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Conditional Statements

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?

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Conditional Statements

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  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

  5. Conditional Truth Table

  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. p  ~q  ~p

  11. p  ~q  ~p

  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 timeor 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 isif ~q then ~p • p  q  ~q  ~p • Conditional and contrapositive are logically equivalent.

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

  18. 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.

  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

  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 conditionfor 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

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