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Chapter 3 Section 5 - Slide 1Copyright © 2009 Pearson Education, Inc.
AND
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 2
Chapter 3
Logic
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 3
WHAT YOU WILL LEARN• Symbolic arguments and standard
forms of arguments• Euler diagrams and syllogistic
arguments
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 4
Section 5
Symbolic Arguments
Chapter 3 Section 5 - Slide 5Copyright © 2009 Pearson Education, Inc.
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.
Chapter 3 Section 5 - Slide 6Copyright © 2009 Pearson Education, Inc.
Law of Detachment
Also called modus ponens. The argument form symbolically written:
Premise 1:
Premise 2:
Conclusion:
If [ premise 1 and premise 2 ] then conclusion
[ (p q) p ] q
p q
p
q
Chapter 3 Section 5 - Slide 7Copyright © 2009 Pearson Education, Inc.
Determine Whether an Argument is Valid
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 Construct a truth table for the statement above. 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.
Chapter 3 Section 5 - Slide 8Copyright © 2009 Pearson Education, Inc.
Example: Determining Validity with a Truth Table
Determine whether the following argument is valid or invalid.
If you score 90% on the final exam, then you will get an A for the course.
You will not get an A for the course.
You do not score 90% on the final exam.
Chapter 3 Section 5 - Slide 9Copyright © 2009 Pearson Education, Inc.
Example: Determining Validity with a Truth Table (continued)
Construct a truth table.
In symbolic form the argument is:
Solution:Let p: You score 90% on the final exam.
q: You will get an A in the course.
p q~q
~p
Chapter 3 Section 5 - Slide 10Copyright © 2009 Pearson Education, Inc.
Example: Determining Validity with a Truth Table (continued)
p q [(p q) ~ q] ~p
TTFF
TFTF
FFTT
TFTT
FFFT
FTFT
Fill-in the table in order, as follows:
Since column 5 has all T’s, the argument is valid.
231 4
TTTT5
Chapter 3 Section 5 - Slide 11Copyright © 2009 Pearson Education, Inc.
Valid Arguments
Law of Detachment
Law of Syllogism
Law of Contraposition
Disjunctive Syllogism
p q
p
q
p q
q r
p r
p q
~q
~ p
p q
~ p
q
Chapter 3 Section 5 - Slide 12Copyright © 2009 Pearson Education, Inc.
Invalid Arguments
Fallacy of the Converse Fallacy of the Inverse
p q
q
p
p q
~ p
~q
Chapter 3 Section 5 - Slide 13Copyright © 2009 Pearson Education, Inc.
Translate the following argument into symbolic form. Determine whether the argument is valid or invalid.If Jenny gets some rest, then she will feel better. If Jenny feels better, then she will help me paint my bedroom. Therefore, if my bedroom is painted, then Jenny must have gotten some rest.
a. p q
q r
p r
Valid
b. p q
q r
p r
Fallacy
c. p q
q r
r p
Valid
d. p q
q r
r p
Fallacy
Chapter 3 Section 5 - Slide 14Copyright © 2009 Pearson Education, Inc.
Translate the following argument into symbolic form. Determine whether the argument is valid or invalid.If Jenny gets some rest, then she will feel better. If Jenny feels better, then she will help me paint my bedroom. Therefore, if my bedroom is painted, then Jenny must have gotten some rest.
a. p q
q r
p r
Valid
b. p q
q r
p r
Fallacy
c. p q
q r
r p
Valid
d. p q
q r
r p
Fallacy
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 15
Section 6
Euler Diagrams and Syllogistic Arguments
Chapter 3 Section 5 - Slide 16Copyright © 2009 Pearson Education, Inc.
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 (pronounced “oiler”) diagrams.
Chapter 3 Section 5 - Slide 17Copyright © 2009 Pearson Education, Inc.
Euler Diagrams
One method used to determine whether an argument is valid or is a fallacy.
Uses circles to represent sets in syllogistic arguments.
Chapter 3 Section 5 - Slide 18Copyright © 2009 Pearson Education, Inc.
Symbolic Arguments Versus Syllogistic Arguments
Euler diagramsall are, some are, none are, some are not
Syllogistic argument
Truth tables or by comparison with standard forms of arguments
and, or, not, if-then, if and only if
Symbolic argument
Methods of determining validity
Words or phrases used
Chapter 3 Section 5 - Slide 19Copyright © 2009 Pearson Education, Inc.
Example: Ballerinas and Athletes
Determine whether the following syllogism is valid or invalid.
All ballerinas are athletic.
Keyshawn is athletic.
Keyshawn is a ballerina.
Chapter 3 Section 5 - Slide 20Copyright © 2009 Pearson Education, Inc.
Example: Ballerinas and Athletes
Keyshawn is athletic, so must be placed in the set of athletic people, which is A. We have a choice, as shown above.
All ballerinas, B, are athletic, A.
AB
U
The conclusion does not necessarily follow from the set of premises. The argument is invalid.
AB
UAB
U
Let A = all Athletes and B = all Ballerinas.Let K represent Keyshawn.
KK
Chapter 3 Section 5 - Slide 21Copyright © 2009 Pearson Education, Inc.
Determine whether the syllogism is valid or is a fallacy.
a. Valid
b. Fallacy
c. Can’t determine
Some teachers teach math.Some teachers teach English.Therefore, some teachers teach
math and English.
Chapter 3 Section 5 - Slide 22Copyright © 2009 Pearson Education, Inc.
Determine whether the syllogism is valid or is a fallacy.
a. Valid
b. Fallacy
c. Can’t determine
Some teachers teach math.Some teachers teach English.Therefore, some teachers teach
math and English.
