2. Propositional Logic
• A proposition is a declarative sentence (a
sentence that declares a fact) that is either
true or false, but not both.
• Are the following sentences propositions?
– Toronto is the capital of Canada.
– Read this carefully.
– 1+2=3
– x+1=2
– What time is it?
• Propositional Logic – the area of logic that deals
with propositions
(No)
(No)
(No)
(Yes)
(Yes)
2
3. Propositional Variables
• Propositional Variables – variables that
represent propositions: p, q, r, s
– E.g. Proposition p – “Today is Friday.”
• Truth values – T, F
3
4. Negation
• Examples
– Find the negation of the proposition “Today is Friday.” and
express this in simple English.
– Find the negation of the proposition “At least 10 inches of
rain fell today in Miami.” and express this in simple English.
DEFINITION 1
Let p be a proposition. The negation of p, denoted by ¬p, is the
statement “It is not the case that p.”
The proposition ¬p is read “not p.” The truth value of the negation of
p, ¬p is the opposite of the truth value of p.
Solution: The negation is “It is not the case that today is Friday.
In simple English, “Today is not Friday.” or “It is not Friday
today.”
Solution: The negation is “It is not the case that at least 10 inches
of rain fell today in Miami.” In simple English, “Less than 10 inches of
rain fell today in Miami.”
4
5. Truth Table
• Truth table:
• Logical operators are used to form new propositions from two or
more existing propositions. The logical operators are also called
connectives.
The Truth Table for the
Negation of a Proposition.
p ¬p
T
F
F
T
5
6. Conjunction
• Examples
– Find the conjunction of the propositions p and q where p is
the proposition “Today is Friday.” and q is the proposition
“It is raining today.”, and the truth value of the
conjunction.
DEFINITION 2
Let p and q be propositions. The conjunction of p and q, denoted
by p Λ q, is the proposition “p and q”. The conjunction p Λ q is true
when both p and q are true and is false otherwise.
Solution: The conjunction is the proposition “Today is Friday
and it is raining today.” The proposition is true on rainy
Fridays.
6
7. Disjunction
• Note:
inclusive or : The disjunction is true when at least one of the
two propositions is true.
– E.g. “Students who have taken calculus or computer science can take
this class.” – those who take one or both classes.
exclusive or : The disjunction is true only when one of the
proposition is true.
– E.g. “Students who have taken calculus or computer science, but not
both, can take this class.” – only those who take one of them.
• Definition 3 uses inclusive or.
DEFINITION 3
Let p and q be propositions. The disjunction of p and q, denoted by
p ν q, is the proposition “p or q”. The conjunction p ν q is false
when both p and q are false and is true otherwise.
7
8. Exclusive
The Truth Table for
the Conjunction of Two
Propositions.
p q p Λ q
T T
T F
F T
F F
T
F
F
F
The Truth Table for
the Disjunction of
Two Propositions.
p q p ν q
T T
T F
F T
F F
T
T
T
F
DEFINITION 4
Let p and q be propositions. The exclusive or of p and q, denoted by
p q, is the proposition that is true when exactly one of p and q is true
and is false otherwise.
The Truth Table for the
Exclusive Or (XOR) of
Two Propositions.
p q p q
T T
T F
F T
F F
F
T
T
F
8
9. Conditional Statements
DEFINITION 5
Let p and q be propositions. The conditional statement p → q, is the
proposition “if p, then q.” The conditional statement is false when p is
true and q is false, and true otherwise. In the conditional statement p
→ q, p is called the hypothesis (or antecedent or premise) and q is
called the conclusion (or consequence).
A conditional statement is also called an implication.
Example: “If I am elected, then I will lower taxes.” p → q
implication:
elected, lower taxes. T T | T
not elected, lower taxes. F T | T
not elected, not lower taxes. F F | T
elected, not lower taxes. T F | F
9
10. Conditional Statement (Cont’)
• Example:
– Let p be the statement “Maria learns discrete
mathematics.” and q the statement “Maria will find a good
job.” Express the statement p → q as a statement in
English.
Solution: Any of the following -
“If Maria learns discrete mathematics, then she will
find a good job.
“Maria will find a good job when she learns discrete
mathematics.”
“For Maria to get a good job, it is sufficient for her
to learn discrete mathematics.”
10
11. Conditional Statement (Cont’)
• Other conditional statements:
– Converse of p → q : q → p
– Contrapositive of p → q : ¬ q → ¬ p
– Inverse of p → q : ¬ p → ¬ q
11
12. Biconditional Statement
• p ↔ q has the same truth value as (p → q) Λ (q → p)
• “if and only if” can be expressed by “iff”
• Example:
– Let p be the statement “You can take the flight” and let q
be the statement “You buy a ticket.” Then p ↔ q is the
statement
“You can take the flight if and only if you buy a ticket.”
Implication:
If you buy a ticket you can take the flight.
If you don’t buy a ticket you cannot take the flight.
DEFINITION 6
Let p and q be propositions. The biconditional statement p ↔ q is
the proposition “p if and only if q.” The biconditional statement p
↔ q is true when p and q have the same truth values, and is false
otherwise. Biconditional statements are also called bi-implications.
12
14. Truth Tables of Compound
Propositions
• We can use connectives to build up complicated compound
propositions involving any number of propositional variables,
then use truth tables to determine the truth value of these
compound propositions.
• Example: Construct the truth table of the compound proposition
(p ν ¬q) → (p Λ q).
The Truth Table of (p ν ¬q) → (p Λ q).
p q ¬q p ν ¬q p Λ q (p ν ¬q) → (p Λ q)
T T
T F
F T
F F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F 14
15. Precedence of Logical Operators
• We can use parentheses to specify the order in which logical
operators in a compound proposition are to be applied.
• To reduce the number of parentheses, the precedence order is
defined for logical operators.
Precedence of Logical Operators.
Operator Precedence
¬ 1
Λ
ν
2
3
→
↔
4
5
E.g. ¬p Λ q = (¬p ) Λ q
p Λ q ν r = (p Λ q ) ν r
p ν q Λ r = p ν (q Λ r)
15
16. Translating English Sentences
• English (and every other human language) is often ambiguous.
Translating sentences into compound statements removes the
ambiguity.
• Example: How can this English sentence be translated into a
logical expression?
“You cannot ride the roller coaster if you are under 4
feet tall unless you are older than 16 years old.”
Solution: Let q, r, and s represent “You can ride the roller
coaster,”
“You are under 4 feet tall,” and “You are older than
16 years old.” The sentence can be translated into:
(r Λ ¬ s) ¬
→ q.
16
17. Translating English Sentences
• Example: How can this English sentence be translated into a
logical expression?
“You can access the Internet from campus only if you
are a computer science major or you are not a freshman.”
Solution: Let a, c, and f represent “You can access the Internet
from campus,” “You are a computer science major,” and “You are a
freshman.” The sentence can be translated into:
a (
→ c ν ¬f).
17
18. Logic and Bit Operations
• Computers represent information using bits.
• A bit is a symbol with two possible values, 0 and 1.
• By convention, 1 represents T (true) and 0 represents F
(false).
• A variable is called a Boolean variable if its value is either
true or false.
• Bit operation – replace true by 1 and false by 0 in logical
operations.
Table for the Bit Operators OR, AND, and XOR.
x y x ν y x Λ y x y
0
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
1
1
0
18
19. Logic and Bit Operations
• Example: Find the bitwise OR, bitwise AND, and bitwise XOR of
the bit string 01 1011 0110 and 11 0001 1101.
DEFINITION 7
A bit string is a sequence of zero or more bits. The length of this
string is the number of bits in the string.
Solution:
01 1011 0110
11 0001 1101
-------------------
11 1011 1111 bitwise OR
01 0001 0100 bitwise AND
10 1010 1011 bitwise XOR
19
20. Propositional Equivalences
DEFINITION 1
A compound proposition that is always true, no matter what the
truth values of the propositions that occurs in it, is called a
tautology. A compound proposition that is always false is called a
contradiction. A compound proposition that is neither a tautology
or a contradiction is called a contingency.
Examples of a Tautology and a Contradiction.
p ¬p p ν ¬p p Λ ¬p
T
F
F
T
T
T
F
F
20
21. Logical Equivalences
DEFINITION 2
The compound propositions p and q are called logically equivalent if p
↔ q is a tautology. The notation p ≡ q denotes that p and q are
logically equivalent.
Truth Tables for ¬p ν q and p → q .
p q ¬p ¬p ν q p → q
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
T
T
F
T
T
• Compound propositions that have the same truth values in all
possible cases are called logically equivalent.
• Example: Show that ¬p ν q and p → q are logically equivalent.
21
23. Constructing New Logical
Equivalences
• Example: Show that ¬(p → q ) and p Λ ¬q are logically equivalent.
Solution:
¬(p → q ) ≡ ¬(¬p ν q) by example on earlier slide
≡ ¬(¬p) Λ ¬q by the second De Morgan law
≡ p Λ ¬q by the double negation law
• Example: Show that (p Λ q) (
→ p ν q) is a tautology.
Solution: To show that this statement is a tautology, we will use logical
equivalences to demonstrate that it is logically equivalent to T.
(p Λ q) (
→ p ν q) ≡ ¬(p Λ q) ν (p ν q) by example on earlier slides
≡ (¬ p ν ¬q) ν (p ν q) by the first De Morgan law
≡ (¬ p ν p) ν (¬ q ν q) by the associative and
communicative law for disjunction
≡ T ν T
≡ T
• Note: The above examples can also be done using truth tables. 23
25. Predicates
• Statements involving variables are neither
true nor false.
• E.g. “x > 3”, “x = y + 3”, “x + y = z”
• “x is greater than 3”
– “x”: subject of the statement
– “is greater than 3”: the predicate
• We can denote the statement “x is greater
than 3” by P(x), where P denotes the
predicate and x is the variable.
• Once a value is assigned to the variable x, the
statement P(x) becomes a proposition and has
a truth value.
25
26. Predicates
• Example: Let P(x) denote the statement “x >
3.” What are the truth values of P(4) and
P(2)?
• Example: Let Q(x,y) denote the statement “x
= y + 3.” What are the truth values of the
propositions Q(1,2) and Q(3,0)?
Solution: P(4) – “4 > 3”, true
P(3) – “2 > 3”, false
Solution: Q(1,2) – “1 = 2 + 3” , false
Q(3,0) – “3 = 0 + 3”, true
26
27. Predicates
• Example: Let A(c,n) denote the statement “Computer c is
connected to network n”, where c is a variable
representing a computer and n is a variable representing
a network. Suppose that the computer MATH1 is
connected to network CAMPUS2, but not to network
CAMPUS1. What are the values of A(MATH1, CAMPUS1)
and A(MATH1, CAMPUS2)?
Solution: A(MATH1, CAMPUS1) – “MATH1 is connect to CAMPUS1”, false
A(MATH1, CAMPUS2) – “MATH1 is connect to CAMPUS2”, true
27
28. Propositional Function (Predicate)
• A statement involving n variables x1, x2,
…, xn can be denoted by P(x1, x2, …, xn).
• A statement of the form P(x1, x2, …, xn)
is the value of the propositional
function P at the n-tuple (x1, x2, …, xn),
and P is also called a n-place predicate
or a n-ary predicate.
28
29. Quantifiers
• Quantification: express the extent to
which a predicate is true over a range
of elements.
• Universal quantification: a predicate is
true for every element under
consideration
• Existential quantification: a predicate
is true for one or more element under
consideration
• A domain must be specified. 29
30. Universal Quantifier
The universal quantification of P(x) is the statement
“P(x) for all values of x in the domain.”
The notation xP(x) denotes the universal quantification of P(x). Here is
called the Universal Quantifier. We read xP(x) as “for all xP(x)” or “for
every xP(x).” An element for which P(x) is false is called a counterexample
of xP(x).
Example: Let P(x) be the statement “x + 1 > x.” What is
the truth value of the quantification xP(x), where
the domain consists of all real numbers?
Solution: Because P(x) is true for all real numbers, the
quantification is true.
30
31. Universal Quantification
• A statement xP(x) is false, if and only if P(x) is not
always true where x is in the domain. One way to show
that is to find a counterexample to the statement
xP(x).
• Example: Let Q(x) be the statement “x < 2”. What is
the truth value of the quantification xQ(x), where
the domain consists of all real numbers?
• xP(x) is the same as the conjunction
P(x1) Λ P(x2) Λ …. Λ P(xn)
Solution: Q(x) is not true for every real numbers, e.g. Q(3) is
false. x = 3 is a counterexample for the statement xQ(x).
Thus the quantification is false.
31
32. Universal Quantifier
• Example: What does the statement xN(x)
mean if N(x) is “Computer x is connected to the
network” and the domain consists of all
computers on campus?
Solution: “Every computer on campus is connected to the
network.”
32
33. Existential Quantification
DEFINITION 2
The existential quantification of P(x) is the statement
“There exists an element x in the domain such that P(x).”
We use the notation xP(x) for the existential quantification of
P(x). Here
is called the Existential Quantifier.
• The existential quantification xP(x) is read as
“There is an x such that P(x),” or
“There is at least one x such that P(x),” or
“For some x, P(x).”
33
34. Existential Quantification
• Example: Let P(x) denote the statement “x > 3”. What
is the truth value of the quantification xP(x), where
the domain consists of all real numbers?
• xP(x) is false if and only if P(x) is false for every
element of the domain.
• Example: Let Q(x) denote the statement “x = x + 1”.
What is the true value of the quantification xQ(x),
where the domain consists for all real numbers?
Solution: “x > 3” is sometimes true – for instance when
x = 4. The existential quantification is true.
Solution: Q(x) is false for every real number. The existential
34
35. Existential Quantification
• If the domain is empty, xQ(x) is false because there
can be no element in the domain for which Q(x) is true.
• The existential quantification xP(x) is the same as
the disjunction P(x1) V P(x2) V … VP(xn)
Quantifiers
Statement When True? When False?
xP(x)
xP(x)
xP(x) is true for every x.
There is an x for which P(x) is
true.
There is an x for which xP(x)
is false.
P(x) is false for every x.
35
36. Uniqueness Quantifier
• Uniqueness quantifier ! or 1
– !xP(x) or 1P(x) states “There exists a unique x such that P(x) is
true.”
• Quantifiers with restricted domains
– Example: What do the following statements mean? The domain in
each case consists of real numbers.
• x < 0 (x2
> 0): For every real number x with x < 0, x2
> 0. “The square of a negative
real number is positive.” It’s the same as x(x < 0 x
→ 2
> 0)
• y ≠ 0 (y3
≠ 0 ): For every real number y with y ≠ 0, y3
≠ 0. “The cube of every non-
zero real number is non-zero.” It’s the same as y(y ≠ 0 y
→ 3
≠ 0 ).
• z > 0 (z2
= 2): There exists a real number z with z > 0, such that z2
= 2. “There is a
positive square root of 2.” It’s the same as z(z > 0 Λ z2
= 2):
36
37. Precedence of Quantifiers
• Precedence of Quantifiers
– and have higher precedence than all logical operators.
– E.g. xP(x) V Q(x) is the same as ( xP(x)) V Q(x)
37
38. Translating from English into Logical
Expressions
• Example: Express the statement “Every student in this
class has studied calculus” using predicates and
quantifiers.
Solution:
If the domain consists of students in the class –
xC(x)
where C(x) is the statement “x has studied calculus.
If the domain consists of all people –
x(S(x) → C(x)
where S(x) represents that person x is in this class.
If we are interested in the backgrounds of people in subjects
besides calculus, we can use the two-variable quantifier Q(x,y)
for the statement “student x has studies subject y.” Then we
would replace C(x) by Q(x, calculus) to obtain xQ(x, calculus)
or x(S(x) → Q(x, calculus))
38
39. Translating from English into Logical
Expressions
• Example: Consider these statements. The first two are
called premises and the third is called the conclusion.
The entire set is called an argument.
“All lions are fierce.”
“Some lions do not drink coffee.”
“Some fierce creatures do not drink coffee.”
Solution: Let P(x) be “x is a lion.”
Q(x) be “x is fierce.”
R(x) be “x drinks coffee.”
x(P(x) → Q(x))
x(P(x) Λ ¬R(x))
x(Q(x) Λ ¬R(x))
39
