1. Conditional Statements M260 2.2

2. Deductive Reasoning Proceeds from a hypothesis to a conclusion. If p then q. p  q hypothesis  conclusion

3. 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.

4. Conditional Truth Table pq pqpq TT TF FT FF

5. pq pqpq TTT TFF FTT FFT

6. Conditional is vacuously true when hypothesis is false.

7. Precedence of Logical Operators ~  and   

8. Precedence Examples p  ~q  ~p Order is ~, ,  (p  (~q))  (~p)

9. p  ~q  ~p

10. pq~p~q p  ~q p  ~q  ~p TT TF FT FF

11. pq~p~q p  ~q p  ~q  ~p TTFFTF TFFTTF FTTFFT FFTTTT

12. Logical Equivalence Statement Forms are logically equivalent if, and only if, they have the same truth tables. P  Q

13. Logical Equivalence Example p  q  r  (p  r)  (q  r)

14. 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.

15. Negation of if p then q ~(p  q)  ~(~p  q)  p  ~q

16. Contrapositive Contrapositive of if p then q is if ~q then ~p p  q  ~q  ~p Conditional and contrapositive are logically equivalent.

17. Converse Converse of if p then q is if q then p Converse (p  q) is (q  p) Conditional and converse are NOT logically equivalent.

18. Inverse Inverse of if p then q is if ~p then ~q Inverse (p  q) is (~p  ~q) Conditional and inverse are NOT logically equivalent. Converse and inverse are logically equivalent.

19. Only If p only if q means if not q then not p id est if p then q

20. Only If Example John will break the world’s record for the mile only if he runs the mile in under four minutes.

21. Biconditional p if, and only if, q Abbreviated: p iff q Notation: p  q pq p  q TTT TFF FTF FFT

22. Precedence of Logical Operators ~  and   and  

23. Rewriting  p  q  (p  q)  (q  p)

24. Sufficient Condition r is a sufficient condition for s If r then s r  s

25. 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

26. Necessary and Sufficient r is a necessary and sufficient condition for s r if, and only if, s r  s

27. 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:

28. Formal vs. Conversational Logic Unrelated conclusions Understood biconditionals