Dilemma: Division Into CasesDilemma: p q

p rq r

rPremises: 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.

Application: Find My Glasses1. 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.

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.

Find My Glasses (cont’d.)Then the original statements become:1. p q 2. r s 3. r t4. ~q 5. u v 6. s pand we can deduce (why?):1. p q 2. s p 3. r s 4. r t ~q ~p ~s r ~p ~s r tHence the glasses are on the coffee table!

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.

Converse ErrorIf 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).

Inverse ErrorIf 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).

Universal Instantiation• Consider the following statement:

All men are mortalSocrates 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 xD and if aD, then P(a) must be true.

Universal Modus Ponens• Formal Version: xD, if P(x), then Q(x).

P(a) for some aD. 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.

Universal Modus Tollens• Formal Version: xD, if P(x), then Q(x).

~Q(a) for some aD. ~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.

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.

Using Diagrams to Show Validity• Does this diagram portray the argument of the

second slide?

Mortals

Men

Socrates

Modus Ponens in Pictures• For all x, P(x) implies Q(x).

P(a).Therefore, Q(a).

{x | Q(x)}

{x | P(x)}

a

A Modus Tollens Example• All humans are mortal.

Zeus is not mortal.Therefore, Zeus is not human.

Mortals

Humans

Zeus

Modus Tollens in Pictures• For all x, P(x) implies Q(x).

~Q(a).Therefore, ~P(a).

{x | Q(x)}

{x | P(x)}

a

Converse Error in Pictures• All humans are mortal.

Felix the cat is mortal.Therefore, Felix the cat is human.

Mortals

HumansFelix?

Felix?

Inverse Error in Pictures• All humans are mortal.

Felix the cat is not human.Therefore, Felix the cat is not mortal.

Mortals

Humans

Felix?

Felix?

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).

An Argument with “No”• Major Premise: No Naturals are negative.• Minor Premise: k is a negative number.• Conclusion: k is not a Natural number.

Natural numbers Negative numbers

k

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.