What You Will Learn
Statements, quantifiers, and
compound statements
3.1-1
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Logic and the English Language
Connectives – “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
3.1-2
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Statements and Logical Connectives
Statement - A sentence that can be judged
either true or false. (only two values!)
Simple Statements - A sentence that
conveys only one idea.
Compound Statements - Sentences that
combine two or more simple statements.
3.1-3
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Negation of a Statement
Negation of a statement – change a
statement to its opposite meaning.
3.1-4
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Quantifiers
Quantifiers - words such as all,
none, no, some, etc…
Be careful when negating statements
that contain quantifiers.
3.1-5
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Negation of Quantified Statements
Form of
statement
All are.
None are.
Some are.
Some are not.
3.1-6
Form of
negation
Some are not.
Some are.
None are.
All are.
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Example 1:
Write the negation of the statement.
Some telephones can take photographs.
Negation: “No telephones can take photographs.”
Write the negation of the statement.
All houses have two stories.
Negation: “Some houses do not Have two stories.”
3.1-7
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Not Statements (Negation)
The symbol used in logic to show the
negation of a statement is ~. It is
read “not”.
The negation of p is: ~p.
3.1-8
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And Statements (Conjunction)
⋀ is the symbol for a conjunction and is read
“and.”
The conjunction of p and q is: p ⋀ q.
The other words used to express a conjunction
are: but, however, and nevertheless.
3.1-9
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Example 2: Write a Conjunction
Write the following conjunction in
symbolic form:
Green Day is not on tour, but Green
Day is recording a new CD.
~t ⋀ r
3.1-10
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Or Statements (Disjunction)
The disjunction is symbolized by ⋁ and
read “or.”
In this book the “or” will be the inclusive or
(unless otherwise indicated).
The disjunction of p and q is: p ⋁ q.
3.1-11
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Example 3: Write a Disjunction
Let
p: Maria will go to the circus.
q: Maria will go to the zoo.
Write the statement in symbolic form.
Maria will go to the zoo or Maria will
not go the circus.
Solution
3.1-12
q ⋁ ~p
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Compound Statements
Use comma to group statments.
When we write the compound statement
symbolically, the simple statements on the
same side of the comma are to be grouped
together within parentheses.
3.1-13
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Example 4: Understand How Commas
Are Used to Group Statements
Let
p: Dinner includes soup.
q: Dinner includes salad.
r: Dinner includes the vegetable of the day
Write the statement in symbolic form.
Dinner includes soup and salad, or vegetable of the
day.
Solution
3.1-14
(p ⋀ q) ⋁ r
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Example 5: Change Symbolic
Statements into Words
Let
p: The house is for sale.
q: We can afford to buy the house.
Write the symbolic statement in words.
Solution
p ⋀ ~q
The house is for sale and we cannot afford
to buy the house.
3.1-15
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Example 5: Change Symbolic
Statements into Words
Let p: The house is for sale.
q: We can afford to buy the house.
Write the symbolic statement in words.
Solution
~(p ⋀ q)
It is false that the house is for sale and
we can afford to buy the house.
3.1-16
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If-Then Statements
The conditional is symbolized by → and is
read “if-then.”
The antecedent is the part of the
statement that comes before the arrow.
The consequent is the part that follows the
arrow.
If p, then q is symbolized as: p → q.
3.1-17
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Example 6: Write Conditional
Statements
Let p: The portrait is a pastel.
q: The portrait is by Beth Anderson.
Write the statement symbolically.
If the portrait is a pastel, then the portrait is
by Beth Anderson.
Solution
3.1-18
p→q
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Example 6: Write Conditional
Statements
Let p: The portrait is pastel.
q: The portrait is by Beth Anderson.
Write the statement symbolically.
It is false that if the portrait is by Beth
Anderson, then the portrait is a pastel.
Solution
3.1-19
~(q → p)
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If and Only If Statements
The biconditional is symbolized by ↔ and is read
“if and only if.”
If and only if is sometimes abbreviated as “iff.”
3.1-20
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Example 8: Write Statements
Using the Biconditional
Let
p: Alex plays goalie on the lacrosse team.
q: The Titans win the Champion’s Cup.
Write the symbolical statement in words.
~(p ↔ ~q)
Solution
It is false that Alex plays goalie on the
lacrosse team if and only if the Titans do
not win the Champion’s Cup.
3.1-21
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Logical Connectives
3.1-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Truth tables for negations,
conjunctions, and disjunctions
3.2-23
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Truth Table
A truth table is used to determine
when a compound statement is true or
false.
3.2-24
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Negation Truth Table
Case 1
Case 2
3.2-25
p
T
F
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~p
F
T
Conjunction Truth Table
Case
Case
Case
Case
3.2-26
1
2
3
4
p
T
T
F
F
q
T
F
T
F
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p⋀q
T
F
F
F
Disjunction Truth Table
Case
Case
Case
Case
3.2-27
1
2
3
4
p
T
T
F
F
q
T
F
T
F
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p⋁q
T
T
T
F
Example 3: Truth Table with a
Negation
Construct a truth table for ~(~q ⋁ p).
Solution
p
q
~
(~q
⋁
p)
T
T
F
F
T
F
T
F
F
F
T
F
4
F
T
F
T
1
T
T
F
T
3
T
T
F
F
2
True only when p is false and q is true.
3.2-28
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Example 7: Use the Alternative
Method to Construct a Truth Table
Construct a truth table for ~p ⋀ ~q.
Solution
3.2-29
p
q
~p
~q
~p ⋀ ~q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
F
F
F
T
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Example 8: Determine the Truth
Value of a Compound Statement
Determine the truth value for each
simple statement. Then, using these
truth values, determine the truth value
of the compound statement.
15 is less than or equal to 9.
The compound statement is false.
3.2-30
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: Determine the Truth
Value of a Compound Statement
Determine the truth value for each simple
statement. Then, using these truth values,
determine the truth value of the compound
statement.
George Washington was the first U.S.
president or Abraham Lincoln was the
second U.S. president, but there has not
been a U.S. president born in Antarctica.
The compound statement is true.
3.2-31
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What You Will Learn
•
Truth tables for conditional and
biconditional
•
Self-contradictions, Tautologies, and
Implications
3.3-32
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Conditional
Case
Case
Case
Case
3.3-33
1
2
3
4
p
T
T
F
F
q
T
F
T
F
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p→q
T
F
T
T
Example:
Construct a truth table for the
statement ~p → ~q.
3.3-34
p
q
~p
→
~q
T
T
F
F
T
F
T
F
F
F
T
T
1
T
T
F
T
3
F
T
F
T
2
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Biconditional
The biconditional statement, p ↔ q
means that p → q and q → p or,
symbolically (p → q) ⋀ (q → p).
case 1
case 2
case 3
case 4
order of
steps
3.3-35
p
T
T
F
F
q (p 
T T T
F T F
T F T
F F T
1 3
q)
T
F
T
F
2
 (q
 p)
T
F
F
T
7
T
T
F
T
6
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T
F
T
F
4
T
T
F
F
5
Example 4: A Truth Table Using
a Biconditional
Construct a truth table for the
statement ~p ↔ (~q → r).
3.3-36
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Example 4: A Truth Table Using
a Biconditional
3.3-37
p
q
r
~p
↔
(~q
→
r)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
F
F
T
T
T
T
1
F
F
F
T
T
T
T
F
5
F
F
T
T
F
F
T
T
2
T
T
T
F
T
T
T
F
4
T
F
T
F
T
F
T
F
3
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Self-Contradiction
A self-contradiction is a compound
statement that is always false.
3.3-38
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Example 8: All Falses, a SelfContradiction
Construct a truth table for the
statement (p ↔ q) ⋀ (p ↔ ~q).
The statement is a self-contradiction
or a logically false statement.
3.3-39
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Tautology
A tautology is a compound statement
that is always true.
3.3-40
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Example 9: All Trues, a Tautology
Construct a truth table for the
statement (p ⋀ q) → (p ⋁ r).
The statement is a tautology or a
logically true statement.
3.3-41
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Implication
An implication is a conditional
statement that is a tautology.
The consequent will be true whenever
the antecedent is true.
3.3-42
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Example 10: An Implication?
Determine whether the conditional
statement [(p ⋀ q) ⋀ q] → q is an
implication.
Yes
3.3-43
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What You Will Learn
Symbolic arguments
Standard forms of arguments
3.5-44
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Symbolic Arguments
A symbolic argument consists of a
set of premises and a conclusion.
3.5-45
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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 when the
conclusion does not necessarily follow
from the given set of premises.
3.5-46
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Law of Detachment
Symbolically, the argument is written:
Premise 1: p → q
Premise 2: p
Conclusion: ∴ q
If [premise 1 and premise 2] then conclusion
[(p → q)
3.5-47
⋀
p
]
→
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q
To Determine Whether an
Argument is Valid
1. Write the argument in symbolic form.
2. Compare the form of the argument with
forms that are known to be either valid
or invalid. Otherwise, go to step 3.
3.5-48
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To Determine Whether an
Argument is Valid
3. If the argument contains two premises,
write a conditional statement of the form
[(premise 1) ⋀ (premise 2)]  conclusion
4. Construct a truth table for the statement
above.
3.5-49
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To Determine Whether an
Argument is Valid
5. If the answer column of the truth table
has all trues, the statement is a
tautology, and the argument is valid.
otherwise, the argument is invalid.
3.5-50
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Example 2: Determining the Validity
of an Argument with a Truth Table
Determine whether the following
argument is valid or invalid.
If you watch Good Morning America,
then you see Robin Roberts.
You did not see Robin Roberts.
∴ You did not watch Good Morning
America.
The argument is valid
3.5-51
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Standard Forms of Valid Arguments
Law of Detachment
pq
p
 ~p
q
Law of Syllogism
pq
qr
p r
3.5-52
Law of Contraposition
pq
~q
Disjunctive Syllogism
pq
~p
 q
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Standard Forms of Invalid
Arguments
Fallacy of the Converse
pq
q
p
Fallacy of the Inverse
pq
~p
 ~q
3.5-53
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Example 4: Identifying a
Standard Argument
Determine whether the following
argument is valid or invalid.
If you are on Facebook, then you see
my pictures.
If you see my pictures, then you
know I have a dog.
∴ If you are on Facebook, then you
know I have a dog. Valid. Law of syllogism
3.5-54
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Example 5: Identifying Common
Fallacies in Arguments
Determine whether the following
argument is valid or invalid.
If it is snowing, then we put salt on
the driveway.
We put salt on the driveway.
∴ It is snowing.
3.5-55
Invalid. Fallacy of the converse.
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Example 5: Identifying Common
Fallacies in Arguments
Determine whether the following
argument is valid or invalid.
If it is snowing, then we put salt on the
driveway.
It is not snowing.
∴We do not put salt on the driveway.
Invalid. Fallacy of the inverse.
3.5-56
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What You Will Learn
Euler diagrams
Syllogistic arguments
3.6-57
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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 diagram
3.6-58
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Euler Diagrams
A method used to determine whether
an argument is valid or is a fallacy.
3.6-59
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Symbolic Arguments Versus
Syllogistic Arguments
Words or phrases
used
Symbolic
argument
Syllogistic
argument
3.6-60
Methods of
determining
validity
and, or, not, if-then, Truth tables or by
if and only if
comparison with
standard forms
of arguments
all are, some are,
Euler diagrams
none are, some are
not
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Example 3: Ballerinas and Athletes
Determine whether the following
syllogism is valid or invalid.
All ballerinas are athletic.
Keyshawn is athletic.
∴ Keyshawn is a ballerina.
The conclusion does not necessarily follow from
the set of premises. The argument is invalid.
3.6-61
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Example 4: Parrots and Chickens
Determine whether the following
syllogism is valid or invalid.
No parrots eat chicken.
Fletch does not eat chicken.
∴ Fletch is a parrot.
The conclusion does not necessarily follow from
the set of premises. The argument is invalid.
3.6-62
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Example 5: A Syllogism Involving
the Word Some
Determine whether the following
syllogism is valid or invalid.
All As are Bs.
Some Bs are Cs.
∴ Some As are Cs.
The conclusion does not necessarily follow from
the set of premises. The argument is invalid.
3.6-63
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Example 6: Fish and Cows
Determine whether the following
syllogism is valid or invalid.
No fish are mammals.
All cows are mammals.
∴ No fish are cows.
The conclusion necessarily follow from the set of
premises. The argument is valid.
3.6-64
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