CAS LX 502

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CAS LX 502

• 10a. A notational holiday

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Sets

• A set is a collection of entities of any kind.
• They can be finite v2, John Saeed, 1984.
• They can be infinite the set of integers greater
  than 5.
• We can specify a finite set by listing its
  members. We can also specify a set by its
  descriptionx x is a book written by Chomsky
• This set contains any x such that x is a book
  written by Chomskyany other x is not in the set.
• A set is defined by its members.n n is an even
  number n n1 is an odd number
• Order makes no difference, repetition is
  meaningless1, 5, 7 5, 7, 1 7, 1, 5
  7, 1, 1, 5, 7

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Set notations

• We can name a set if we wish A1, 3, 8.
• If A1, 3, 8 and B A, then B 1, 3, 8.
• a ? A means a is a member of the set A.
• a ? A means a is not a member of A.
• 1 ? 1, 3, 8 4 ? 1, 3, 8.
• Ø is the empty set a set that has no members.
  It could also be written like this .
• ?x x ? Ø

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Set notations

• A ? B means that everything in A is also in B,
  although there may be more stuff in B (A is a
  subset of B).
• ?x x ? A ? x ? B
• A ? B means that not only is everything in A in
  B, but there is something else in B that isnt in
  A (A is a proper subset of B).
• ?x x ? A ? x ? B ? ?xx ? B ? x ? A
• A B means that everything in A is in B and
  everything in B is in A.
• ?x (x ? A ? x ? B) ? (x ? B ? x ? A)

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Set notations

• A ? B is the set of elements that are both in A
  and B (the intersection of A and B).
• A ? B x x ? A ? x ? B
• A ? B is the set of elements that are either in A
  or in B, or in both (the union of A and B).
• A ? B x x ? A ? x ? B
• If two sets have no overlap, they are said to be
  disjoint.
• A and B are disjoint iff A ? B Ø.

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Ordered pairs

• Ordered pairs are like a two-membered set in
  which order matters
• lt1, 3gt ? lt3, 1gt lt1, 1gt ? lt1gt
• 1, 3 3, 1 1, 1 1
• An ordered n-tuple is sometimes called a vector
  lt1, 3, 5, 7, 9gt

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Labeled brackets

• The labeled bracket notation is another way to
  write hierarchical information.
• A B C is a group, containing B and C, that is
  labeled collectively as A.

A
B
C
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Labeled brackets
S

• So, the tree on the rightprovides the
  sameinformation as
• S NP Np Bond VP Vt likes NP Np Loren
  

NP
VP
Vt
NP
NP
likes
Bond
NP
Loren
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Functions

• Back to functions for a bit.
• A function is a mapping from an input to an
  output.
• Generally, the output that results depends on the
  input received (e.g., f(x) 2 ? x) (although one
  could define a constant function that maps any
  input into the same output e.g. f(x) 2).

f
input
output
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Functions

• We have several way to write a function. Suppose
  we want to write the not function that reverses a
  truth value. So, a truth value is provided as an
  input and a truth value is returned as an output.
• fnot is a function that, for any x?true,false,
  returns ?x.
• We can also write the mapping explicitly, using
  ordered pairs of the form ltinput, outputgt
• fnot lttrue, falsegt, ltfalse, truegt
• Or, we can write in the form ?inputoutput
• fnot ?x?x.

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Applying functions to arguments

• The input that we provide to a function is called
  its argument.
• fnot(true) the output fnot maps true to.
• fnot(true) ?x?x (true) true ?x?x ?true
  false.

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Functions returning functions

• A function can return not a single value (like
  false or Bond), but a function that is ready to
  apply to a different argument.
• fand ?y ?x x ? y
• This function takes a single argument, returns a
  function.
• fand (true) ?y ?x x ? y (true) true
  ?y ?x x ? y ?x x ? true
• The resulting function maps true to true and
  false to false.
• fand (false) ?y ?x x ? y (false)
  false ?y ?x x ? y ?x x ? false
• The resulting function maps true to false and
  false to false.

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Formally stating ?-conversion

• value ? vbl result or ? vbl result
  (value)
• When we evaluate this we get result, but with all
  of the instances of variable within result
  replaced with value.
• 8 ? x 12 x ? x 12 8 ? 8 12 64 76

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Modified functions

• Suppose fsquare ? x x ? x .
• fsquare(2) 4, fsquare(3) 9, fsquare(4) 16,
  etc.
• Now, suppose we want a new function, fnew, that
  is just like fsquare except that if you give it
  3, it gives you 5 instead of 9.
• fnew(2) 4, fnew(3) 5, fnew(4) 16, etc.
• We can write it like this fnew fsquare3/5.
• This means fnew(x) 5 if x3, fsquare(x)
  otherwise.

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Returning to the task

• Now, we can get back to the task at hand.
• We have intuitions about the truth conditions of
  sentenceswe know what it would take for a
  sentence to be true or false.
• Limiting ourselves to a small portion of English,
  and a very restricted set of situations, we try
  to put together a system to predict those
  intuitions and try to understand how they come
  about.

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Our little fragment (F2)

• With our fragment of English, we have a set of
  syntactic rules that can generate a small number
  of recognizably English sentences.
• Sentences have subjects and verb phrases (S ? NP
  VP), verb phrases can either have intransitive
  verbs (VP ? Vi) or transitive verbs with an
  object (VP ? Vt NP). Noun phrases can either be
  proper names (NP ? NP) or consist of a determiner
  and a common noun (NP ? Det NC). Sentences can be
  negated (S ? Neg S) or coordinated (S ? S ConjP
  ConjP ? Conj S).

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Syntactic base rules
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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Situations and truth

• Since the truth of Bond likes Loren depends on
  the situation, we evaluate a sentence like Bond
  likes Loren against a model of the situation.
• The model (M) tells us who the individuals are
  (the universe U) and their properties and
  relations (F). We write that like MltU,Fgt.

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The denotation function F

• The F function (denotation function) applies to
  lexical items (the nodes at the bottom of the
  tree) and returns a semantic value (the
  denotation).
• F(Bond) returns the individual from U that we
  refer to with the proper name Bond.
• F(is boring) returns a set of the boring
  individuals from U in the situation modeled by M.
• F(likes) returns a set of ordered pairs ltx,ygt,
  where x and y are individuals from U, and x likes
  y in the situation modeled by M.

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Semantic rules

• Our semantic rules are based on the assumption
  that the meaning of any constituent part of the
  sentence structure should arise from the meanings
  of its own parts and how they are combined
  (meaning is compositional).
• We have a set of rules to interpret the lexical
  items (the bottom nodes on the tree), and then
  two general rules Pass-up (the meaning of a
  constituent with only one part is the same as the
  meaning of that one part) and Functional
  Application (the meaning of a constituent with
  two parts is the result of providing one as the
  argument of the other, where one is always a
  function).

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Semantic types

• We can classify the kinds of functions each node
  represents by using a notation that tells us what
  kind of argument the function takes, and what
  kind of result it provideslt argument type,
  result type gt
• We add two basic types ltegt individual, lttgt
  truth value (true or false). These can only be
  arguments or results, they are not functions.
• A function like ? x x is boring in M is type
  lte,tgt because it is true or false of individuals.

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Lexical interpretation rules

• To interpret proper names, we use the denotation
  provided by F.
• PavarottiM,g F(Pavarotti) ltegt
• To interpret intransitive verbs or common nouns,
  we use a function that is true of an individual
  when the individual is in the denotation provided
  by F.
• is boringM,g ? x x ? F(is boring) lte,tgt
• fishM,g ? x x ? F(fish) lte,tgt

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Bond is hungry
is hungryM,g ? x x ? F(is hungry) lte,tgt
S
BondM,g F(Bond)ltegt
NP
VP
Vi
NP
is hungry
Bond
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Bond is hungry
is hungryM,g ? x x ? F(is hungry) lte,tgt
S
BondM,g F(Bond)ltegt
NP
VP
Vi
NP
is hungry
Bond

• Pass-up tells us that the meaning of a node
  connected by just a single branch to a single
  daughter node has the same interpretation as the
  daughter node.

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Bond is hungry
is hungryM,g ? x x ? F(is hungry) lte,tgt
S
BondM,g F(Bond)ltegt
NP
VP
Vi
NP
is hungry
Bond

• S has two daughters, one is a function (lte,tgt)
  and one can serve as an argument (ltegt) for it.
  So, SM,g is the result of applying the function
  VPM,g to the argument NPM,g (Functional
  Application)

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Bond is hungry
is hungryM,g ? x x ? F(is hungry) lte,tgt
S
BondM,g F(Bond)ltegt
NP
VP
Vi
NP
is hungry
Bond

• SM,g VPM,g ( NPM,g ) ? x x ? F(is
  hungry) ( F(Bond) ) F(Bond) ? x x ? F(is
  hungry) lttgt F(Bond) ? F(is hungry)

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Lexical interpretation rules

• To interpret transitive verbs, we use a function
  that, given the object as an argument, returns a
  predicate (lte,tgt) that can be applied to the
  subject, and is ultimately true if the ordered
  pair ltsubject,objectgt is in the denotation
  provided by F.
• likesM,g ? y ? x ltx,ygt ? F(likes)
  lte,lte,tgtgt

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Bond likes Loren
S
BondM,g F(Bond)ltegt
LorenM,g F(Loren)ltegt
NP
VP
Vt
NP
NP
likes
Bond
NP
Loren

• ? y ? x ltx,ygt ? F(likes) lte,lte,tgtgt

Lexical interpretation rules
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Bond likes Loren
S
BondM,g F(Bond)ltegt
LorenM,g F(Loren)ltegt
NP
VP
Vt
NP
NP
likes
Bond
NP
Loren

• ? y ? x ltx,ygt ? F(likes) lte,lte,tgtgt

Pass-up
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Bond likes Loren
? x ltx,F(Loren)gt ? F(likes) lte,tgt
S
BondM,g F(Bond)ltegt
LorenM,g F(Loren)ltegt
NP
VP
Vt
NP
NP
likes
Bond
NP

• ? y ? x ltx,ygt ? F(likes) lte,lte,tgtgt

Loren
Functional Application VtM,g is type lte,lte,tgtgt
(takes an ltegt argument), and NPM,g is type ltegt.
So VPM,g VtM,g (NPM,g) ? y ? x
ltx,ygt ? F(likes) ( F(Loren) ) F(Loren) ? y
? x ltx,ygt ? F(likes) ? x ltx,F(Loren)gt
? F(likes)
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Bond likes Loren
ltF(Bond),F(Loren)gt ? F(likes)lttgt
? x ltx,F(Loren)gt ? F(likes) lte,tgt
S
BondM,g F(Bond)ltegt
LorenM,g F(Loren)ltegt
NP
VP
Vt
NP
NP

• ? y ? x ltx,ygt ? F(likes) lte,lte,tgtgt

likes
Bond
NP
Loren
Functional Application VPM,g is type lte,tgt
(takes an ltegt argument), and NPM,g is type ltegt.
So SM,g VPM,g (NPM,g) ? x
ltx,F(Loren)gt ? F(likes) ( F(Bond) ) F(Bond) ?
x ltx,F(Loren)gt ? F(likes)
ltF(Bond),F(Loren)gt ? F(likes) lttgt
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The problem with object quantifiers

• Proceeding on, the definition of every is
• everyM,g ?P ?Q ?x?U P(x) ? Q(x)
• The idea is that every P Q is true whenever being
  P implies being Q. That is, every fish is boring
  means for all x (in the universe U), if x is a
  fish, then x is boring.
• So, every needs two predicates to be true or
  false. One is the common noun, the other is the
  sentence with a variable in place of the
  quantifier (every (fish)) (something B likes).

S
NP
VP
Vt
NP
NP
ltlte,tgt,tgt
lte,lte,tgtgt
NC
likes
Bond
Det
every
fish
lte,tgt
ltlte,tgt,ltlte,tgt,tgtgt
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The problem with object quantifiers

• The problem with having a quantifier like every
  fish in object position is that it leaves no way
  to interpret VPM,g.
• Neither daughter is a function that can take the
  other as an argument.
• One needs ltegt(but the other is ltlte,tgt,tgt).
• One needs lte,tgt(but the other is lte,lte,tgtgt).
• This is why QR is needed. QR replaces the NP with
  ti, interpreted as type ltegt, which can be taken
  by VtM,g as an argument.

S
NP
VP
Vt
NP
NP
ltlte,tgt,tgt
lte,lte,tgtgt
NC
likes
Bond
Det
every
fish
ltlte,tgt,ltlte,tgt,tgtgt
lte,tgt
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Quantifier Raising
S
S
NP
S?
NP
VP
S
NC
1
Det
Vt
NP
NP
NP
VP
every
fish
NC
likes
Bond
Det
Vt
NP
t1
every
fish
likes
Bond
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Quantifier Raising
S
NP
S?
S
NC
1
lttgt
Det
NP
VP
every
fish
lte,tgt
ltegt
Vt
NP
t1
ltegt
lte,lte,tgtgt
likes
Bond

• ?iM,g g(i) the individual that i points to
• SM,g ltF(Bond), g(1)gt ? F(likes) Bond
  likes him1

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Quantifier Raising
S

• The goal is to provide every with two predicates
  it has gotten fish. It needs one that means
  something Bond likes.
• We can deduce that S?M,g must be apredicate
  (typelte,tgt), and that therefore 1M,g must be
  type ltt,lte,tgtgt.
• We want 1M,g to turn Bond likes it1 into
  something Bond likes.

lttgt
NP
ltlte,tgt,tgt
S?
S
NC
1
lttgt
Det
lte,tgt
NP
VP
every
fish
lte,tgt
ltegt
ltlte,tgt,ltlte,tgt,tgtgt
ltegt
Vt
NP
t1
lte,lte,tgtgt
likes
Bond
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Lambda abstraction

• We have something that means Bond likes it1
  (ltF(Bond), g(1)gt ? F(likes) ).
• We want something that means something Bond
  likes ? x ltF(Bond), xgt ? F(likes)
• So, what we want to do is add a ? and a variable
  (x) to make it a function, and then replace g(1)
  with x inside the result.

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Modified assignment functions

• How do we change g(1) inside S somewhere to x?
• One way to do it is to evaluate S with a modified
  assignment function. What we want is to have this
  function map 1 to whatever x is, but otherwise
  act like g.
• We write that assignment function like g1/x.

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Quantifier Raising
S
lttgt
NP
ltlte,tgt,tgt
S?
S
NC
1
lttgt
Det
lte,tgt
NP
VP
every
fish
lte,tgt
ltegt
ltlte,tgt,ltlte,tgt,tgtgt
ltegt
Vt
NP
t1
lte,lte,tgtgt
likes
Bond

• SM,g ltF(Bond), g(1)gt ? F(likes)
• SM,g1/x ltF(Bond), g1/x(1)gt ? F(likes)
  ltF(Bond), xgt ? F(likes)

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Quantifier Raising
S
lttgt
NP
ltlte,tgt,tgt
S?
S
NC
1
lttgt
Det
lte,tgt
NP
VP
every
fish
lte,tgt
ltegt
ltlte,tgt,ltlte,tgt,tgtgt
ltegt
Vt
NP
t1
lte,lte,tgtgt
likes
Bond

• So we define 1M,g as ? S ? x SM,g1/x
  and we end up with S?M,g ? x ltF(Bond), xgt
  ? F(likes) , as desired.

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Quantifier Raising
?x ltF(Bond), xgt ? F(likes)
?Q ?x?U x ? F(fish) ? Q(x)
S
lttgt
NP
lte,tgt
ltlte,tgt,tgt
S?
S
NC
1
lttgt
Det
lte,tgt
ltt,lte,tgtgt
NP
VP
every
fish
lte,tgt
ltegt
ltlte,tgt,ltlte,tgt,tgtgt
ltegt
Vt
NP
t1
lte,lte,tgtgt
?P ?Q ?x?U P(x) ? Q(x)
likes
Bond

• SM,g ?Q ?x?U x ? F(fish) ? Q(x) ( ?x
  ltF(Bond), xgt ? F(likes) ) ?x?U x ? F(fish) ?
  ltF(Bond), xgt ? F(likes)

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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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?

• ? ?
• ?
• ? ?
• ? ?
• ?
• ?