Logic
Statements, Connectives, and Quantifiers

In symbolic logic, we only care whether statements
are true or false – not their content.

In logic, a statement or a proposition is a declarative
sentence that is either true or false.

We often represent statements by lowercase letters
such as p, q, r or s.
Examples of statements:

Today is Wednesday

2+2=4

Kobe Bryant will play in the Super Bowl

Chris Bosh plays for the Toronto Raptors
Examples of non-statements:

What day is today? - this is not a declarative
statement – it is a question.

Come here – this is a command.

This statement is false – this is a paradox. It
cannot be either true or false. Why?

That was a good movie – this is an ambiguous
sentence – there may be no agreement on what
makes a movie good.
Try
Classify each of the following sentences as a statement or not
a statement.
1.
When is your next class?
2.
That was a hard test!
3.
Four plus three is eight.
4.
Gordon Campbell is a great Premier.
5.
Vancouver is the capital of BC.
6.
Is Vancouver the capital of BC?
Truth Value

Because we deal only with statements that
can be classified as “true” or “false”, we
can assign a truth value to a statement p.

We use T to represent the value “true”
and F to represent the value “false”.
Compound Statements

A compound statement may be formed by
combining two or more statements or by
negating a single statement.

Example:

Today is not Tuesday.

Nanaimo has no mayor but Victoria has
one.
Connectives

The words or phrases used to form
compound statements are called connectives.

Some of the connectives used in English are:
Or; either…or; and; but; if…then.
Compound Statements

Decide whether each statement is
compound.
1.
If Jim wrote the test, then he passed.
2.
The car was fixed by Jack and Jill.
3.
He either brought it to your house or he sent it to school.
4.
Craig loves Math and Psychology.
Connectives

The connectives used in logic generally fall
into five categories:
Negation
 Conjunction
 Disjunction
 Conditional
 Biconditional

Negations

The negation of a given statement p is a
statement that is true when p is false and is
false when p is true.

We denote the negation of p by ~p.

Example:- p: Victoria is the capital of BC
~p: Victoria is not the capital of BC.
Connectives

Negation Example:
Quantifiers

A quantifier tells us “how many” and fall
into two categories.

Universal quantifiers

Existential quantifiers
Quantifiers
Quantifiers
Negating Quantifiers

Suppose we want to negate the statement “All
professional athletes are wealthy.” (universal)
Correct: “Some athletes are not wealthy” or “Not all
athletes are wealthy.” (both existential)
Incorrect: “All athletes are not wealthy.” (universal)
Negating Quantifiers

Negate the statement “Some students will get a
scholarship.” (existential)
Correct: “No students will get a scholarship.” (universal)
Incorrect: “Some students will not get a scholarship.”
(existential)
Negating Quantifiers
Write a negation of each statement.
1. The flowers are not watered.
2. Some people have all the luck.
3. Everyone loves a winner.
4. The Olympics will start on 12th February.
5. Everybody loves somebody sometime.
6. All the balls are red.
7. Some students did not write the test.
Negating Inequalities
Connectives

Example:
Connectives

Example:
Connectives

DEFINITION Conditional
A conditional or an implication expresses the notion of if … then. We
use an arrow, , to represent a conditional.

Example:
Suppose that p represents “The Raptors win the NBA Playoffs” and
q represents “Bosh will win the MVP”.
We would read the statement p  q as “If The Raptors win the Playoffs, then Bosh will win the MVP.
Write the statement “If The Raptors do not win the Playoffs, then
Bosh will not win the MVP symbolically. ~p ~q
Connectives - Conditional
We read p  q as “p implies q” or
“if p, then q”.
In the conditional p  q, the statement p is
the antecedent or hypothesis, while q is the
consequent or conclusion.
Connectives - Conditional
Examples:
If it rains, then I carry my umbrella.
If US President Obama comes to the Olympics, then
security will be tight.
If the package does not arrive today, I will call to find out
why.
Connectives - Conditional
Examples:
Statement: All equilateral triangles have acute
angles.
If-then form: If a triangle is equilateral, then it has
acute angles.
Connectives - Conditional
Try: Write each statement in if-then form.
1.
“Winners never quit”
If you are a winner, then you never quit.
2.
It is difficult to study when you are distracted.
If you are distracted, then it is difficult to study.
Connectives - Conditional
Given
p: “I have a Math test”
q: “I do not have time for breakfast”
Write the following statements in symbolic form:
1.
If I have time for breakfast, then I have a Math test.
2.
If I have a Math test, then I do not have time for breakfast.
3.
If I don’t have a Math test, then I have time for breakfast.
Truth Tables
Truth tables are often used to show all possible true-false
patterns for statements.
For example, when statement p is true, ~p is false and
when p is false, ~p is true. Thus the truth table for negation
is given by
Statement
p
T
F
Negation
~p
F
T
Truth Table for the Conjunction p and q
Let us decide on the truth values of the conjunction
p  q . Recall that a conjunction represents the idea of
“both”.
The compound statement 2 + 2 = 4 and Victoria is the capital
of BC is true because each component statement is true.
However, the compound statement 2 + 2 = 4 and Vancouver
is the capital of BC is false, even though part of the
statement is false.
Truth Table for the Conjunction p and q
For the conjunction p  q to be true, both p and q must
be true. The truth table for the conjunction is given below.
P
T
T
F
F
p and q
pq
q
T
F
T
F
T
F
F
F
Truth Table for the Disjunction p or q
Let us decide on the truth values of the disjunction
p  q . Recall that a disjunction represents the idea of
“either”.
The compound statement: I have a quarter or I have a dime
is true whenever I have either a quarter, a dime, or both.
The only way this disjunction could be false would be if I
had neither a quarter nor a dime.
Truth Table for the Disjunction p or q
For the disjunction p  q to be false, both p and q must
be false. The truth table for the disjunction is given below.
P
p or q
q
p
q
T
T
F
F
T
F
T
F
T
T
T
F
Truth Tables
Example: Let p: “5 > 3” and q: “ - 3 > 0”.
Find the truth value of:
1.
pq
2.
pq
3.
~ pq
Truth Tables
Example: Suppose that p is false, q is true and r is false.
What is the truth value of the compound statement
(Parentheses first)?
1.
~ p  q  ~ r 
2.
~ p  q  ~ r 
Truth Tables
Example: Let p: “3 > 2”, q: “ 5 < 4” and r: 3 < 8.
Determine whether each statement is true or false.
1.
~ p ~ q
2.
~  p  q
3.
~ p  r   ~ q ~ p
~ p is false so the and is false.
pq
p is true and q is false so
is false
and hence the statement is TRUE.
Constructing Truth Tables
Consider the statement
~ p  q ~ q .
a)
Construct a truth table.
b)
Suppose both p and q are true. Find the truth value of the statement.
We begin by listing all possible combinations of truth values
for p and q. We then list the truth values of ~p
p
q
~p
T
T
F
F
T
F
T
F
F
F
T
T
Constructing Truth Tables
~ p  q ~ q
p
q
~p
T
T
F
T
F
F
F
T
T
F
F
T
Now we use the columns for ~p and q along with the and truth
table to find the truth values of ~ p  q .
p
q
T
T
F
F
T
F
T
F
~p ~ p  q
F
F
T
T
F
F
T
F
Constructing Truth Tables
~ p  q ~ q
p
q
~p
~ pq
T
T
F
F
T
F
F
F
F
T
T
T
F
F
T
F
Finally, we include a column for ~q and use the or truth table to
combine ~ p  q with ~q.
~ p  q ~q ~ p  q ~ q
p
q
~p
T
T
F
F
F
F
T
F
F
F
T
T
F
T
T
T
F
T
F
F
T
F
T
T
b) When both p and
q are true, the
statement is FALSE.
Constructing Truth Tables
TRY: Construct truth tables for
a)
b)
~ p ~q
~  p  q
p q
T
T
F
F
What do you observe?
They have the same truth values.
T
F
T
F
~ p ~q
F
F
F
T
The Lady or the Tiger
~ p
The Lady or the Tiger
~ p
The Lady or the Tiger
~ p
The Island of Questioners
The Island of Questioners
The Island of Questioners
The Island of Questioners
Equivalent Statements

Logically equivalent statements express
the same meaning.
DeMorgan’s Laws