1. LING 581: Advanced Computational Linguistics
Lecture Notes
May 1st

2. Last time how do we define every?
(Montague-style) λP1.[λP2.[∀X (P1(X) -> P2(X))]]
(Barwise & Cooper-style) {X: P1(X)} ⊆ {X: P2(X)}
Prolog versions (with threshold=2):
(Montague-style)
lambda(P1,lambda(P2,(\+ (call(P1,X),\+ call(P2,X)), call(P1,Y), call(P1,Z), Y \== Z)))
(Barwise & Cooper-style)
lambda(P1,lambda(P2,(setof(X,call(P1,X),L1), setof(Y,call(P2,Y),L2), subset(L1,L2), length(L1,N), N>1)))

3. Modify psg2.pl

4. psg3.pl (Montague-style) atLeast(P1,X,2)
lambda(P1,lambda(P2,(\+ (call(P1,X),\+ call(P2,X)), call(P1,Y), call(P1,Z), Y \== Z)))

5. psg3.pl
Prolog code; not a grammar rule

6. psg3.pl

7. psg3.pl

8. psg3.pl
Let's check it out…

9. psg3.pl
psg3.pl
psg2.pl
Took out the findall; moved it into the S rule

10. psg3.pl

11. psg3.pl

12. psg3.pl

13. Quantifiers Part 3: Coordination Extend the grammars to handle
Every man and every woman likes James
cf. Every man likes James and every woman likes James
[NP[NP Every man] and [NP every woman]] likes James
rewrite:
likes(sophia, james)
as:
likes(james, sophia)

14. Quantifiers Part 3: Coordination Every man and every woman likes James
[NP[NP Every man] and [NP every woman]] likes James

15. Quantifiers Part 3: Coordination Every man and every woman likes James
[NP[NP Every man] and [NP every woman]] likes James

16. Other Quantifiers
Other quantifiers can also be expressed using set relations between two predicates:
Example:
no: {X: P1(X)} ∩ {Y: P2(Y)} = ∅
∩ = set intersection
∅ = empty set
no man smokes
{X: man(X)} ∩ {Y: smokes(Y)} = ∅
should evaluate to true for all possible worlds where there is no overlap between men and smokers
men
smokers

17. Other Quantifiers
Other quantifiers can also be expressed using set relations between two predicates:
Example:
some: {X: P1(X)} ∩ {Y: P2(Y)} ≠ ∅
∩ = set intersection
∅ = empty set
some men smoke
{X: man(X)} ∩ {Y: smokes(Y)} ≠ ∅
men
smokers

18. Names as Generalized Quantifiers
we’ve mentioned that names directly refer
here is another idea…
Conjunction
X and Y
both X and Y have to be of the same type
in particular, semantically...
we want them to have the same semantic type
what is the semantic type of every baby?
Example
every baby and John likes ice cream
[NP[NP every baby] and [NP John]] likes ice cream
every baby likes ice cream
{X: baby(X)} ⊆ {Y: likes(Y,ice_cream)}
John likes ice cream
??? ⊆ {Y: likes(Y,ice_cream)}
John ∈ {Y: likes(Y,ice_cream)}
want everything to be a set (to be consistent)
i.e. want to state something like
({X: baby(X)} ∪{X: john(X)}) ⊆ {Y: likes(Y,ice_cream)}
note: set union (∪) is the translation of “and”

19. Downwards and Upwards Entailment (DE & UE)
Quantifier every has semantics
{X: P1(X)} ⊆ {Y: P2(Y)}
e.g. every woman likes ice cream
{X: woman(X)} ⊆ {Y:likes(Y,ice_cream)}
Every is DE for P1 and UE for P2
Examples:
(25) a. Every dog barks
b. Every Keeshond barks (valid)
c. Every animal barks (invalid)
semantically, “Keeshond” is a sub-property or subset with respect to the set “dog”
animal
dog
Keeshond
animal
dog
Keeshond

20. Downwards and Upwards Entailment (DE & UE)
Quantifier every has semantics
{X: P1(X)} ⊆ {Y: P2(Y)}
e.g. every woman likes ice cream
{X: woman(X)} ⊆ {Y:likes(Y,ice_cream)}
Every is DE for P1 and UE for P2
Examples:
(25) a. Every dog barks
d. Every dog barks loudly (invalid)
c. Every dog makes noise (valid)
semantically, “barks loudly” is a subset with respect to the set “barks”, which (in turn) is a subset of the set “makes noise”
make noise
barks
barks loudly
make
noise
barks
loud

21. Downwards and Upwards Entailment (DE & UE)
Contrast every with some:
every man smokes
every person smokes
every man in the state smokes
some is upwards entailing (UE) in P1:
some man smokes
some person smokes
some man in the state smokes