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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