1. QUANTIFICATION THEORY
Definition: (Open Proposition or Propositional Function)
An open proposition is a declarative sentence which
1. contains one or more variables
2. is not a proposition, and
3. produces a proposition when each of its variables is replaced by a specific object from a
designated set.
The set of objects which the variables in an open proposition can represent is the UNIVERSE OF
DISCOURSE of the open proposition. To be precise it is necessary to establish the universe explicitly but
frequently the universe is left implicit.
examples of open propositions
1. the number x + 1 is an even integer.
2. 2x + y = 5 ∧ x - 3y = 8
3. x1
2
+ x2
2
+ x3
2
= 14
4. x is a rational number.
5. y > 5 .
6. x + y = 5
7. x climbed Mount Everest.
8. He is a lawyer and she is a scientist.
Something to think about:
Implicitly, the universe of discourse for open propositions 1, 2, 3, 4, 5, and 6 is the set of integers. What
about propositions 7 and 8?
FUNCTIONAL NOTATIONS
( The above propositional functions can be denoted using capital letters and literal variables )
Example:
1. P( x ) can denote “x is a rational number”
2. Q( x, y ) can denote x+y=5
3. R( X1, X2, X3 ) can denote “x1
2
+ x2
2
+ x3
2
= 14”
BINDING A VARIABLE
To change an open proposition (propositional function) into a proposition, each individual variable
must be bound. This can be done in two ways:
a)By assigning values:
ex. 1) P ( 1 ) is true
When x = 1, the number x + 1 is an even integer.
2) Q ( 2, 1 ) is false
When x=2 and y=1, ( 2x + y = 5 ∧ x - 3y = 8 )
3) R ( 1, 2, 3 ) is true
When X1 = 1, X2 = 2 and X3 = 3, X1
2
+ X2
2
+ X3
2
= 14
B)By Quantification
1. Universal Quantification
Notation: ∀x P( x ) read as “For all x, P( x )”

2. For an open sentence P(x) with variable x, the sentence ∀x P( x ) is read “for all x, P(x)”
and is true precisely when the truth set for P(x) is the entire universe. The symbol ∀ is called the
universal quantifier.
similar forms:
For all x, All x such that
For every x, Every x such that
For each x, Each x such that
ex:
1) Let P( x ) denote X2
≥ 4
consider U = R
set of real nos.
universe of discourse
∀x P( x ) is false
For all real numbers x, It is false that x2
≥ 4
2) P( x ) : x2
> 0
U = Z+
set of positive integers
∀x P( x ) is true
For all positive integers x , x2
> 0 is true
2. Existential Quantification
Notation : ∃x P( x ) read as “there exist an x such that P(x)”
The sentence ∃x P( x ) is read “there exist an x such that P( x )” and is true precisely
when the truth set for P(x) is nonempty. The symbol ∃ is called the existential quantifier
similar forms:
There exist an x such that…
There is an x such that…
For some x …
There is at least one x such that…
Some x is such that …
ex: Let P( x ) denote x + 75 = 80
U = Z+
set of positive integers
∃x P( x ) is true
“There exist a positive integer x such that x+75=80” is true
DETERMINING THE TRUTH VALUE OF QUANTIFIED PROPOSITIONS
STATEMENT WHEN IS IT TRUE? WHEN IS IT FALSE?
∃x P( x ) if there is at least one c ∈ U For all c ∈ U
such that P( x ) is satisfied P(c) is false
∀x P( x ) If all elements of U If there is at least one c ∈ U
satisfies P( x ) such that P( x ) is false
Quantified Propositions with 2 variables

3. In general, if P(x,y) is any predicate involving the two variables x and y, then the following
possibilities exist:
(∀x)(∀y)P(x,y) (∀x)(∃y)P(x,y)
(∃x)(∀y)P(x,y) (∃x)(∃y)P(x,y)
(∀y)(∀x)P(x,y) (∃y)(∀x)P(x,y)
(∀y)(∃x)P(x,y) (∃y)(∃x)P(x,y)
If a sentence involves both the universal and the existential quantifiers, one must be careful about
the order in which they are written. One always works from left to right.
For instance, consider the two sentences concerning real numbers:
1. (∀x)(∃y)[x+y=5]
2. (∃y)(∀x)[x+y=5]
Statement 1 is true. Why?
Statement 2 is false. Why?
Question:
Consider (∀x)(∀y)[x∈R ∧ y∈R ⇒ xy ∈ R]
a. What does it mean?(R=set of real numbers)
b. What is its truth value?
Exercises:
A. Determine the truth values:
let U = { 1,2,3... } = Z+
(set of positive integers)
1) ∀x ( x is a prime ⇒ x is odd )
Ans: false ( 2 is prime but not odd )
2) ∀x ∀y ( x is odd ∧ y is odd ) ⇒ x ∗ y is odd
Ans: true
3) ∃w ( 2w + 1 = 5 )
Ans: true
4) ∃x ( 2x + 1 = 5 ∧ x 2
= 9 )
Ans: false
B. Determine the truth value of the following sentences where the universe is the set of integers
1. ∀x,[x2
-2 ≥ 0]
2. ∀x,[x2
-10x+21 = 0]
3. ∃x,[x2
-10x+21 = 0]
4. ∀x,[x2
-x-1 ≠ 0]
5. ∃x,[x2
-3 = 0]
6. ∃x,[(x2
>10) ∧ (x is even)]
4 TYPES OF QUANTIFIED PROPOSITIONS
• let : H(x) denote x is human
• M(x) - x is mortal
1. Universal Affirmative
ex: All humans are mortal. ∀x ( H(x) ⇒ M(x) )
2. Universal Negative
ex: No human is mortal. ∀x ( H(x) ⇒ ¬M(x) )
3. Existential Affirmative
ex: Some humans are mortal. ∃x ( H(x) ∧ M(x) )
4. Existential Negative
ex: Some humans are not mortal. ∃x( H(x) ∧ ¬M(x))

4. Examples and Exercises:
Translate each of the following statements into symbols, using quantifiers, variables, and predicate
symbols.
1. All members are either parents or teachers.
Ans: ∀x [ M(x) ⇒ ( P(x) ∨ T(x) ) ]
2. Some politicians are either disloyal or misguided.
Ans: ∃x [ P(x) ∧ ( D(x) ∨ M(x) ) ]
3. All birds can fly
4. Not all birds can fly
5. All babies are illogical
6. Some babies are illogical
7. If Joseph is a man, then Joseph is a giant
8. There is a student who likes mathematics but not history
QUANTIFICATION RULES( Rules of Inference):
1. Universal Instantiation or Universal Specification
∀x P( x )__
∴P( c ) * c is an arbitrary element of U
example : universe of discourse,U = set of humans
∀x P( x ) : All humans are mortals
P( c ) : Ronald is mortal.
2. Existential Instantiation or Existential Specification
∃x P( x )__
∴P( c ) *where c is some element of U
3. Universal Generalization ( UG )
P ( c )____
∴∀x P( x )
4. Existential Generalization ( EG )
P( c )____
∴∃x P( x )
Construct a proof of validity:
1. No mortals are perfect.
All humans are mortals.
Therefore, no humans are perfect.
1) ∀x ( M(x) ⇒ ¬P(x) )
2) ∀x ( H(x) ⇒ M(x) )____
∴ ∀x ( H(x) ⇒ ¬P(x) )
3) M(c) ⇒ ¬P(c) Universal Instantiation ( 1 )
4) H(c) ⇒ M(c) Universal Instantiation ( 2 )
5) H(c) ⇒ ¬P(c) Hypothetical Syllogism ( 4 )( 3 )
6) ∀x ( H(x) ⇒ ¬P(x) ) Universal Generalization ( 5 )
2. Tigers are fierce and dangerous.
Some tigers are beautiful.
Therefore, some dangerous things are beautiful.
1) ∀x [ T(x) ⇒ ( F(x) ∧ D(x) ) ]
2) ∃x [ T(x) ∧ B(x) ]____
∴∃x ( D(x) ∧ B(x))
3) T(c) ∧ B(c) Existential Instantiation ( 2 )
4) T(c) ⇒ ( F(c) ∧ D(c) ) Universal Instantiation ( 1 )

5. 5) T(c) Simplification ( 3 )
6) F(c) ∧ D(c) Modus Ponens ( 4 )( 5 )
7) D(c) Simplification ( 6 )
8) B(c) Simplification ( 3 )
9) D(c) ∧ B(c) Conjunction ( 7 )( 8 )
10) ∃x [ D(x) ∧ B(x)] Existential Generalization ( 9 )
3. Some cats are animals.
Some dogs are animals.
Therefore, some cats are dogs.
Find the mistake in the given proof.
1) ∃x ( C(x) ∧ A(x) ) [ans: error at line 4, we should not use w to instantiate set D.]
2) ∃x( D(x) ∧ A(x) ) [ w is a cat as used in line 3]
∴∃x( C(x) ∧ D(x) )
3) C(w) ∧ A(w) Existential Instantiation ( 1 )
4 ) D(w) ∧ A(w) Existential Instantiation ( 2 )
5) C(w) Simplification ( 3 )
6) D(w) Simplification ( 4 )
7) C(w) ∧ D(w) Conjunction ( 5 )( 6 )
8) ∃x ( C(x) ∧ D(x) ) Existential Generalization ( 7 )
4. Required: Proof of validity
Mathematicians are neither prophets nor wizards
Hence, if Einstein is a mathematician, he is not a prophet.
1) ∀x [ M(x) ⇒ ¬( P(x) ∨ W(x) ) ]
∴M(e) ⇒ ¬P(e)
2) M(e) Rule of Conditional Proof
∴¬P(e)
3) M(e) ⇒ ¬( P(e) ∨ W(e) ) Universal Instantiation ( 1 )
4) ¬( P(e) ∨ W(e) ) Modus Ponens ( 2 )( 3 )
5) ¬P(e) ∧ ¬W(e) de Morgan’s law ( 4 )
6) ¬P(e) Simplification ( 5 )
QUANTIFICATION NEGATION ( use ≡ instead of ⇔ in the ff )
1. ¬∃xA(x) ⇔ ∀x ¬( A(x) )
2. ¬∀xA(x) ⇔ ∃x ¬( A(x) )
3. ¬∃x ( A(x) ∧ B(x) ) ⇔ ∀x ( A(x) ⇒ ¬B(x) )
4. ¬∀x ( A(x) ⇒ B(x) ) ⇔ ∃x ( A(x) ∧ ¬B(x) )
State the negation of the following :
1. All officers are fighters.
∀x ( O(x) ⇒ F(x) )
negation: ¬∀x ( O(x) ⇒ F(x) ) ⇔ ∃x [ O(x) ∧ ¬F(x) ]
Some officers are not fighters.
2. Some members are fighters.
∃x [ M(x) ∧ F(x) ]
negation: ¬∃x [ M(x) ∧ F(x) ] ⇔ ∀x [ M(x) ⇒ ¬F(x) ]
All members are not fighter or No members are fighters.
Symbolizing Relations
The following expresses relations between two individuals:
1. Plato was a student of Socrates.
2. Baguio is north of Manila.

6. 3. Chicago is smaller than New York.
These are called “BINARY” or “ DYADIC” relations
Let A( x, y ) : x attracts y then
1) b attracts everything ∀x A( b, x )
2 ) b attracts something ∃x A( b, x )
3) everything attracts b ∀x A( x, b )
4) something attracts b ∃x A( x, b )
5) everything attracts everything ∀x∀y( x, y )
example: Helen likes David
Whoever likes David likes Tom
Helen likes only good looking men
Therefore, Tom is a good looking man
let : h = Helen; d = David; t = Tom
G(x) = x is a good looking man
L( x, y ) : x likes y
1) L( h, d )
2) ∀x [ L( x, d ) ⇒ L( x, t ) ]
3) ∀x[ L( h, x ) ⇒ G(x) ]
∴G(t)
4) L( h, d ) ⇒ L( h, t ) Universal Instantiation( 2 )
5) L( h, t ) Modus Ponens( 1 )( 4 )
6) L( h, t ) ⇒ G(t) Universal Instantiation ( 3 )
7) G(t) Modus Ponens( 5 )( 6 )
“There is a man whom all men despise.
Therefore, at least one man despises himself.”
Let D( x, y ) : x despises y
M(x) : x is a man
1) ∃x [ M(x) ∧ ∀y ( M(y) ⇒ D( y, x ) ]
∴∃x [ M(x) ∧ D( x, x ) ]
2) M(c) ∧ ∀y [ M(y) ⇒ D( y, c ) ] Existential Instantiation ( 1 )
3) M(c) ∧ [ M(c) ⇒ D( c, c ) ] Universal Instantiation ( 2 )
4) M(c) ⇒ D( c, c ) Simplification ( 3 )
8) ∃x [ M(x) ∧ D( x, x ) ] Existential Generalization ( )
Transcribe the following assertions into logical notations
Let P(x,y,z) denote x - y = z
1. For every x and y, there are some z such that x-z=y
∀x ∀y [ ∃zP( x, z, y ) ]
2. There is an x such that for all y, y - x = y
∃x [ ∀y P( y, x, y ) ]

7. Quantification Negation
a) For all x [ x is prime implies (x2
+ 1) is even ]
∀x ( x is prime ⇒ x2
+ 1 is even )
negation:
≡ ¬∀x ( x is prime ⇒ x2
+ 1 is even )
≡ ∃x ( x is prime ∧ x2
+ 1 is not even )
“There exists an x such that x is prime and x2
+1 is odd”
b) There exists an x ( x is rational ∧ x2
= 3 )
∃x ( x is rational ∧ x2
= 3 )
negation:
≡ ¬∃x ( x is rational ∧ x2
= 3 )
≡ ∀x ( x is rational ⇒ x2
< > 3 )
“ For all x, x is rational and 2x is not equal to three”
c)There exist an x such that for all y ( xy = y )
∃x [ ∀y ( xy = y ) ]
negation:
≡ ¬∃x [ ∀y ( xy = y ) ]
≡ ∀x [ ∃y ( xy < > y ) ]
“For all x , there exists a y such that xy is not equal to y .”
A Peek at PROLOG
ü The following are statements in Prolog. The statements describe a pile of colored blocks.
isabove(g,b1)
isabove(b1,w1)
isabove(w2,b2)
color(g,gray)
color(b1,blue)
color(b2,blue)
color(b3,blue)
color(w1,white)
color(w2,white)
isabove(X,Z) if isabove(X,Y) and isabove(Y,Z)
ü The following are statements in Prolog. The statements ask questions about the piled blocks
?color(b1,blue)
?isabove(X,w1