Conditional Statements

M260 2.2

Deductive Reasoning

• Proceeds from a hypothesis to a conclusion.

• If p then q.

• p q

• hypothesis conclusion

Conditional Example

• If you show up for work on Monday morning, then you will get the job.

• When is the statement false?

• Answer--Only when the hypothesis is true and the conclusion is false.

Conditional Truth Table

p q pq

T T

T F

F T

F F

Conditional Truth Table

p q pq

T T T

T F F

F T T

F F T

Conditional is vacuously true when hypothesis is false.

Precedence of Logical Operators

• ~ and

Precedence Examples

• p ~q ~p

• Order is ~, , • (p (~q)) (~p)

p ~q ~p

p ~q ~p

p q ~p ~q p ~qp ~q

~p

T T

T F

F T

F F

p ~q ~p

p q ~p ~q p ~qp ~q

~p

T T F F T F

T F F T T F

F T T F F T

F F T T T T

Logical Equivalence

• Statement Forms are logically equivalent if, and only if, they have the same truth tables.

• P Q

Logical Equivalence Example

• p q r (pr) (qr)

Rewriting

• p q ~p q• Either you get to work on time

or you are fired

• If you do not get to work on time,then you are fired.

Negation of if p then q

• ~(p q) ~(~p q) p ~q

Contrapositive

• Contrapositive of if p then q isif ~q then ~p

• p q ~q ~p

• Conditional and contrapositive are logically equivalent.

Converse

• Converse of if p then q isif q then p

• Converse (p q) is (q p)

• Conditional and converse are NOT logically equivalent.

Inverse

• Inverse of if p then q isif ~p then ~q

• Inverse (p q) is (~p ~q)

• Conditional and inverse are NOT logically equivalent.

• Converse and inverse are logically equivalent.

Only If

• p only if q means if not q then not p

• id est if p then q

Only If Example

• John will break the world’s record for the mile only if

• he runs the mile in under four minutes.

Biconditional

• p if, and only if, q

• Abbreviated: p iff q

• Notation: p q p q p q

T T T

T F F

F T F

F F T

Precedence of Logical Operators

• ~ and and

Rewriting

• p q (p q) (q p)

Sufficient Condition

• r is a sufficient condition for s

• If r then s

• rs

Necessary Condition

• r is a necessary condition for s

• If not r then not s

• ~r ~s

• s only if r

• If s then r

Necessary and Sufficient

• r is a necessary and sufficient conditionfor s

• r if, and only if, s

• r s

Practice Necessary/Sufficient

• Use “John is eligible to vote” and “John is at least 18 years old” to make

• A conditional statement:

• A necessary statement:

• A sufficient statement:

Formal vs. Conversational Logic

• Unrelated conclusions

• Understood biconditionals