
2
) ) => Φ Π1) man (knows Dexter)
4/ (CB
2
)(   =>) Φ Π1 man (knows Dexter)
5/ B
2
Π1 (   =>)  man (Φ knows Dexter)
6/ Π1 ((   =>)  man (Φ knows Dexter))
7/ ((   =>)  man (Φ knows Dexter)) x
8/  =>  (man x) ((knows Dexter) x)
i) Several girls carried a box
Answer: It has two readings
1. There is one box who is carried by several girls → A box was carried by several girls
∃ y [ box (y) & ∃x [ girls (x) & carries (x,y) ] ]
 1/(a box) (was­carried­by several girls)
2/ Σ2 box (was­carried­by several girls)
3/ ((B(CB
2
) ) &  Φ Σ2) box (was­carried­by several girls)
4/ (CB
2
)(   &) Φ Σ2 box (was­carried­by several girls)
5/ B
2
Σ2 (   &)  Φ box (was­carried­by several girls)
6/ Σ2 ((   &)  Φ box (was­carried­by several girls)
7/ ((   &)  Φ box (was­carried­by several girls) x
8/  &  ( box x) ((was­carried­by several girls) x)
2. For several girls, there is one box which they carries
∃x [ girls (x) &  ∃y [ box (y) & carries (x,y) ] ]
 1/(several girls) (carried­a­box)
2/ Σ2 girls (carried­a­box)
3/ ((B(CB
2
) ) &  Φ Σ2) girls (carried­a­box)
4/ (CB
2
)(   &) Φ Σ2 girls (carried­a­box)
5/ B
2
Σ2 (   &)  Φ girls (carried­a­box)
6/ Σ2 ((   &)  Φ girls (carried­a­box)
Spring 2012 IA165 Combinatory Logic for Computational Semantics by Juyeon Kang
7/ ((   &)  Φ girls (carried­a­box) x
8/  &  ( girls x) ((carried­a­box) x)