Dilemma: Division Into Cases

1
Dilemma Division Into Cases

• Dilemma p Ú q
• p r
• q r
• \ r
• Premises x is positive or x is negative.
• If x is positive , then x2 is positive.
• If x is negative, then x2 is positive.
• Conclusion x2 is positive.

2
Application Find My Glasses

• 1. If my glasses are on the kitchen table, then I
  saw them at breakfast.
• 2. I was reading in the kitchen or I was reading
  in the living room.
• 3. If I was reading in the living room, then my
  glasses are on the coffee table.
• 4. I did not see my glasses at breakfast.
• 5. If I was reading in bed, then my glasses are
  on the bed table.
• 6. If I was reading in the kitchen, then my
  glasses are on the kitchen table.

3
Find My Glasses (contd.)

• Let p My glasses are on the kitchen table.
• q I saw my glasses at breakfast.
• r I was reading in the living room.
• s I was reading in the kitchen.
• t My glasses are on the coffee table.
• u I was reading in bed.
• v My glasses are on the bed table.

4
Find My Glasses (contd.)

• Then the original statements become
• 1. p q 2. r Ú s 3. r t
• 4. q 5. u v 6. s p
• and we can deduce (why?)
• 1. p q 2. s p 3. r Ú s 4. r t
• q p s r
• \ p \ s \ r \ t
• Hence the glasses are on the coffee table!

5
Fallacies

• A fallacy is an error in reasoning that results
  in an invalid argument.
• Three common fallacies
• Using vague or ambiguous premises
• Begging the question
• Jumping to a conclusion.
• Two dangerous fallacies
• Converse error
• Inverse error.

6
Converse Error

• If Zeke cheats, then he sits in the back row.
• Zeke sits in the back row.
• \ Zeke cheats.
• The fallacy here is caused by replacing the
  impication (Zeke cheats sits in back) with its
  biconditional form (Zeke cheats sits in back),
  implying the converse (sits in back Zeke
  cheats).

7
Inverse Error

• If Zeke cheats, then he sits in the back row.
• Zeke does not cheat.
• \ Zeke does not sit in the back row.
• The fallacy here is caused by replacing the
  impication (Zeke cheats sits in back) with its
  inverse form (Zeke does not cheat does not sit
  in back), instead of the contrapositive (does not
  sit in back Zeke does not cheat).

8
Universal Instantiation

• Consider the following statement All men are
  mortal Socrates is a man. Therefore, Socrates
  is mortal.
• This argument form is valid and is called
  universal instantiation.
• In summary, it states that if P(x) is true for
  all xÎD and if aÎD, then P(a) must be true.

9
Universal Modus Ponens

• Formal Version " xÎD, if P(x), then
  Q(x). P(a) for some aÎD. \ Q(a).
• Informal Version If x makes P(x) true, then x
  makes Q(x) true. a makes P(x) true. \ a makes
  Q(x) true.
• The first line is called the major premise and
  the second line is the minor premise.

10
Universal Modus Tollens

• Formal Version " xÎD, if P(x), then
  Q(x). Q(a) for some aÎD. \ P(a).
• Informal Version If x makes P(x) true, then x
  makes Q(x) true. a makes Q(x) false. \ a makes
  P(x) false.

11
Examples

• Universal Modus Ponens or Tollens???
• If a number is even, then its square is even.
• 10 is even.
• Therefore, 100 is even.
• If a number is even, then its square is even.
• 25 is odd.
• Therefore, 5 is odd.

12
Using Diagrams to Show Validity

• Does this diagram portray the argument of the
  second slide?

Mortals
Men
Socrates
13
Modus Ponens in Pictures

• For all x, P(x) implies Q(x).P(a).Therefore,
  Q(a).

x Q(x)
x P(x)
a
14
A Modus Tollens Example

• All humans are mortal.Zeus is not
  mortal.Therefore, Zeus is not human.

Zeus
Mortals
Humans
15
Modus Tollens in Pictures

• For all x, P(x) implies Q(x).Q(a).Therefore,
  P(a).

x Q(x)
a
x P(x)
16
Converse Error in Pictures

• All humans are mortal.Felix the cat is
  mortal.Therefore, Felix the cat is human.

Mortals
Felix?
Humans
Felix?
17
Inverse Error in Pictures

• All humans are mortal.Felix the cat is not
  human.Therefore, Felix the cat is not mortal.

Mortals
Felix?
Felix?
Humans
18
Quantified Form of Converseand Inverse Errors

• Converse Error " x, P(x) implies Q(x). Q(a),
  for a particular a. \ P(a).
• Inverse Error " x, P(x) implies Q(x). P(a),
  for a particular a. \ Q(a).

19
An Argument with No

• Major Premise No Naturals are negative.
• Minor Premise k is a negative number.
• Conclusion k is not a Natural number.

Negative numbers
Natural numbers
k
20
Abduction

• Major Premise All thieves go to Pauls Bar.
• Minor Premise Tom goes to Pauls Bar.
• Converse Error Therefore, Tom is a thief.
• Although we cant conclude decisively if Tom is a
  thief or not, if we have further information that
  99 of the 100 people in Pauls Bar are thieves,
  then the odds are that Tom is a thief and the
  converse error is actually valid here.
• This is called abduction by Artificial
  Intelligence researchers.