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Chapter 3 Section 4 – Slide 1Copyright © 2009 Pearson Education, Inc.
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Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 2
Chapter 3
Logic
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 3
WHAT YOU WILL LEARN• Statements, quantifiers, and
compound statements• Statements involving the words not,
and, or, if… then…, and if and only if• Truth tables for negations,
conjunctions, disjunctions, conditional statements, and biconditional statements
• Self-contradictions, tautologies, and implications
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 4
WHAT YOU WILL LEARN• Equivalent statements, De Morgan’s
law, and variations of conditional statements
• Symbolic arguments and standard forms of arguments
• Euler diagrams and syllogistic arguments
• Using logic to analyze switching circuits
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 5
Section 4
Equivalent Statements
Chapter 3 Section 4 – Slide 6Copyright © 2009 Pearson Education, Inc.
Equivalent Statements
Two statements are equivalent if both statements have exactly the same truth values in the answer columns of the truth tables.
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.
Sometimes the words logically equivalent are used in place of the word equivalent.
Symbols: or
Chapter 3 Section 4 – Slide 7Copyright © 2009 Pearson Education, Inc.
De Morgan’s Laws
~ (p q) ~ p ~ q
~ (p q) ~ p ~ q
Chapter 3 Section 4 – Slide 8Copyright © 2009 Pearson Education, Inc.
Example: Using De Morgan’s Laws to Write an Equivalent Statement
Use De Morgan’s laws to write a statement logically equivalent to “Benjamin Franklin was not a U.S. president, but he signed the Declaration of Independence.”
Solution: Let
p: Benjamin Franklin was a U.S. president
The statement symbolically is ~p V q.
q: Benjamin Franklin signed the Declaration of Independence
Chapter 3 Section 4 – Slide 9Copyright © 2009 Pearson Education, Inc.
Example: Using De Morgan’s Laws to Write an Equivalent Statement (continued)
Therefore, the logically equivalent statement to the given statement is:“It is false that Benjamin Franklin was a U.S. president or Benjamin Franklin did not sign the Declaration of Independence.”
The logically equivalent statement in symbolic form is
~ p q ~ p ~ q
Chapter 3 Section 4 – Slide 10Copyright © 2009 Pearson Education, Inc.
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.
Switching Between a Conditional and a Disjunction
p q ~p q
Chapter 3 Section 4 – Slide 11Copyright © 2009 Pearson Education, Inc.
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.
Chapter 3 Section 4 – Slide 12Copyright © 2009 Pearson Education, Inc.
“if not q, then not p”~p~qContrapositive of
the conditional
“if not p, then not q”~q~pInverse of the conditional
“if q, then p”pqConverse of the conditional
“if p, then q”qpConditional
ReadSymbolic FormName
Variations of the Conditional Statement
Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 4 – Slide 13
Section 5
Symbolic Arguments
Chapter 3 Section 4 – Slide 14Copyright © 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 4 – Slide 15Copyright © 2009 Pearson Education, Inc.
Law of Detachment
Also called modus ponens. The argument form symbolically written:
Premise 1:
Premise 2:
If [ premise 1 and premise 2 ] then conclusion
Conclusion:
[ (p q)
p q
p
q
p ] q
Chapter 3 Section 4 – Slide 16Copyright © 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 4 – Slide 17Copyright © 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 4 – Slide 18Copyright © 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 4 – Slide 19Copyright © 2009 Pearson Education, Inc.
Example: Determining Validity with a Truth Table (continued)
Fill-in the table in order, as follows:
Since column 7 has all T’s, the argument is valid.
p q [(p q) ~q] ~p
T T T T T F F T F
T F T F F F T T F
F T F T T F F T T
F F F T F T T T T
1 3 2 5 4 7 6
Chapter 3 Section 4 – Slide 20Copyright © 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 4 – Slide 21Copyright © 2009 Pearson Education, Inc.
Invalid Arguments
Fallacy of the Converse Fallacy of the Inverse
p q
q
p
p q
~ p
~q