Chapter 5: Indirect Rules

80-210: Logic & ProofsJuly 9, 2009Karin Howe

Falsum (In)troduction = falsum• Truth table for falsum:

• Restriction: can never appear as a part of a compound formula; i.e., W ≠ WFF

F

Negation

I

pp_________________

Not (In)troduction and Not Elimination (Out)

Negation

I E

p A.. __________________p

p A.. __________________p

Strategies

• Reductio Strategy– Assume the opposite of your goal and try to

derive the falsum.• Reductio Variant on Arrow-In Strategy

– Use when your goal is a conditional - begin with an Arrow-In Strategy and within the Arrow-In Strategy, use the Reductio Strategy.

A note about strategies (tactics)• Tactic 1: Extraction (HAMMER!!)

– Apply elimination rules forward in order to extract goals that occur as positive subformulae of the formulae on available lines.

• Tactic 2: Conversion (Poof Variant on Or-Out Strategy)– Use disjunction elimination in order to obtain goal disjunctions.

• Tactic 3: Inversion (Backwards Strategy)– Invert non-atomic goals by applying introduction rules backward to them.

• Tactic 4: Division (Or-Out Strategy)– Use disjunction elimination on any goals for which the previous three tactics have

either not applied, or not been successful.• Tactic 5: Refutation (Reductio)

– Apply negation elimination backward to goals that cannot be obtained by any other means.

• Two more (mine):– Arrow In Strategy– Reductio Variant on Arrow-In Strategy

Double Negation

Negation

DNI DNE

p___________________p

p________________p

Practice Makes Perfect!!

• Practice Using Negation Introduction• Practice Using Negation Elimination• Practice Using the Basic Rules• Using the Double Negation Rules

• More practice =>

There was only one catch and that was Catch-22, which specified that a concern for one's own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.

Orr's asking to be grounded would be proof of his sanity. Asking to be grounded is a necessary condition for being grounded. If Orr is sane, then he cannot be grounded. All of which shows that Orr cannot be grounded.

A S, G A, S G G

"Son of the big man that killed Nag," she hissed, "stay still. I am not ready yet. Wait a little. Keep very still … If you move I strike, and if you do not move I strike."

• If you move I strike, and if you do not move I strike.

• Suppressed conclusion: I strike.

• (M S) & (M S) S

Why Worry?• There are only two things to worry about, either you're healthy or you're sick.

If you're healthy, there's nothing to worry about and if you're sick … there are two things to worry about … either you'll get well or you won't. If you get well there is nothing to worry about, but if you don't. you'll have two things to worry about … either you'll go to heaven or to hell. If you go to heaven you have nothing to worry about and if you go to hell you'll be so busy shaking hands with all of us that you'll have no time to worry.

• Suppressed conclusion: Either you have nothing to worry about or you'll have no time to worry.

• H = you are healthy, S = you are sick, W = you have something to worry about, A = you will get well, B = you will go to heaven, C = you will go to hell, T= you will have time to worry

• H S, (H W) & [S (A A)], (A W) & [A (B C)], (B W) & (C T) W T

My Three SonsMan 1: Yes, I’m married and have three

fine sons.Man 2: That’s wonderful! How old are

they?Man 1: Well, the product of their ages

is equal to 36.Man 2: Hmm. That doesn’t tell me

enough. Give me another clue.Man 1: O.K. The sum of their ages is

the number on that building across the street.

Man 2: Ah ha! I’ve almost got the answer, but I still need another clue.

Man 1: Very well. The oldest one has red hair.

Man 2: I’ve got it!

• If the answer has a unique sum, then Man 2 would not need another clue [beyond the clue about the building number]. But he did need another clue. If the answer does not have a unique sum, then it is either 9-2-2 or 6-6-1. If the answer is 6-6-1, then there is no oldest son. But there is an oldest son. Hence the answer is 9-2-2.

• A = the answer is 9-2-2, B = the answer is 6-6-1

• U C, C, U (A B), B O, O A