Indefinite Noun Phrases as Restricted Free
Variables
Casey Woolwine
December 16, 2016
Indefinites
N
1 / 42
Overview
1
Two Puzzles
Scope Islands
Donkey Anaphora
2
Free Variable Approach
Heim
3
Restricted Free Variables
Indefinites
N
2 / 42
Two Puzzles
Scope Islands
Initial examples:
I No movement
i Someone reported that Max and all the ladies disappeared.
ii Someone will be offended if we don’t invite most philosophers.
iii Many students believe anything every teacher says.
Indefinites
N
3 / 42
Two Puzzles
Scope Islands
Initial examples:
I No movement
i Someone reported that Max and all the ladies disappeared.
ii Someone will be offended if we don’t invite most philosophers.
iii Many students believe anything every teacher says.
II Movement
i Everyone reported that Max and some lady disappeared.
ii Most guests will be offended if we don’t invite some philosopher.
iii All students believe anything that many teachers say.
Indefinites
N
3 / 42
Two Puzzles
Scope Islands
Against pragmatic strengthening:
I Every man loves a dog.
i [All men, x] [some dog, y] [x loves y]
ii [Some dog, y] [all men, x] [x loves y]
Indefinites
N
4 / 42
Two Puzzles
Scope Islands
Against pragmatic strengthening:
I Every man loves a dog.
i [All men, x] [some dog, y] [x loves y]
ii [Some dog, y] [all men, x] [x loves y]
II If some relative of mine dies, I’ll inherit a house.
i If [[some relative of mine, x] [x dies]] then [[some house, y] [I inherit y]]
ii [Some relative of mine, x] [if [x dies] then [[some house, y] [I inherit y]]]
Indefinites
N
4 / 42
Two Puzzles
Scope Islands
Against referential explanations:
I If some relative of mine dies, I’ll inherit a house.
i If [[some relative of mine, x] [x dies]] then [[some house, y] [I inherit y]]
ii [if [x1 dies] then [[some house, y] [I inherit y]]]
Indefinites
N
5 / 42
Two Puzzles
Scope Islands
Against referential explanations:
I If some relative of mine dies, I’ll inherit a house.
i If [[some relative of mine, x] [x dies]] then [[some house, y] [I inherit y]]
ii [if [x1 dies] then [[some house, y] [I inherit y]]]
II Most linguists have looked at every analysis that solves some problem.
i [Most linguists, x] [[some problem, y] [every analysis that solves y, z] [x looked
at z]]
Indefinites
N
5 / 42
Two Puzzles
Donkey Anaphora
I If John owns a donkey he feeds it.
i * If [[some donkey, x] [Johns owns x]] then [John feeds x]
Indefinites
N
6 / 42
Two Puzzles
Donkey Anaphora
I If John owns a donkey he feeds it.
i * If [[some donkey, x] [Johns owns x]] then [John feeds x]
ii [some donkey, x] [if [John owns x] then [John feeds x]]
Indefinites
N
6 / 42
Two Puzzles
Donkey Anaphora
I If John owns a donkey he feeds it.
i * If [[some donkey, x] [Johns owns x]] then [John feeds x]
ii [some donkey, x] [if [John owns x] then [John feeds x]]
II Every farmer who owns a donkey feeds it.
i *[Every farmer that [some donkey, y] [owns y], x] [x feeds y]
Indefinites
N
6 / 42
Two Puzzles
Donkey Anaphora
I If John owns a donkey he feeds it.
i * If [[some donkey, x] [Johns owns x]] then [John feeds x]
ii [some donkey, x] [if [John owns x] then [John feeds x]]
II Every farmer who owns a donkey feeds it.
i *[Every farmer that [some donkey, y] [owns y], x] [x feeds y]
ii [Some donkey, y] [[Every farmer that owns y, x] [x feeds y]]
Indefinites
N
6 / 42
Free Variable Approach
Heim
I A man arrived.
Indefinites
N
7 / 42
Free Variable Approach
Heim
I A man arrived.
i man(x) and arrived(x)
Indefinites
N
7 / 42
Free Variable Approach
Heim
I A man arrived.
i man(x) and arrived(x)
ii ∃x(man(x) and arrived(x))
Indefinites
N
7 / 42
Free Variable Approach
Heim
I A man arrived.
i man(x) and arrived(x)
ii ∃x(man(x) and arrived(x))
II Every farmer who owns a donkey feeds it.
i [every donkey(x) and farmer(y) such that owns(y)(x)][feeds(y)(x)]
Indefinites
N
7 / 42
Free Variable Approach
Logical Forms
I NP Indexing: Assign every NP a referential index.
II
S
NP
a
man
VP
arrived
Indefinites
N
8 / 42
Free Variable Approach
Logical Forms
I NP Indexing: Assign every NP a referential index.
II
S
S
NP
a
man
NP1
VP
arrived
a
man
VP
arrived
Indefinites
N
8 / 42
Free Variable Approach
Logical Forms
I NP Indexing: Assign every NP a referential index.
II
S
S
NP
a
man
NP1
VP
arrived
a
man
VP
arrived
III NP Prefixing: Adjoin every non-pronominal NP to S, leaving behind a
coindexed empty NP.
Indefinites
N
8 / 42
Free Variable Approach
Logical Forms
I NP Indexing: Assign every NP a referential index.
II
S
NP
a
man
S
VP
arrived
NP1
a
man
S
VP
arrived
NP1
a
man
S
arrived(e1 )
e1
arrived
III NP Prefixing: Adjoin every non-pronominal NP to S, leaving behind a
coindexed empty NP.
Indefinites
N
9 / 42
Free Variable Approach
Logical Forms
I NP Indexing: Assign every NP a referential index.
II
S
S
VP
NP
every
man
arrived
NP1
every
man
S
VP
arrived
NP1
every
man
S
arrived(e1 )
e1
arrived
III NP Prefixing: Adjoin every non-pronominal NP to S, leaving behind a
coindexed empty NP.
Indefinites
N
10 / 42
Free Variable Approach
Logical Forms
I Quantifier Construal: Attach every quantifier as a leftmost immediate
constituent of S.
Indefinites
N
11 / 42
Free Variable Approach
Logical Forms
I Quantifier Construal: Attach every quantifier as a leftmost immediate
constituent of S.
II
S
NP1
every
man
S
arrived(e1 )
e1
arrived
Indefinites
N
11 / 42
Free Variable Approach
Logical Forms
I Quantifier Construal: Attach every quantifier as a leftmost immediate
constituent of S.
II
S
NP1
every
S
arrived(e1 )
man
e1
III
arrived
S
every
NP1
–
man
S
arrived(e1 )
e1
arrived
Indefinites
N
11 / 42
Free Variable Approach
NP Semantics
I [[–N̄n ]]=[[aN̄n ]] = N̄(xn )
Indefinites
N
12 / 42
Free Variable Approach
NP Semantics
I [[–N̄n ]]=[[aN̄n ]] = N̄(xn )
II
S
NP1
S
man(x1 )
arrived(e1 )
a
e1
man
III
arrived
S
every
NP1
S
man(x1 )
arrived(e1 )
–
e1
man
arrived
Indefinites
N
12 / 42
Free Variable Approach
Binding
I Existential Closure
i
S
man(x1 ) and arrived(e1 )
NP2
a
S
man
arrived(e1 )
e1
arrived
Indefinites
N
13 / 42
Free Variable Approach
Binding
I Existential Closure
i
S
S
man(x1 ) and arrived(e1 )
∃x1 (man(x1 ) and arrived(x1 ))
NP2
a
S
man
arrived(e1 )
e1
∃
S
man(x1 ) and arrived(e1 )
arrived
NP2
a
man
S
arrived(e1 )
e1
arrived
Indefinites
N
13 / 42
Free Variable Approach
Binding
I Unselective Binding
i
S
NP1
every
S
feeds(e1 )(e2 )
NP1
–
farmer
S
who1
NP2
a
donkey
S
owns(e1 )(e2 )
Indefinites
N
14 / 42
Free Variable Approach
Binding
I Unselective Binding
i
S
∀x1 ∀x2 (if (farmer(x1 ) and donkey(x2 ) and owns(x1 )(x2 )) then feeds(x1 )(x2 ))
NP1
every
S
feeds(e1 )(e2 )
NP1
–
farmer
S
who1
NP2
a
N
donkey
S
owns(e1 )(e2 )
Indefinites
15 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
Indefinites
N
16 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
i
S
NP1
every
S
TWS(e1 )(e2 )
NP1
–
professor
S
who1
NP2
a
SOM
S
met(e1 )(e2 )
Indefinites
N
16 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
i
S
∃
S
every
NP1
S
TWS(e1 )(e2 )
NP1
–
professor
S
who1
NP2
a
SOM
S
met(e1 )(e2 )
Indefinites
N
17 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
i
S
∃
S
every
NP1
S
TWS(e1 )(e2 )
NP1
–
professor
S
who1
NP2
a
SOM
S
met(e1 )(e2 )
ii ∃x2 ∀x1 (if (x2 is a student and x1 is a professor and x2 met x1 ) then
(x1 thought x2 was sharp) )
Indefinites
N
17 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
i
S
NP2
a
S
SOM
NP1
every
S
TWS(e1 )(e2 )
NP1
–
professor
who1
S
met(e1 )(e2 )
Indefinites
N
18 / 42
Free Variable Approach
Scoping Problems
I Every professor who met a student of mine thought she was sharp.
i
S
∃
S
NP2
a
S
SOM
NP1
every
S
TWS(e1 )(e2 )
NP1
–
professor
who1
S
met(e1 )(e2 )
Indefinites
N
19 / 42
Restricted Free Variables
Modifying Heim
I Keeping:
i Indefinites are free variables.
Indefinites
N
20 / 42
Restricted Free Variables
Modifying Heim
I Keeping:
i Indefinites are free variables.
ii They may be bound through Existential Closure, overt quantifiers in whose
restrictor they occur, and conditionals.
Indefinites
N
20 / 42
Restricted Free Variables
Modifying Heim
I Keeping:
i Indefinites are free variables.
ii They may be bound through Existential Closure, overt quantifiers in whose
restrictor they occur, and conditionals.
II Changing:
i There is no movement of indefinite NPs.
Indefinites
N
20 / 42
Restricted Free Variables
Modifying Heim
I Keeping:
i Indefinites are free variables.
ii They may be bound through Existential Closure, overt quantifiers in whose
restrictor they occur, and conditionals.
II Changing:
i There is no movement of indefinite NPs.
ii Overt quantifiers combine first with their restrictor NP and then with their
nuclear scope.
Indefinites
N
20 / 42
Restricted Free Variables
Modifying Heim
I
S
every1
S
NP1
-
N
S
NP2
person
a
N
VP
t1
dog
V
t2
loves
II
S
DP
1
every
S
person
VP
t1
DP
loves
a
dog
Indefinites
N
21 / 42
Restricted Free Variables
Variable Restriction Sets
I
α
β
II
γ
α; ψ1
β; ψ2
γ; ψ3
Indefinites
N
22 / 42
Restricted Free Variables
Variable Restrictions
I <ρ, φ, ψ>
i ρ = tn , hen , or xn
ii φ = a set of ϕ ∈D<e,t>
iii ψ = a set of variable restrictions for the free variables occurring in elements of φ
Indefinites
N
23 / 42
Restricted Free Variables
Variable Restrictions
I <ρ, φ, ψ>
i ρ = tn , hen , or xn
ii φ = a set of ϕ ∈D<e,t>
iii ψ = a set of variable restrictions for the free variables occurring in elements of φ
II Examples
i < x1 , { λy.man(y) }, ∅ >
Indefinites
N
23 / 42
Restricted Free Variables
Variable Restrictions
I <ρ, φ, ψ>
i ρ = tn , hen , or xn
ii φ = a set of ϕ ∈D<e,t>
iii ψ = a set of variable restrictions for the free variables occurring in elements of φ
II Examples
i < x1 , { λy.man(y) }, ∅ >
ii < x1 , { λy.loves(y)(x2 ) & man(y)}, { <x2 , { λy.woman(y) }, ∅> } >
Indefinites
N
23 / 42
Restricted Free Variables
Variable Restrictions
I <ρ, φ, ψ>
i ρ = tn , hen , or xn
ii φ = a set of ϕ ∈D<e,t>
iii ψ = a set of variable restrictions for the free variables occurring in elements of φ
II Examples
i < x1 , { λy.man(y) }, ∅ >
ii < x1 , { λy.loves(y)(x2 ) & man(y)}, { <x2 , { λy.woman(y) }, ∅> } >
iii <t1 , ∅, ∅>
Indefinites
N
23 / 42
Restricted Free Variables
Variable Restrictions
I <ρ, φ, ψ>
i ρ = tn , hen , or xn
ii φ = a set of ϕ ∈D<e,t>
iii ψ = a set of variable restrictions for the free variables occurring in elements of φ
II Examples
i < x1 , { λy.man(y) }, ∅ >
ii < x1 , { λy.loves(y)(x2 ) & man(y)}, { <x2 , { λy.woman(y) }, ∅> } >
iii <t1 , ∅, ∅>
III Pronouns and Traces
i t1 ; { <t1 , ∅, ∅> }
ii he1 ; { <he1 , ∅, ∅> }
Indefinites
N
23 / 42
Restricted Free Variables
Variable Restrictions
I [[a]]= λϕ∈D<e,t> ; ψ . xn ; {< xn , {ϕ}, ψ >} where n is new to the tree.
II a man
DP
x1 ; {< x1 , {λx.man(x)}, ∅ >}
Det
NP
a
man
λϕ∈D<e,t> ;ψ. xn ; {< xn , {ϕ}, ψ >}
λx.man(x); ∅
Indefinites
N
24 / 42
Restricted Free Variables
Variable Restrictions
S
DP
every
person
1
S
λy.person(y); ∅
t1 ;{ <t1 , ∅, ∅> }
VP
loves
DP
λx.λy.loves(y)(x); ∅
x2 , { < x2 , { λy.dog(y) }, ∅ > }
a
dog
λy.dog(y); ∅
Indefinites
N
25 / 42
Restricted Free Variables
Functional Application
I Functional Application: If α is a branching node and {β ; ψ 0 , γ ; ψ 00 } the set
of its daughter nodes, then for any assignment a, if [[β]]a is a function whose
domain contains [[γ]]a , then [[α]]a = [[β]]a ([[γ]]a ) ; ψ 0 ∪ ψ 00 .
II
S
sleeps(x1 ); {< x1 , {λx.man(x)}, ∅ >}
DP
VP
x1 ; {< x1 , {λx.man(x)}, ∅ >}
V
sleeps
Det
NP
a
man
λϕ∈D<e,t> ;ψ. xn ; {< xn , {ϕ}, ψ >}
λx.man(x); ∅
λx.sleeps(x); ∅
Indefinites
N
26 / 42
Restricted Free Variables
Functional Application
I Functional Application: If α is a branching node and {β ; ψ 0 , γ ; ψ 00 } the set
of its daughter nodes, then for any assignment a, if [[β]]a is a function whose
domain contains [[γ]]a , then [[α]]a = [[β]]a ([[γ]]a ) ; ψ 0 ∪ ψ 00 .
II
S
writes(x1 )(x2 );
< x1 , { λx.woman(x) } , ∅ >
< x2 , { λx.book(x) } , ∅ >
DP
VP
x1 ; {< x1 , {λx.woman(x)}, ∅ >}
λy.writes(y)(x2 ); {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
woman
V
DP
writes
x2 ; {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
book
Indefinites
N
27 / 42
Restricted Free Variables
Functional Application
who chased a pig
CP
who2
C̄
S
C
that
chased (t2 )(x1 );
< x1 , { λx.pig(x) } , ∅ >
< t2 , ∅ , ∅ >
VP
t2
λy.chased(y)(x1 ); {< x1 , {λx.pig (x)}, ∅ >}
V
DP
chased
x1 ; {< x1 , {λx.pig (x)}, ∅ >}
Det
N
a
pig
Indefinites
N
28 / 42
Restricted Free Variables
Predicate Abstraction
I Predicate Abstraction: Let α be a branching node with daughters β and γ ;
ψ, where β dominates only a numerical index i and there is some nested
variable of ψ that is indexed by i. Then, for any assignment a, [[α]]a = λx:
x/i
x∈D. [[γ]]a ;ψ −i .
II
CP
λy.chased(y)(x1 ); {< x1 , {λx.pig (x)}, ∅ >}
who2
C̄
chased (t2 )(x1 );
< x1 , { λx.pig(x) } , ∅ >
< t2 , ∅ , ∅ >
C
that
S
chased (t2 )(x1 );
< x1 , { λx.pig(x) } , ∅ >
< t2 , ∅ , ∅ >
.
.
.
Indefinites
N
29 / 42
Restricted Free Variables
Predicate Modification
I Predicate Modification: If α is a branching node and {β ; ψ 0 , γ ; ψ 00 } the set
of its daughter nodes, then for any assignment a, α is in the domain of [[]]a if
both β and γ are, and both [[β]]a and [[γ]]a are of type <e,t>. In that case
[[α]]a = λx: x∈D and x is in the domain of [[β]]a and [[γ]]a . [[β]]a (x) =
[[γ]]a (x)= 1 ; ψ 0 ∪ψ 00 .
II
DP
x2 ; {< x2 , {λy.dog(y) & chased(y)(x1 )}, {< x1 , {λx.pig (x)}, ∅ >} >}
Det
NP
a
λy.dog(y) & chased(y)(x1 ); {< x1 , {λx.pig (x)}, ∅ >}
N
CP
dog
λy.chased(y)(x1 ); {< x1 , {λx.pig (x)}, ∅ >}
.
.
.
Indefinites
N
30 / 42
Restricted Free Variables
Open Formulas
S
writes(x1 )(x2 );
< x1 , { λx.woman(x) } , ∅ >
< x2 , { λx.book(x) } , ∅ >
DP
VP
x1 ; {< x1 , {λx.woman(x)}, ∅ >}
λy.writes(y)(x2 ); {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
woman
V
DP
writes
x2 ; {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
book
Indefinites
N
31 / 42
Restricted Free Variables
Existential Closure
I Existential Closure: Let α be a branching node with daughters β and γ ; ψ,
where β dominates only an existential closure operator indexed with i, and
there is some nested indefinite variable, xi of ψ, that is also indexed by i.
Then, for any assignment a, [[α]]a = ∃xi ([[ϕ]]a (xi ) & [[γ]]a ) ; ψ −i , where ϕ
is the only element of the set of restricting formulas for the variable
restriction with xi as its first element.
II
S
∃x1 (man(x1 ) & sleeps(x1 ))
∃1
S
sleeps(x1 ); {< x1 , {λx.man(x)}, ∅ >}
DP
VP
x1 ; {< x1 , {λx.man(x)}, ∅ >}
Det
NP
a
man
V
sleeps
Indefinites
N
32 / 42
Restricted Free Variables
Existential Closure
S
∃x1 ∃x2 (woman(x1 ) & book(x2 ) & writes(x1 )(x2 ))
∃1,2
S
writes(x1 )(x2 );
< x1 , { λx.woman(x) } , ∅ >
< x2 , { λx.book(x) } , ∅ >
DP
VP
x1 ; {< x1 , {λx.woman(x)}, ∅ >}
λy.writes(y)(x2 ); {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
woman
V
DP
writes
x2 ; {< x2 , {λx.book(x)}, ∅ >}
Det
NP
a
book
Indefinites
N
33 / 42
Restricted Free Variables
Scope
I Every man loves a dog. (wide scope)
II
S
∃x1 (dog(x1 ) & ∀y(if man(y) then loves(y)(x1 ))
∃1
S
∀y(if man(y) then loves(y)(x1 )); { < x1 , { λy. dog(y)}, ∅ > }
DP
VP
Det
N
V
every
man
loves
DP
Det
N
a
dog
Indefinites
N
34 / 42
Restricted Free Variables
Scope
I Every man loves a dog. (narrow scope)
II
S
∀y(if man(y) then ∃x2 (dog(x2 ) & loves(y)(x2 )))
DP
VP
Det
N
every
man
λy.∃x2 (dog(x2 ) & loves(y)(x2 )); ∅
1
S
∃x2 (dog(x2 ) & loves(t1 )(x2 )); { < t1 , ∅, ∅ > }
∃2
S
t1
VP
V
loves
DP
Det
a
N
dog
N
Indefinites
35 / 42
Restricted Free Variables
Donkey Anaphora
Every farmer who owns a donkey feeds it.
S
DP
every
VP
2
NP
S
t2
farmer
CP
who1
VP
feeds
it?
S
t1
VP
owns
DP
a
donkey
Indefinites
N
36 / 42
Restricted Free Variables
Donkey Anaphora
I [[Everyn ]] = λϕ∈D <e,t> ; ψ∈{ψ00 :xn ∈ρψ00 } . λϕ’∈D <e,t> ; ψ’. {y: ∃xn (ϕn (xn )
= ϕ(y) = 1 and ϕ’(y) = 0) } = ∅ ; ψ −n ∪ ψ’−n where ϕn is the only
element of the non-empty set of restricting formulas for the variable
restriction with xn as its first element.
II
S
{y:∃x3 (donkey(x3 ) = (farmer(y) & owns(y)(x3 ))=1 & feeds(y)(x3 )=0)}=∅; ∅
λy.feeds(y)(it3 ); {< it3 , ∅, ∅ >}
DP
λϕ0 ;ψ’.{y:∃x3 (donkey(x3 )=(farmer(y) & owns(y)(x3 ))=1 and ϕ’(y)=0)}=∅; ψ’−3
1
S
t1
Det
NP
every3
λy.farmer(y) & owns(y)(x3 ); { < x3 , {λx.donkey(x)}, ∅ >}
.
.
.
VP
V
feeds
Indefinites
N
it3
37 / 42
Restricted Free Variables
Donkey Anaphora
I [[Non ]] = λϕ∈D <e,t> ; ψ∈{ψ00 :xn ∈ρψ00 } . λϕ’∈D <e,t> ; ψ’. {y: ∃xn (ϕn (xn ) =
ϕ(y) = ϕ’(y) = 1) } = ∅ ; ψ −n ∪ ψ’−n where ϕn is the only element of the
non-empty set of restricting formulas for the variable restriction with xn as its
first element.
Indefinites
N
38 / 42
Restricted Free Variables
Donkey Anaphora
I [[Non ]] = λϕ∈D <e,t> ; ψ∈{ψ00 :xn ∈ρψ00 } . λϕ’∈D <e,t> ; ψ’. {y: ∃xn (ϕn (xn ) =
ϕ(y) = ϕ’(y) = 1) } = ∅ ; ψ −n ∪ ψ’−n where ϕn is the only element of the
non-empty set of restricting formulas for the variable restriction with xn as its
first element.
II [[Mostn ]] = λϕ∈D <e,t> ; ψ∈{ψ00 :xn ∈ρψ00 } . λϕ’∈D <e,t> ; ψ’. |{y: ∃xn (ϕn (xn )
= ϕ(y) = ϕ’(y) = 1) }| > |{y: ∃xn (ϕn (xn ) = ϕ(y) = 1 and ϕ’(y) = 0) }| ;
ψ −n ∪ ψ’−n where ϕn is the only element of the non-empty set of restricting
formulas for the variable restriction with xn as its first element.
Indefinites
N
38 / 42
Restricted Free Variables
Nested Restriction Sets
I The set of nested restriction sets, ψ̄, of ψ= { ψ 0 : there is some sequence of
variable restrictions <ρn , φn , ψn >n ... <ρn+p , φn+p , ψn+p >n+p such that
ψ n+p =ψ 0 , <ρn , φn , ψn >n ∈ ψ, and for every <ρk , φk , ψk >k in this
sequence, <ρk , φk , ψk >k ∈ ψk−1 }
i ψ = {<ρ, φ, {<ρ0 , φ0 , {<ρ00 , φ00 , ∅>}>}>}
ii ψ̄ { {<ρ0 , φ0 , {<ρ00 , φ00 , ∅>}>}, {<ρ00 , φ00 , ∅>}, ∅ }
Indefinites
N
39 / 42
Restricted Free Variables
Nested Variables
I The set of nested variables, ρψ , of ψ = { ρ : for some φ and ψ 0 , <ρ, φ, ψ 0 >
∈ ψ or <ρ, φ, ψ 0 > ∈ ψ 00 for some ψ 00 ∈ ψ̄ }
i ψ = {<ρ, φ, {<ρ0 , φ0 , {<ρ00 , φ00 , ∅>}>}>}
ii ρψ = { ρ, ρ0 , ρ00 }
Indefinites
N
40 / 42
Restricted Free Variables
ψ −i
I ψ −i = The result of replacing every nested restriction set, ψ 0 , of ψ that
contains a variable restriction, <ρ, φ, ψ 00 > where ρ is a variable indexed by i,
with (ψ 0 \{ <ρ, φ, ψ 00 > }) ∪ ψ 00
i ψ = {<ρ, φ, {<ρ0 , φ0 , {<ρ00 , φ00 , ∅>}>}>}
ii ψ −n for ρ0 = xn { < ρ, φ, { < ρ00 φ00 , ∅ > } > }
Indefinites
N
41 / 42
Restricted Free Variables
Abusch
Figure: Every person who likes everyone who likes a cat likes the cat.
Abusch(1993-94)
Indefinites
N
42 / 42