Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
● Tom is mortal is-mortal(Tom)→
● Dick is mortal is-mortal (Dick)→
● Fido is mortal is mortal (Fido)→
--------------> Everything is mortal is-mortal(everything)→
→ ”?” is-mortal(x)
Quantification_Introduction1
● Universal quantifier
The expression: ∀x P(x), denotes the universal quantification of the
atomic formula P(x).
➢ ∀ is called the universal quantifier, and x means all the objects
x in the universe. If this is followed by P(x) then the
meaning is that P(x) is true for every object x in the
universe.
➢ For example, "All cars have wheels" could be transformed into
the propositional form, ∀x P(x), where:
✗ P(x) is the predicate denoting: x has wheels, and
✗ the universe of discourse is only populated by cars.
● Socretes is handsome is-handsome(Socretes)→
● Tom is handsome is-handsome (Tom)→
● Harry is handsome is handsome (Harry)→
-------------> Something is handsome is-handsome(something)→
→ ”?” is-handsome(x)
Quantification_Introduction2
● Existential quantifier
The expression: ∃xP(x), denotes the existential quantification of P(x).
➢ "There exists an x such that P(x)" or "There is at least one x
such that P(x)".
➢ ∃ is called the existential quantifier, and x means at least one
object x in the universe. If this is followed by P(x) then
the meaning is that P(x) is true for at least one object x
of the universe.
➢ For example, "Someone loves you" could be transformed into
the propositional form, ∃xP(x), where:
✗ P(x) is the predicate meaning: x loves you,
✗ The universe of discourse contains (but is not limited to) all
living creatures.
Quantification_Preliminary work1
● Quantifiers: ”universal” and ”existential”
Natural language quantifiers have traditionally been categorised as either
type <a> or type <b> quantifiers.
<a>: Quantifiers of type <a> are properties of sets and are expressed
through pronouns like nothing, everybody or no one. They combine with
a verb phrase to form a sentence:
Everybody enjoyed the party.Everybody enjoyed the party.
<b>: Quantifiers of type <b> are binary relations between sets and are
expressed through determiners like some, all or no. They combine with a
noun phrase (the restriction of the quantifier) and a verb phrase (its
scope) to form a sentence:
All guests enjoyed the party.All guests enjoyed the party.
Quantification_Preliminary work2
● Theories of quantification
a. Fregean teories with bound variables
1. Classical theory in First­Order Language
2. Montague’s quantification expressed in Church’s λ­Calculus
b. Fregean theory without bound variables 
3. Illative theory expressed in Curry’s Combinatory Logic
a. Fregean teories with bound variables
1. Classical theory in First­Order Language
2. Montague’s quantification expressed in Church’s λ­Calculus
b. Fregean theory without bound variables 
3. Illative theory expressed in Curry’s Combinatory Logic
Examples
a) Fregean analysis of Quantifiers in First-order language
b) Logical representations of quantifiers using Church’s λ-calculus
Everybody is pretty
Every girl is pretty
Some is pretty
Some girl is pretty
a)  (∀x)[is­pretty’(x)]
b)  (λP.((∀x)[P(x)])(is­pretty’))
a)  (∀x)[girl’(x) => is­pretty’(x)]
b)  (λP.λQ((∀x)[P(x) => Q(x)])(girl’s)(is­pretty’))
a)  (∃x)[is­pretty’(x)]
b)  (λP.((∃x)[P(x)])(is­pretty’))
a)  (∃x)[girl’(x) & is­pretty’(x)]
b)  (λP.λQ((∃x)[P(x) & Q(x)])(girl’s)(is­pretty’))
Quantification_Formal analysis
● Illative quantifiers in CL framework
➢ Illative operators “represent” classical quantifiers inside Curry’s
Combinatory Logic formalism.
➢ Illative operators are adjoined to the “pure” applicative formalism
and their actions are defined, by means of elimination and
introduction rules in Gentzen’s Natural Deduction style, without
using bound variables.
● Illative universal quantifiers: Π1
and Π2
 Π1
f: every is f
 Π2
fg: every f is g
The two quantifiers Π1
and Π2
are not independent since it is possible
to define Π2
, inside Combinatory Logic, from Π1
by the following
relation between operators:
These are propositions
Definition of the universal quantifier
[ Π2
 =def
 ((B(CB2
) ) => Φ Π1
) ]
Definition of the universal quantifier
[ Π2
 =def
 ((B(CB2
) ) => Φ Π1
) ]
● This relation shows that the restricted illative quantifier Π2
is defined by
means of a Combinator B(CB2
) that combines the implication operatorΦ
=> with the quantifier Π1
.
1/ Π2
fg hyp.
2/ [ Π2
=def (B(CB2
)Φ) => Π1
]  def. de Π2
3/ ((B(CB2
)Φ) => Π1
) fg  rempl. 2., 1.
4/ (CB2
)(Φ =>) Π1
fg  [e­B]
5/ B2
 Π1
(Φ =>) fg  [e­C]
6/ Π1
 (Φ => fg)  [e­B2
]
The elimination rule [e-Π2
] is deduced from [e-Π1
]:
1/ Π1
(Φ => fg)  hyp.
2/ fx  hyp.
3/ (Φ => fg) x  [e­Π1
],1.
4/ =>(fx)(gx)  [e­Φ],3.
Definitions of the [e­Π2
] and [e­Π1
]:
Π1
f      Π2
fg         f(x)
­­­­­­­­­­­[e­Π1
] ­­­­­­­­­­­­­­­­­­­­[e­Π2
]
     fx g(x)
Definitions of the [e­Π2
] and [e­Π1
]:
Π1
f      Π2
fg         f(x)
­­­­­­­­­­­[e­Π1
] ­­­­­­­­­­­­­­­­­­­­[e­Π2
]
     fx g(x)
Modus ponens
P   Q , P→
­­­­­­­­­­­­­­
Q
Comment: whenever an instance of ”P   Q” and ”P” appear →
by themselves on lines of a logical proof, ”Q” can validly be 
placed on a subsequent line.  
Modus ponens
P   Q , P→
­­­­­­­­­­­­­­
Q
Comment: whenever an instance of ”P   Q” and ”P” appear →
by themselves on lines of a logical proof, ”Q” can validly be 
placed on a subsequent line.  
● Illative existential quantifiers: Σ1
and Σ2
 “Σ1
f” (“there is a f”)
 “Σ2
fg” (“there is a f which is g”)
● Expression of Σ2
in terms of & (conjunction) and 1 :Σ
These are propositions
Definition of the existential quantifier
[ Σ2
=def
(B(CB2
)Φ) & Σ1
]
Definition of the existential quantifier
[ Σ2
=def
(B(CB2
)Φ) & Σ1
]
● Examples
Jane is pretty → (C*Jane)(is-pretty)
Everybody is pretty → Π1
(is-pretty)
Every girl is pretty → (Π2
(girl))(is-pretty)
Somebody runs → Σ1
(runs)
Some girl is pretty → (Σ2
(girl))(is-pretty)
Every boy love some girl → (Π2
(boy))(love(Σ2
(girl)))
● Every man like itself
1/(every man) (like itself)
2/ Π2
man (like itself)
3/ ((B(CB2
) ) => Φ Π1
) man (like itself)
4/ (CB2
)(   =>) Φ Π1
 man (like itself)
5/ B2
Π1
(   =>)  man (like itself)Φ
6/ Π1
((   =>)  man (like itself)Φ )
7/ ((   =>)  man (like itself)Φ ) x
8/  
=>  (man x) ((like itself) x)
[ Π2
 =def
 ((B(CB2
) ) => Φ Π1
) ][ Π2
 =def
 ((B(CB2
) ) => Φ Π1
) ]
Definitions of the [e­Π2
] and [e­Π1
]:
Π1
f      Π2
fg         f(x)
­­­­­­­­­­­[e­Π1
] ­­­­­­­­­­­­­­­­­­­­­­[e­Π2
]
     fx g(x)
Definitions of the [e­Π2
] and [e­Π1
]:
Π1
f      Π2
fg         f(x)
­­­­­­­­­­­[e­Π1
] ­­­­­­­­­­­­­­­­­­­­­­[e­Π2
]
     fx g(x)
● Some girl is pretty    there is (exist at least one) a girl who is pretty→
1/ (some(girl))(is­pretty)
2/(Σ2
(girl))(is­pretty)
3/ ((B(CB2
)Φ) & Σ1
(girl)) (is­pretty)
4/ ((CB2
) (Φ  &) Σ1
(girl)) (is­pretty)
5/ (B2
 Σ1 
(Φ  &) (girl)) (is­pretty)
6/ Σ1  
((Φ  &) (girl) (is­pretty))
7/  
((Φ  &) (girl) (is­pretty)) x
8/ & (girl(x)) ((is­pretty) x)
[ Σ2
=def
(B(CB2
)Φ) & Σ1
][ Σ2
=def
(B(CB2
)Φ) & Σ1
]
Definitions of the [e­Σ2
] and [e­Σ1
]:
    Σ1
f             Σ2
fg         f(x)
­­­­­­­­­­­[e­Σ1
] ­­­­­­­­­­­­­­­­­­­­[e­Σ2
]
     fx          g(x)
Definitions of the [e­Σ2
] and [e­Σ1
]:
    Σ1
f             Σ2
fg         f(x)
­­­­­­­­­­­[e­Σ1
] ­­­­­­­­­­­­­­­­­­­­[e­Σ2
]
     fx          g(x)
18
Next week...
● Continue about the application of the combinators to natural
language analysis: RevisionRevision