Proof Strategies PHIL 122 Fall 2012 Karin Howe

My Bag of Tricks: Proof Strategies HAMMER!! Backwards Strategy Abracadabra! "Proof Bits" Strategy Conditional Proof Strategy Reductio Strategy DN Variant on Reductio Strategy

HAMMER!! Strategy Use inference rules from the top down to break the premises into little bloody pieces. Klenk refers to this as the "top down method" (Unit 7, p. 131). Demonstrate using Ex 9b from Unit 7: (B v C) > A, A > (S v T), B & ~S, (T & A) > (W > S) / ~W (see next slide)

Backwards Strategy Specify a goal which you want to work towards using the Backwards Strategy. This is usually your conclusion, but might not be if using the Backwards Strategy from somewhere in the middle of your proof. Place 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. Klenk refers to this as the "bottom-up method", or the method of *working backwards* (p. 135). Use Ex 5k in Unit 8 to demonstrate this method: A, B / ~(A  ~B) (see next slide)

Abracadabra! Strategy Don't like a formula in your premises? Use the rules of replacement to transform it into something "friendlier" (i.e., easier to take apart) This strategy can also be used in conjunction with the Backwards Strategy -- don't like that conclusion? Then use rules of replacement backwards to transform the conclusion into something easier to prove!

Some useful Abracadabra! moves

"Proof Bits" Strategy There are a number of relatively easy proofs that represent frequently re-occurring patterns of inference (what Klenk calls "proof bits"). These "proof bits" can be used as modules within a longer proof, allowing you to break down complicated proofs into smaller, more manageable pieces. Klenk, Unit 8, pp. 159-161

Some useful "proof bits" Use as example of "proof bits" strategy: Ex. 7j from Unit 8: F  (~G  ~F), ~(H  G), ~(H  W), A  (S  W)/ ~(A  F). Uses Abracadabra! Strategies 1,2 and 3, and "proof bits" 1 and 2. (see next slide) More useful "proof bits" and abracadabra! Moves can be found in Ex 5, Unit 8 of Klenk and in the exercises at the end of this powerpoint.

Conditional Proof Strategy When your goal is a conditional, try assuming its antecedent in the hope of deriving its consequent (using CP). Use as example for CP Strategy: (A  B)  ~(C v D), (~C  E)  (F  ~O), (F  ~H)  (J  ~K) / (A  E)  ~K (p. 167 in Klenk (Unit 8)) (see next slide)

Reductio Strategy The traditional name for Indirect Proof is reductio ad absurdum Reductio Strategy is good to use to use when your goal is a negation, and there is no obvious way to derive it using the rules of inference. Reductio Strategy: Assume the unnegated form of your goal and try to derive a contradiction of the form p  ~p.

DN Variant on Reductio Strategy Use when your goal does not have a negation as its main operator, and there is no obvious way to derive your goal directly using the rules of inference. Assume the negation of your goal, and try to arrive at a SFC. If successful, you will then have derived your goal in its double negated form by IP. Then you can simply apply DN to get your goal! SFC = Standard Form Contradiction (contradiction of the form p & ~p)

Basic Structure: Reductio Strategy and DN Variant on Reductio Strategy Use Ex5b, Unit 9 to illustrate reductio: A & ~B / ~(A = B)

Interestingly, when this proof is done using the Backwards Strategy, it can be done in only 7 lines. Try it!

Practice: "Proof Bits" Prove the following useful short proofs as "proof bits" for later use in longer proofs: C / (C  A) A, B / (A  B) (A  B) / (A  (B  C)) A / (B  (A  A)) ((A  B)  (C  D)), ((E  B)  A) / ((E  B)  (C  D))

(A  B), A / B (A  B) / ((A  C)  B) (A  (B  C)) / (A  B) (A  B), (A  C) / (A  (B  C)) Q / (((X  Y) v Z)  Q) P / (P  ((X  Y) v Z)) (A  B), (~B  C) / (A  C) (P   Q) / (P  Q) (P  Q), ( P  Q) / Q

"Proofs Bits" and Abracadabra! Prove the following useful short proofs using only the rules of replacement: (A  B) / (A  B) (P  (Q  P)) / (P  Q) (P  Q) / ((P  Q)  (Q  P)) (A  B) / (A  B) (P  Q) / (Q  P)

(S  (T  U)) / (T  (S  U)) (A  (B  C) ) / ((A  B)  (A  C)) A / (A  A) ((A  C)  B) / ((A  B)  (B  C)) ((A  B)  (C  D)) / ((A  C)  (B  D))

Practice Using Strategies! Try using the Backwards Strategy on these proofs: ((A  B)  (C  D)), ((A  B)  (Y  Z)) / ((C  D)  (Y  Z)) ((A  D)  W), (B  C), (R  (C  T)), (R  A) / (W  C) [note: some "proof bits" may also prove useful] Use the Reductio Strategy (T  (A  Q)), (P  (Q  T)), ((S  W)  A), ((S  (T  W))  (A  D)) / P [some "proof bits" may also be useful]

Use DN Variant on Reductio Strategy (H  S), ((H  W)  (S  (A  A))), ((A  W)  (A  (B  C))), ((B  W)  (C  T)) / (W  T) Use a combination of Backwards Strategy and Conditional Proof Strategy (A  (B  C)), (E  (C  P)), C / ((B  P)  (A  E)) ((A  B)  (A  B)) / (A  B)

Use a combination of Backwards Strategy and Reductio (C  (D  C)), (C  D) / (C  D) Use a combination of Conditional Proof Strategy and Reductio ((A  B)  (C  D)), (C  (E  F)), (F  E), ((D  A)  (P  Q)), ( P  ~Q) / (A  B)