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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 x 2 is positive. If x is negative, then x 2 is positive. Conclusion: x 2 is positive. Application: Find My Glasses.

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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.


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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.


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Find My Glasses (cont’d.)

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.


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Find My Glasses (cont’d.)

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!


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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.


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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).


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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).


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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.


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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.


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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.


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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.


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Using Diagrams to Show Validity

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

Mortals

Men

Socrates


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Modus Ponens in Pictures

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

{x | Q(x)}

{x | P(x)}

a


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A Modus Tollens Example

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

Zeus

Mortals

Humans


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Modus Tollens in Pictures

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

{x | Q(x)}

a

{x | P(x)}


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Converse Error in Pictures

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

Mortals

Felix?

Humans

Felix?


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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


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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).


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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


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Abduction

  • Major Premise: All thieves go to Paul’s Bar.

    Minor Premise: Tom goes to Paul’s Bar.

    Converse Error: Therefore, Tom is a thief.

  • Although we can’t conclude decisively if Tom is a thief or not, if we have further information that 99 of the 100 people in Paul’s 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.


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