Review

1
Review

• We can use truth tables to determine under what
  conditions a compound statement is true or false.
• For example, consider the claim If Mark and
  Sarah go to Florida for Spring break, then Kelly
  wont

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Truth Table

• (M S)?K
• T T T F FT
• T T T T TF
• T F F T FT
• T F F T TF
• F F T T FT
• F F T T TF
• F F F T FT
• F F F T TF

3
Classifying Statements

• A compound statement is logically true
  (tautologous) if it is true regardless of the
  truth values of the basic statements
• Example If Justin and Steve go to the bar, then
  Justin goes to the bar

4
See?

• (J S)?J
• T T T T T
• T F F T T
• F F T T F
• F F F T F

5
More

• A compound statement is logically false
  (self-contradictory) if it is false regardless of
  the truth values of the basic statements
• Example Mary and Lucy went to the store, but
  Lucy didnt

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See?

• (M L) L
• T T T F FT
• T F F F TF
• F F T F FT
• F F F F TF

7
More More

• A compound statement is contingent if it is true
  under some assignments of truth values to the
  basic statements, false under others
• Example If Illinois beats Michigan, then either
  Illinois goes to the Rose Bowl or Ohio State does

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See?

• I?(R v O)
• TT T T T
• TT T T F
• TT F T T
• TF F F F
• FT T T T
• FT T T F
• FT F T T
• FT T T F

9
Comparing Statements

• Two statements are logically equivalent if they
  have the same truth value on every row of the
  truth table
• Example Either Mary doesnt go to France or Joe
  doesnt and Its not the case that both Mary
  and Joe go to France

10
See?

• M v J (M J)
• FT F FT F T T T
• FT T TF T T F F
• TF T FT T F F T
• TF T TF T F F F

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More More More

• Two statements are contradictory if they have
  opposite truth values on every row of the truth
  table
• Example If Kwame goes to Yale, then Kwame is
  smart and Kwame goes to Yale but he is not
  smart

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See?

• Y?S Y S
• T T T T F FT
• T F F T T TF
• F T T F F FT
• F T F F F TF

13
Ever More

• Two or more statements are consistent if there is
  at least one row on the truth table where all the
  statements are true
• Example If Yuri goes to the ball, then Jin
  goes, Jin goes to the ball and Michelle does
  not, and Yuri goes to the ball or Michelle goes

14
See?

• Y?J J M Y v M
• T TT T FFT T T T
• T TT T TTF T T F

15
More!

• Two or more statements are inconsistent if there
  are no rows on the truth table where all the
  statements are true
• Example Azar and Sam go to the party, Sam and
  Maria go to the party, Azar goes to the party,
  but Maria does not

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See?

• A S S M A M
• T T T T T T T F FT
• T T T T F F T T TF
• T F F F F T T F FT
• T F F F F F T T TF
• F F T T T T F F FT
• F F T T F F F F TF
• F F F F F T F F FT
• F F F F F F F F TF

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Note

• All tautologies are logically equivalent to each
  other (they always have the same truth value,
  namely, true), and are consistent
• All self-contradictions are logically equivalent
  to each other (they always have the same truth
  value, namely, false), and are inconsistent
• All pairs of claims that are contradictory are
  inconsistent (because they always have differing
  truth values)
• Some, but not all, consistent claims are
  logically equivalent (for example, tautologies)
• Some, but not all, inconsistent claims are
  logically equivalent (for example,
  self-contradictions)

18
Testing Validity with Truth Tables

• An argument is valid if there are NO rows on the
  truth table for the argument where the premises
  are true and the conclusion false. Otherwise,
  the argument is invalid
• Example of validity Bob is going to Taco Bell
  for lunch, and you know if he goes there, he will
  have nachos. So Bob will have nachos

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See?

• B / B?N // N
• T T T T T
• T T F F F
• F F T T T
• F F T F F

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Example of Invalidity

• If Senator Jones is a republican, then he will
  vote for an increase in spending in the war in
  Iraq. But he is not a republican. Therefore, he
  wont vote for an increase in spending in the war
  in Iraq

21
See?

• J?W / J // W
• T T T FT FT
• T F F FT TF
• F T T TF FT
• F T F TF TF