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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 won’t”

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

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

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 didn’t”

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”

Comparing Statements

- Two statements are logically equivalent if they have the same truth value on every row of the truth table
Example: “Either Mary doesn’t go to France or Joe doesn’t” and “It’s not the case that both Mary and Joe go to France”

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”

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”

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”

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

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)

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”

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 won’t vote for an increase in spending in the war in Iraq”

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