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SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 3-1
Chapter 3
Logic
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3.1
Statements and Logical
Connectives
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HISTORY—The Greeks:

Aristotelian logic: The ancient Greeks were the
first people to look at the way humans think and
draw conclusions. Aristotle (384-322 B.C.) is
called the father of logic. This logic has been
taught and studied for more than 2000 years.
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Mathematicians

Gottfried Wilhelm Leibniz (1646-1716) believed that all
mathematical and scientific concepts could be derived
from logic. He was the first to seriously study symbolic
logic. In this type of logic, written statements use
symbols and letters.

George Boole (1815 – 1864) is said to be the founder of
symbolic logic because he had such impressive work in
this area.
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Logic and the English Language


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Connectives - words such as and, or, if, then
Exclusive or - one or the other of the given
events can happen, but not both.
Inclusive or - one or the other or both of the
given events can happen.
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Statements and Logical Connectives

Statement - A sentence that can be judged either true
or false.



Labeling a statement true or false is called assigning a
truth value.
Simple Statements - A sentence that conveys only one
idea.
Compound Statements - Sentences that combine two
or more ideas and can be assigned a truth value.
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Quantifiers

Negation of a statement - the opposite
meaning of a statement.



The negation of a false statement is always a true
statement.
The negation of a true statement is always false.
Quantifiers - words such as all, none, no,
some, etc…
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Slide 3-8
Example: Write Negations
Write the negation of the statement.
Some candy bars contain nuts.

Since some means “at least one” this
statement is true. The negation is “No candy
bars contain nuts,” which is a false statement.
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Slide 3-9
Example: Write Negations continued
Write the negation of the statement.
All tables are oval.

This is a false statement since some tables
are round, rectangular, or other shapes. The
negation could be “Some tables are not oval.”
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Slide 3-10
Compound Statements

Statements consisting of two or more simple
statements are called compound statements.

The connectives often used to join two simple
statements are and, or, if…then…, and if and
only if.
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Slide 3-11
Not Statements
The symbol used in logic to show the negation
of a statement is ~. It is read “not”.
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And Statements


 is the symbol for a conjunction and is read
“and.”
The other words that may be used to express a
conjunction are: but, however, and
nevertheless.
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Slide 3-13
Example: Write a Conjunction
Write the conjunction in symbolic form.
The dog is gray, but the dog is not old.
Solution:
Let p and q represent the simple statements.
p: The dog is gray.
q: The dog is old.
In symbol form, the compound statement is
p
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q.
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Or Statements:

The disjunction is symbolized by  and read
“or.”
Example:
Write the statement in symbolic form.
Carl will not go to the movies or Carl with not go
to the baseball game.
Solution:
p q
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If-Then Statements

The conditional is symbolized by
and is read “if-then.”
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
The antecedent is the part of the statement that
comes before the arrow.
The consequent is the part that follows the arrow.
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Slide 3-16
Example: Write a Conditional Statement
Let p: Nathan goes to the park.
q: Nathan will swing.
Write the following statements symbolically.
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If Nathan goes to the park, then he will swing.

If Nathan does not go to the park, then he will not
swing.
Solutions:
p q

a) p  q
b)
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Slide 3-17
If and Only If Statements

The biconditional is symbolized by  and is
read “if and only if.”
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If and only if is sometimes abbreviated as “iff.”
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Slide 3-18
Example: Write a Statement Using the
Biconditional
Let p: The dryer is running.
q: There are clothes in the dryer.
Write the following symbolic statements in words.
a)
qp
b)
p 
q
Solutions:

The clothes are in the dryer if and only if the dryer is
running.

It is false that the dryer is running if and only if the
clothes are not in the dryer.
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Slide 3-19
3.2
Truth Tables for Negation,
Conjunction, and Disjunction
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Truth Table

A truth table is used to determine when a
compound statement is true or false.
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Negation Truth Table
p
~p
Case 1
T
F
Case 2
F
T
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Slide 3-22
Conjunction Truth Table
p
q
pq
Case 1
T
T
T
Case 2
T
F
F
Case 3
F
T
F
Case 4
F
F
F
The conjunction is true only when both p and q are true.
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Slide 3-23
Disjunction
p
q
pq
Case 1
T
T
T
Case 2
T
F
T
Case 3
F
T
T
Case 4
F
F
F
The disjunction is true when either p is true, q is true, or
both p and q are true.
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Slide 3-24
3.3
Truth Tables for the
Conditional and Biconditional
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Conditional
p
q
pq
Case 1
T
T
T
Case 2
T
F
F
Case 3
F
T
T
Case 4
F
F
T
The conditional statement p  q is true in every case
except when p is a true statement and q is a false
statement.
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Slide 3-26
Biconditional

The biconditional statement, p  q means that p
or, symbolically p  q  q  p .

 

 q and q  p,
p
q
(p

q)

(q

p)
case 1
T
T
T
T
T
T
T
T
T
case 2
T
F
T
F
F
F
F
T
T
case 3
F
T
F
T
T
F
T
F
F
case 4
F
F
F
T
F
T
F
T
F
1
3
2
7
4
6
5
order of steps
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Slide 3-27
Self-Contradiction

A self-contradiction is a compound statement
that is always false.

When every truth value in the answer column of
the truth table is false, then the statement is a
self-contradiction.
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Slide 3-28
Tautology

A tautology is a compound statement that is
always true.

When every truth value in the answer column of
the truth table is true, the statement is a
tautology.
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Slide 3-29
Implication

An implication is a condition statement that is a
tautology.

The consequent will be true whenever the
antecedent is true.
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Slide 3-30
3.4
Equivalent Statements
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Equivalent Statements

Two statements are equivalent if both statements have
exactly the same truth values in the answer columns of
the truth tables.
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

Symbols:  or 
In a truth table, if the answer columns are identical, the
statements are equivalent.
If the answer columns are not identical, the statements
are not equivalent.
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Slide 3-32
De Morgan’s Laws
~ ( p  q)  ~ p  ~ q
~ ( p  q)  ~ p  ~ q
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Slide 3-33
p  q  ~p  q
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To change a conditional statement into a disjunction,
negate the antecedent, change the conditional symbol
to a disjunction symbol, and keep the consequent the
same.
To change a disjunction statement to a conditional
statement, negate the first statement, change the
disjunction symbol to a conditional symbol, and keep the
second statement the same.
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Slide 3-34
Variations of the Conditional Statement

The variations of conditional statements are the
converse of the conditional, the inverse of the
conditional, and the contrapositive of the
conditional.
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Slide 3-35
Variations of the Conditional Statement
Name
Symbolic
Form
Read
Conditional
p
 q
“if p, then q”
Converse of the conditional
q
 p
“if q, then p”
Inverse of the conditional
~p  ~q
Contrapositive of the
conditional
~q
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“if not p, then not q”
 ~p “if not q, then not p”
Slide 3-36
3.5
Symbolic Arguments
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Symbolic Arguments


An argument is valid when its conclusion
necessarily follows from a given set of
premises.
An argument is invalid (or a fallacy) when the
conclusion does not necessarily follow from the
given set of premises.
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Slide 3-38
Valid or Invalid?


If the truth table answer column is true in every
case, then the statement is a tautology, and the
argument is valid.
If the truth table answer column is not true in
every case then the statement is not a
tautology, and the argument is invalid.
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Slide 3-39
Law of Detachment


Also called modus ponens.
The argument form:
pq
p
q
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Slide 3-40
Determining Whether an Argument is
Valid
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Write the argument in symbolic form.
Compare the form with forms that are known
to be either valid or invalid.
If the argument contains two premises, write a
conditional statement of the form
[(premise 1)  (premise 2)]  conclusion
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Slide 3-41
Determining Whether an Argument is
Valid continued

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Construct a truth table for the statement in
step 3.
If the answer column of the table has all trues,
the statement is a tautology, and the argument
is valid. If the answer column of the table does
not have all trues, the argument is invalid.
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Slide 3-42
Valid Arguments
Law of Detachment
pq
p
q
Law of Syllogism
Law of Contraposition
pq
~q
 ~p
Disjunctive Syllogism
pq
qr
pq
~p
p  r
 q
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Slide 3-43
Invalid Arguments

Fallacy of the Converse

Fallacy of the Inverse
pq
q
pq
~p
p
 ~q
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Slide 3-44
3.6
Euler Diagrams and
Syllogistic Arguments
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Syllogistic Arguments


Another form of argument is called a syllogistic
argument, better known as syllogism.
The validity of a syllogistic argument is
determined by using Euler diagrams.
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Slide 3-46
Euler Diagrams


One method used to determine whether an
argument is valid or is a fallacy.
Uses circles to represent sets in syllogistic
arguments.
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Slide 3-47
Symbolic Arguments Versus Syllogistic
Arguments
Words or phrases used
Methods of determining
validity
Symbolic
argument
and, or, not, if-then, if and
only if
Truth tables or by
comparison with
standard forms of
arguments
Syllogistic
argument
all are, some are, none are,
some are not
Euler diagrams
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Slide 3-48