Chapter 4: Derivations

PHIL 121: Methods of ReasoningApril 10, 2013

Instructor:Karin HoweBinghamton University

Basic Introduction Rules

Conjunction (&I) Disjunction ( I) p

q__________ p & q &I

p________ q p IL

p________ p q IR

Basic Elimination RulesConjunction

(&E)Conditional

(E)Biconditional

(E)

p & q_____ p &EL

p q p________ q E

p qq____________p EL

p & q_____ q &ER

p qp____________q ER

Fill in the Blank Proofs

1. (A & D) H Pr.2. C & D Pr.3. A & B Pr. / H4. A

____5. _____ &E, 26. A & D &I, 4,57. _____ E, 1,6

1. A (B C) Pr.2. (C D) A Pr.3. B & D Pr. / C4. B _____5. D _____6. C D _______7. _______ E, 2, 68. ______

_______9. C E, 8,

4

Strategies

• HAMMER!!– Use the Elimination Rules to break apart the premises

into small bloody pieces• Backwards Strategy

– Start by placing your goal at the bottom and then try to specify a line or lines from which you can get your goal in a single step. Then do the same for the lines you specified and repeat until you work your way up to the premises.

Practice Using Basic Elimination and Introduction Rules in the Proof Lab

• Practice Using Conjunction Introduction &I), p. 49

• Practice Using Disjunction Introduction (I), p. 50• Practice Using Conditional Elimination (E), p.

54 [skip Problem 3 for now]• Practice Using Conjunction Elimination (&E), p.

55• Practice Using Biconditional Elimination (E), p.

101, Problem 1

Rules Requiring AssumptionsConditional Biconditional Disjunction

I I E

p q I

p q I

p q

r

p..q

Ap A

r

q A

r

p A

q

q A

p

Fill in the Blank Proofs1. A B Pr.2. A C Pr. / A (B & C)3. A A

(I)4. B

______5. _____ E, 2,36. B & C _______7. A (B & C) _______

1. A (B C) Pr. / (A B) (A C)2. A B A (I)3. ________ A (I)4. _______ E, 1, 35. B ________6. C ________7. A C I, 58. (A B) (A C)

______

Fill in the Blank Proofs1. A B Pr.2. C D Pr. 3. E (A C) Pr.4. E Pr. / B D5. ________ E, 3,46. A A (E)7. _____ E, 1,68. B D _______9. C A (E)10. D _______11. _________ I, 1012. B D E, 5, 8, 11

1. (B & C) D Pr. 2. C Pr. / B

D3. B A (I)4. B & C ________5. _______ E, 1,4 6. D ________7. B & C E, 1,68. ______ &E, 7 9. B D I, 5, 8

More Strategies• Arrow In Strategy

– When your goal is a conditional, try assuming its antecedent in the hope of deriving its consequent.

• Alternatively, if you have the consequent of the desired conditional, you can simply derive the conditional directly.

• Version 2 of I rule: q____p q I

Strategies, con't• Or Elimination Strategy

– Use when you have a disjunction available whose disjuncts might imply your goal. Assume each disjunct separately and try to derive your goal from each. If you succeed, derive your goal from the disjunction and the goal derived from each disjunct by Or Elimination.

• Poof Variant on Or Elimination Strategy– Use when your goal is a disjunction and when you have a disjunction

available whose disjuncts might imply your goal. Assume each disjunct of the original disjunction separately and use it to try to derive one side of the disjunction which is your goal. Poof in the other side of the disjunction. If you succeed in arriving at your goal disjunction for both cases, derive the disjunction which is your goal from the original disjunction and the goal derived from each disjunct by Or Elimination.

Another strategic note…positive subformulae

1. If is an atomic formula, its only positive subformula is itself; i.e.,

2. If is any negated formula , its only positive subformula is itself; i.e.,

3. If is a conjunction ( & ), its positive subformulae are itself, and all positive subformula of each conjunct

4. If is a disjunction ( ), its positive subformulae are itself, and all positive subformula of each disjunct

5. If is a conditional ( ), its positive subformulae are itself, and all positive subformula of its consequent

6. If is a biconditional ( ), its positive subformulae are itself, and all positive subformula of the formulae on either side of the biconditional

Practice Using Rules Requiring Assumptions in the Proof Lab

• Practice Using Conditional Introduction (I), p. 51

• Practice Using Biconditional Introduction (I), p. 101, Problems 2-4

• Practice Using Disjunction Elimination (E), p. 56

Putting It All Together …

• Practice Using Introduction Rules, p. 52• Practice Using Elimination Rules, p. 57• Practice Using the Binary Rules, p. 58

• Any more questions??

• If not, then how about some more practice?

"Aha!" said Pooh (Rum-tum-tiddle-um-tum.) "If I know anything about anything, that hole means Rabbit," he said, "and Rabbit means Company," he said, "and Company means Food…"

• There is a rabbit HOLE here. A rabbit hole means that there is a RABBIT nearby. If there is a rabbit nearby, then there is COMPANY around. If there is company around, then there is FOOD available. Thus, there is food available.

• H, R, R C, C F F

"My friend, you'll help in this thing–for my sake–that's why you're here–I mightn't be able alone. If you flinch, I'll kill you. Do you understand that? And if I have to kill you, I'll kill her–and then I reckon nobody'll ever know much about this little business."

• If you FLINCH, I'll kill YOU. And if I have to kill you, I'll kill HER. Therefore, I'll kill both of you, if you flinch.

• F Y, Y H F (Y & H)

The cake will either make me LARGER or SMALLER. If it makes me larger, I can reach the KEY; and if it makes me smaller, I can CREEP under the door. If I reach the key, I'll get into the GARDEN; and if I creep under the door, I'll get into the garden. So [either way] I'll get into the garden.

L S, (L K) & (S C), (K G) & (C G) G

Soon her eye fell on a little glass box that was lying under the table: she opened it, and found in it a very small cake, on which the words "EAT ME" were beautifully marked in currants. "Well, I'll eat it." said Alice, "and if it makes me larger, I can reach the key; and if it makes me smaller, I can creep under the door; so either way I'll get into the garden, and I don't care which happens!"