CAS LX 502

1
CAS LX 502

• 10b. Binding

2
Syntactic base rules (F2)
S ? NP VP VP ? Vt NP
S ? S ConjP VP ? Vi
ConjP ? Conj S NP ? Det NC
S ? Neg S NP ? NP
Det ? the, a, every NP ? Pavarotti, Loren, Bond, Nemo, Dory, Blinky, Semantics, The Last Juror, hen, shen, itn, himn, hern, himselfn, herselfn, itselfn.
Conj ? and, or NP ? Pavarotti, Loren, Bond, Nemo, Dory, Blinky, Semantics, The Last Juror, hen, shen, itn, himn, hern, himselfn, herselfn, itselfn.
Vt ? likes, hates NP ? Pavarotti, Loren, Bond, Nemo, Dory, Blinky, Semantics, The Last Juror, hen, shen, itn, himn, hern, himselfn, herselfn, itselfn.
Vi ? is boring, is hungry NP ? Pavarotti, Loren, Bond, Nemo, Dory, Blinky, Semantics, The Last Juror, hen, shen, itn, himn, hern, himselfn, herselfn, itselfn.
Neg ? it is not the case that NC ? book, fish, man, woman
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Quantifier RaisingS X NP Y ? S NP S? i S X ti Y
Pass-Upb aM,g aM,g Functional applicationg a b M,g bM,g ( aM,g ) or aM,g ( bM,g ) whichever is defined
PavarottiM,g F(Pavarotti) (any NP)
is boringM,g ? x x ? F(is boring) (any NC or Vi)
likesM,g ? y ? x ltx,ygt ? F(likes) (any Vt)
andM,g ? y ? x x ? y (analogous for or)
it is not the case thatM,g ? x ?x
?iM,g g(i)
iM,g ?S ?x SM,gi/x
everyM,g ?P ?Q ?x?U P(x) ? Q(x)
aM,g ?P ?Q ?x?U P(x) ? Q(x)
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Binding

• Among our NPs, we have a couple of special
  types
• self anaphora himselfn, herselfn, itselfn
• pronouns himn, hern, itn, hen, shen.
• Unlike Bond, Loren, Pavarotti, these NPs dont
  denote a specific individual in U, regardless of
  the model they pick up their reference from
  elsewhere.

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Binding

• We think of the denotation of these elements as
  dependenttheir reference is provided by
  something else.
• In the case of pronouns, it is provided by
  pointing (courtesy of the assignment function
  g).
• With self anaphora, the reference of self is
  bound to the reference of another (generally
  preceding) NP.
• Bond1 likes himself1.

He1 is hungry
1 ?
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Bond1 likes himself1

• It is common, when discussing reference of
  pronouns and self anaphora to write sentences
  with a subscript on each NP under consideration.
• What Bond1 likes himself1 really means is
• The NPs Bond and himself both point to the same
  individual (hence the coindexation).
• Bond intrinsically names a particular individual.
• Himself has no intrinsic reference, its all in
  the connections (via the index) to other NPs.
• Therefore, himself also points to the individual
  Bond points to.

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Bond1 likes himself1

• This is a useful notation for describing the
  intended meanings of sentences, but this is not
  something F2 can give us.
• NP ? Bond
• NP ? himselfn
• The intended meaning of NP ? himselfn is that NP
  can be rewritten as himselfn for some index n.
  Doesnt matter which, pick one. In fact, pick 1.
• Notice the lack of any NP ? Bondn rule.

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Every fish likes itself

• In fact, self anaphora (and pronouns too) can
  pick up their reference from quantifiers like
  every fish too. We might write this informally
  as
• Every fish1 likes itself1.
• although this is just as un-directly-generable
  by F2 as Bond1 likes himself1 was.
• An example of a bound pronoun, slightly outside
  of F2, would be every fish1 thinks that Loren
  likes him1.

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Pavarotti likes himself
S

• Our syntactic rules allow us to build and
  evaluate this sentence, but with a twist. Lets
  see.

NP
VP
S ? NP VP


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Pavarotti likes himself
S

• Our syntactic rules allow us to build and
  evaluate this sentence, but with a twist. Lets
  see.

NP
VP
Vt
NP
S ? NP VP
VP ? Vt NP

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Pavarotti likes himself
S

• Our syntactic rules allow us to build and
  evaluate this sentence, but with a twist. Lets
  see.

NP
VP
Vt
NP
NP
NP
S ? NP VP
VP ? Vt NP
NP ? NP
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Pavarotti likes himself
S

• Our syntactic rules allow us to build and
  evaluate this sentence, but with a twist. Lets
  see.

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
S ? NP VP Vt ? likes
VP ? Vt NP NP ? Pavarotti
NP ? NP
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Pavarotti likes himself
S

• NP ? himselfn
• The subscript n means pick an index, any index.
• Lets pick 3.

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
himself3
S ? NP VP Vt ? likes
VP ? Vt NP NP ? Pavarotti
NP ? NP NP ? himselfn
14
Pavarotti likes himself
S

• What should this mean?

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
himself3
15
Pavarotti likes himself
S

• What should this mean?
• So, we need himself3 to denote Pavarotti.
• Our interpretation rules say it refers to
  g(3)whoever 3 points to.

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
himself3
?iM,g g(i)himself3M,g g(3)
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Pavarotti likes himself
S

• In less formal circumstances, this interpretation
  is written likePavarotti3 likes
  himself3.Where we give the proper name
  Pavarotti a matching index.
• In F2, we cant give Pavarotti a subscript.
  First, no rule provides for that. Second, the
  interpretation of anything3 is g(3). We lose the
  connection to Pavarotti. Right?

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
himself3
?iM,g g(i)himself3M,g g(3)
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Pavarotti likes himself
S

• So thats the puzzle How can Pavarotti still
  refer to this guy and simultaneously bind the
  anaphor himself3 by matching its index?
• It turns out, theres a straightforward way to do
  this, and we can do it with the F2 we already
  have.

NP
VP
Vt
NP
NP
likes
Pavarotti
NP
himself3
?iM,g g(i)himself3M,g g(3)
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Pavarotti likes himself
S

• The answer? QR.
• All that QR says is findan NP inside an S,
  pickan index, attach the indexand the NP to the
  top, andreplace the original NPwith a t that
  shares the index.Pavarotti is an NP. So, hey,
  why not?
• Quantifier RaisingS X NP Y ? S NP S? i S X
  ti Y

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
SM,g ltg(3), g(3)gt ? F(likes)lttgt
S

• Using a little foresight, well pick 3 for our
  index when we do QR as well.
• So, what does that leave us with?
• ?iM,g g(i)
• So, himself3M,g and t3M,g both g(3).
• Using the rules as usual, the lower SM,g is
  true if g(3) likes g(3).

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
SM,g ltg(3), g(3)gt ? F(likes)lttgt
S

• We have a node that is true if g(3) likes g(3).
• We want to combine it with the index node 3.
• We want the result to be the property of liking
  oneself.
• We can then attribute to Pavarotti the property
  of liking oneself, and the sentence will be true
  if Pavarotti likes himself.

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
SM,g ltg(3), g(3)gt ? F(likes)lttgt
S

• iM,g ?S ?x SM,gi/x
• 3M,g ?S ?x SM,g3/x
• We know thatSM,g ltg(3), g(3)gt ? F(likes)
• What if we evaluate that not with the assignment
  function g but with g3/x instead? That is, with
  an assignment function where 3 points to x?
• g3/x (1) g(1)
• g3/x (2) g(2)
• g3/x (3) x
• g3/x (4) g(4)

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
We know this for surethats what g3/x means.
himself3
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Pavarotti likes himself
SM,g ltg(3), g(3)gt ? F(likes)lttgt
S

• iM,g ?S ?x SM,gi/x
• 3M,g ?S ?x SM,g3/x
• We know thatSM,g ltg(3), g(3)gt ? F(likes)
• SM,g3/x ltg3/x (3), g3/x (3)gt ?
  F(likes) ltx, xgt ? F(likes)

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt
S

• iM,g ?S ?x SM,gi/x
• 3M,g ?S ?x SM,g3/x
• SM,g3/x ltx, xgt ? F(likes)
• So, now, we can figure out what S?M,g
  is3M,g is a function, it takes an S as its
  argument.3M,g ( SM,g ) ?x SM,g3/x
  ?x ltx, xgt ? F(likes)

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt

• A point of pedantic clarification
• Strictly speaking, SM,g is theargument, so
  in3M,g ?S ?x SM,g3/x S is
  really going to be SM,g.
• So, what were evaluating is really SM,g
  M,g3/x. We take SM,g tomean the value of
  S under the current model and assignment
  function. If we are evaluating the whole sentence
  under M,g, thats the current model and
  assignment function. But by evaluating SM,g
  under M,g3/x, weve changed the current
  assignment function inside the brackets to be
  g3/x. So,this is effectively the same as
  SM,g 3/xM,g3/x. You can safely ignore this
  technicality, and just do things as they were
  done on the previous slide. This is here only for
  completeness.

S
NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Pavarotti likes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt
S

• S?M,g ?x ltx, xgt ? F(likes)
• Perfect, now were set.
• Combining the lte,tgt functionrepresented by
  S?M,g andthe ltegt individual representedby the
  top NP, we have
• SM,g S?M,g ( NPM,g ) ?x ltx, xgt ?
  F(likes) ( F(Pavarotti) ) ltF(Pavarotti),
  F(Pavarotti) gt ? F(likes)

NP
S?
S
3
NP
t3
VP
Pavarotti
Vt
NP
likes
NP
himself3
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Every manlikes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt
S

• Whats more, we can easily now interpret Every
  man likes himself. Same structure, only now with
  every man instead of Pavarotti. We perform QR
  just as we did, we interpret every node up to the
  S? just as we didso S?M,g is the property of
  liking oneself.
• everyM,g takes first the predicate man, then
  the predicate likes oneself, and the sentence is
  true if being a man implies liking oneself.

NP
S?
S
3
NC
Det
man
every
t3
VP
Vt
NP
likes
NP
himself3
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Every manlikes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt
S

• everyM,g ?P ?Q ?x?U P(x) ? Q(x)
• NCM,g ?x x ? F(man)
• NPM,g DetM,g ( NCM,g )
• In terms of the definition ofevery, we will
  replace P with NCM,g.
• We also need P(x), which will be NCM,g (x),
  or?x x ? F(man) (x),so (replacing xes with
  xes),P(x) is just x ? F(man).
• NPM,g ?Q ?x?U x ? F(man) ? Q(x)

NP
S?
S
3
NC
Det
man
every
t3
VP
Vt
NP
likes
NP
himself3
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Every manlikes himself
S?M,g ?x ltx, xgt ? F(likes) lte,tgt
S

• NPM,g ?Q ?x?U x ? F(man) ? Q(x)
  ltlte,tgt,tgt
• Now, the last step, combiningthe NP and the S?.
• S?M,g is the predicatelikes oneself, type
  lte,tgt.
• NPM,g needs a predicate to call Q. So, we
  replace Q with S?M,g. We also need Q(x), which
  is ?x ltx, xgt ? F(likes) (x), or (replacing
  xes with xes), ltx, xgt ? F(likes) .
• SM,g NPM,g ( S?M,g ) ?x?U x ? F(man) ?
  ltx, xgt ? F(likes)

NP
S?
S
3
NC
Det
man
every
t3
VP
Vt
NP
likes
NP
himself3
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Himselfn, herselfn, itselfn

• Additional restrictions on self anaphors
• Himself must denote a man, herself must denotes a
  woman, itself must denote something that is
  neither a man nor a woman.
• Pavarotti3 likes herself3.
• There is also a restriction that the thing that
  has a matching index must be nearby and higher
  in the tree
• Pavarotti3 is hungry and Loren likes himself3.
• Himself3 likes Pavarotti3.

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Binding theory

• The nearby condition, approximately
• A self anaphor must have an index that matches
  the index of something within the smallest S that
  contains it.
• LX522 Binding theory, Principle A.
• The higher in the tree condition,
  approximately
• The node with an index matching a self anaphor
  must be the sister of a node that contains the
  self anaphor.
• LX522 The binder must c-command the self
  anaphor.
• You can verify for yourself that these conditions
  hold, and youll hear more about them in LX522.
  These are not semantic conditionsour system can
  interpret sentences violating these conditions
  just fine. These are syntactic.

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Pronouns vs. self anaphors

• Often, when you want co-reference but cant use a
  self anaphor, you can use a pronoun instead
• Pavarotti3 is hungry and Loren likes him3.
• Pavarotti3 likes him3.
• F2 does not distinguish between pronouns and
  self anaphors, the interpretation is the same
  him3M,g himself3M,g g(3).
• So, where a pronoun is appropriate, where a self
  anaphor is appropriate, these are governed by
  separate constraints on what structures are valid
  as structuressyntactic constraints.

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VP anaphora

• In a manner somewhat similar to the way herself
  picks up the referent of a higher NP, English has
  a phrase do too that seems to pick up the
  denotation of a preceding VP.
• Loren likes Bond and Pavarotti does too.
• The VP in the first sentence is a predicate,
  meaning is a Bond-liker ?x ltx, F(Bond)gt ?
  F(like)
• The VP in the second sentence, does too, is
  interpreted as if it were just the same as the VP
  in the first sentence
• Loren is a Bond-liker and Pavarotti is a
  Bond-liker.

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VP anaphora

• Every man likes Loren and Nemo does too.
• Being a man implies being a Loren-liker, and Nemo
  is a Loren-liker.
• A fish likes every book.
• There is a fish z such that for every book x, z
  likes x
• For every book x, there is a fish z such that z
  likes x

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A fish likes every book

• Every book is a quantifier in object position
  (type ltlte,tgt,ltlte,tgt,tgtgt) and must undergo QR in
  order to be interpretable.
• A fish can either undergo QR or not. If it does
  not, we have the one-uncritical-fish
  interpretation. If it does, we have the
  to-each-its-own interpretation.

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and Loren does too

• A fish likes every book.
• A fish 1 every book 2 t1 likes t2
• every book 2 a fish likes t2
• Loren likes every book.
• every book 2 Loren likes t2
• A fish likes every book and Loren does too.
• Parallelism Where a VP is anaphoric to another,
  the two sentences must have a parallel structure.
• Scope economy Only do QR if it makes a
  difference in the meaning.

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And were done!Weve described English!

• Well, not exactly. Actually, there are plenty of
  English sentences we dont have a formal
  interpretation procedure for yet.
• Heres one
• Most fish are boring.

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Most fish are boring

• The first steps are simple enough. Most looks
  like it is a determiner like every, so we can
  add Det ? mostto our syntactic base rules.
• And it seems to mean something kind of similar.
  It should take two predicates and return a truth
  value that is, most takes fish then are boring
  and says something about individuals for which
  fish is true being individuals for which is
  boring is true too.

38
Most fish are boring

• Every says given predicates P and Q, for each
  individual x in U, P(x) ? Q(x).
• Most is less than every, so maybe
• Given predicates P and Q, for over half of the
  individuals x in U, P(x) ? Q(x).
• That was easy. But wait.
• What if there are 2 fish in U and 47 books?
  Certainly if x is a book, fish(x) ? Q(x) is true
  for any Q, given that false?anything is true.
• If thats the case, then every fish is happy can
  be true, even while most fish are unhappy is true
  (after all, 47 of 49 xes are such that being a
  fish implies being unhappy). Hmm.

39
Most fish are boring

• What we need is to be checking only the fish, not
  all of the individuals in the universe. We dont
  care about non-fish when we are evaluating most.
  Of the fish, are most such that they are also
  boring?

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Thinking in terms of sets

• A predicate like is boring or fish defines a set,
  the set of boring individuals, or the set of
  fish.
• In fact, the F function already gave us that set,
  and our interpretation rule turned it into a
  function.
• is boringM,g ?x x ? F(is boring)

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Most fish are boring
F(fish)
F(is boring)

• When we think interms of sets, whatmost fish
  are boringis saying is that theboring fish
  outnumberthe non-boring fish.
• The boring fish areindividuals in bothF(fish)
  and F(is boring).
• The nonboring fish areindividuals in F(fish)but
  not in F(is boring).
• Most(A,B) A?B gt A-B

U
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Most fish are boring
F(fish)
F(is boring)

• For predicates P and Q,
• P x P(x)
• Q x Q(x)
• P ? Q x P(x) ? Q(x)
• P - Q x P(x) ??Q(x)
• Most(A,B) A?B gt A-B
• So, we can write most as
• mostM,g ?P ?Q x P(x) ? Q(x) gt
  x P(x) ? ?Q(x)

U
43
?

• ? ?
• ?
• ? ?
• ? ?
• ?
• ?