LOGIC

1
(No Transcript)
2
(No Transcript)
3
(No Transcript)
4
(No Transcript)
5
Exercises

• Find the negations of the following propositions.
  
• (a)p "22 4 (b)q "1 0
• (c)r "Diamonds are a pearl's best friend.
• (d)s "All the politicians in this town are
  crooks.
• Solution
• (a) p is the statement "it is not true that 22
  4," or more simply,
• p "22 4."
• (b) q "1 0."
• (c) r "Diamonds are not a pearl's best friend."
  
• (d) s " Not all the politicians in this town
  are crooks."

6
Before we go on...

• Notice that p is false, because p is true.
  However, q is true, because q is false. A
  statement of the form q can very well be true
  it is a common mistake to think it must be false.
  
• To say that diamonds are not a pearl's best
  friend is not to say that diamonds are a pearl's
  worst enemy. The negation is not the polar
  opposite, but whatever would deny the truth of
  the original statement. Similarly, saying that
  not all politicians are crooks is not the same as
  saying that no politicians are crooks, but is the
  same as saying that some politicians are not
  crooks. Negations of statements involving the
  quantifiers "all" or "some" are tricky. We'll
  study quantifiers in more depth when we discuss
  the predicate calculus.

7
More exercises

1. p "There is life on Mars. p
2. p "All lacrosse players are tall. p
3. p "My computer has some new software.
   p

8
Which of the following are statements? Comment on
the truth values of all the statements you
encounter if a sentence fails to be a statement,
explain why.

• 1. All swans are white.
• 2. The fat cat sat on the mat.
• 3. Look in thy glass and tell whose face thou
  viewest.  
• 4. My glass shall not persuade me I am old.  
• 5. Father Nikolsky penned his dying confession to
  Patriarch Arsen III Charnoyevich of Peç in the
  pitch dark, somewhere in Poland, using a mixture
  of gunpowder and saliva, and a quick Cyrillic
  hand, while the innkeeper's wife scolded and
  cursed him through the bolted door.  

9

• 6. 1,000,000,000 is the largest number.
• 7. There is no largest number.
• 8. There may or may not be a largest number.
• 9. Intelligent life abouds in the universe.
• 10. This definitely is a statement.
• 11. The speaker is lying.
• 12. This is exercise number 12.

10

• 14. "potato" is spelled p-o-t-a-t-o-e.
• Let
• p "Our mayor is trustworthy,"
• q "Our mayor is a good speller,"
• r "Our mayor is a patriot."
• Express each of the following statements in
  logical form
• 15. Although our mayor is not trustworthy, he is
  a good speller.
• 16. Either our mayor is trustworthy, or he is a
  good speller.
• 17. Our mayor is a trustworthy patriot who spells
  well.
• 18. While our mayor is both trustworthy and
  patriotic, he is not a good speller.