# section_3.16_answers - 1(Ex 8.28 1 P ↔ ¬ P 2 ⊥ Proof 1...

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PHIL12A Section answers, 16 March 2011 Julian Jonker 1 How much do you know? Construct formal proofs for the following arguments: 1. (Ex 8.19) 1 A B 2 ¬ B 3 ¬ A Proof: 1 A B 2 ¬ B 3 A 4 B Intro: 1, 3 5 Intro: 2, 4 6 ¬ A ¬ Intro: 3-5 2. (Ex 8.24) 1 A B 2 A C 3 B D 4 C D 1

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Proof: 1 A B 2 A C 3 B D 4 A 5 C Elim: 2, 4 6 C D Intro: 5 7 B 8 D : 3, 7 9 C D Intro: 8 10 C D Elim: 1, 4-6, 7-9 3. (Ex 8.25) 1 A B 2 B C 3 A C Proof: 2
1 A B 2 B C 3 A 4 B Elim: 1, 3 5 C Elim: 2, 4 6 C 7 B Elim: 2, 6 8 A Elim: 1, 7 9 A C Intro: 3-5, 6-8 4. (Ex 8.46) 1 Cube(a) (Cube(b) Tet(c)) 2 Tet(c) Small(c) 3 (Cube(b) Small(c)) Small(b) 4 ¬ Cube(a) Small(b) Proof: 3

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1 Cube(a) (Cube(b) (Tet(c)) 2 Tet(c) Small(c) 3 (Cube(b) Small(c)) Small(b) 4 ¬ Cube(a) 5 Cube(a) 6 Intro: 4, 5 7 Small(b) Elim: 6 8 Cube(b) Tet(c) 9 Cube(b) 10 Tet(c) Elim: 8, 9 11 Small(c) Elim: 2, 10 12 Cube(b) Small(c) Intro: 9-11 13 Small(b) Elim: 3, 12 14 Small(b) Elim: 1, 5-7, 8-13 15 ¬ Cube(a) Small(b) Intro: 4-14 4
2 Something slightly harder, if there’s time.

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Unformatted text preview: 1. (Ex 8.28) 1 P ↔ ¬ P 2 ⊥ Proof: 1 P ↔ ¬ P 2 P 3 ¬ P ↔ Elim: 1, 2 4 ⊥ ⊥ Intro: 2, 3 5 ¬ P ¬ Intro: 2-4 6 P ↔ Elim: 1, 5 7 ⊥ ⊥ Intro: 5, 6 2. (Ex 8.38) Give formal proofs of both ( P ∧ Q ) → P and ¬ ( P ∧ Q ) ∨ P . These formulas are equivalent, so the point of this exercise is to show how useful it is to have → . The long way: 5 1 2 ¬ ( ¬ ( P ∧ Q ) ∨ P ) 3 P 4 ¬ ( P ∧ Q ) ∨ P 5 ⊥ 6 ¬ P 7 ¬ P 8 P ∧ Q 9 P 10 ⊥ 11 ¬ ( P ∧ Q ) 12 ¬ ( P ∧ Q ) ∨ P 13 ⊥ 14 ¬¬ P 15 ⊥ 16 ¬¬ ( ¬ ( P ∧ Q ) ∨ P ) 17 ¬ ( P ∧ Q ) ∨ P The short way: 1 2 P ∧ Q 3 P ∧ Elim: 2 4 ( P ∧ Q ) → P → Intro: 2-3 6...
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## This note was uploaded on 09/19/2011 for the course PHILOS 12A taught by Professor Fitelson during the Spring '08 term at Berkeley.

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section_3.16_answers - 1(Ex 8.28 1 P ↔ ¬ P 2 ⊥ Proof 1...

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