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# notes_14_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & \$ % Announcements and Such • Today’s Music: Wishbone Ash • I have posted my solutions to HW #4 and HW #5. • HW #6 is due on Thursday @ 4pm. + The final is in class on Thursday. You’ll be given 3 hours to do it. • I’ve posted two important handouts concerning the final exam: – The (Complete) Natural Deduction Rules Handout (provided at final). – A sample final exam, which has the same structure as the actual final. This sample was discussed, in detail, in lecture last week. • Today: Chapter 6, finalé, and Chapters 7 & 8 Intro. – We’ll finish-up chapter 6 (LMPL) today, and move on to Chs. 7&8. – I will only be covering (some of) the L2PL parts of Chapters 7 & 8. UCB Philosophy Chapter 6 Finalé & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 2 ' & The Rule of ∃-Elimination: Nine Examples • Here are 9 examples of proofs involving all four quantifier rules. 1. ( ∃ x) ∼ Fx ` ∼ ( ∀ x)Fx [ p. 200, example 5] 2. ( ∃ x)(Fx → A) ` ( ∀ x)Fx → A [ p. 201, example 6] 3. ( ∀ x)( ∀ y)(Gy → Fx) ` ( ∀ x)[( ∃ y)Gy → Fx] [ p. 203, I. # 19 ⇒ ] 4. ( ∃ x)[Fx → ( ∀ y)Gy] ` ( ∃ x)( ∀ y)(Fx → Gy) [ p. 203, I. # 20 ⇐ ] 5. A ∨ ( ∃ x)Fx ` ( ∃ x)(A ∨ Fx) [ p. 203, II. # 2 ⇐ ] 6. ( ∃ x)(Fx & ∼ Fx) ` ( ∀ x)(Gx & ∼ Gx) [ p. 203, I. # 12 ⇒ ] 7. ( ∀ x)[Fx → ( ∀ y) ∼ Fy] ` ∼ ( ∃ x)Fx [ p. 203, I. # 5] 8. ( ∀ x)( ∃ y)(Fx & Gy) ` ( ∃ y)( ∀ x)(Fx & Gy) [ p. 201, example 7] 9. ( ∃ y)( ∀ x)(Fx & Gy) ` ( ∀ x)( ∃ y)(Fx & Gy) [other direction] UCB Philosophy Chapter 6 Finalé & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 3 ' & \$ % Proof of (8) Problem is: (Ëx)(‰y)(Fx&Gy) Ê (‰y)(Ëx)(Fx&Gy) 1 (1) (Ëx)(‰y)(Fx&Gy)„„„ÊÊÊÊ Premise 1 (2) (‰y)(Fa&Gy)„„„ÊÊÊÊÊÊÊ 1 ËE 3 (3) Fa&GbÊÊÊÊÊÊÊÊÊÊÊÊ Assumption 1 (4) (‰y)(Fc&Gy)„„„ÊÊÊÊÊÊÊ 1 ËE 5 (5) Fc&GdÊÊÊÊÊÊÊÊÊÊÊÊ Assumption 5 (6) FcÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ 5 &E 1 (7) FcÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ 4,5,6 ‰E 3 (8) Gb„„„„ÊÊÊÊÊÊÊÊÊÊÊÊÊÊ 3 &E 1,3 (9) Fc&GbÊÊÊÊÊÊÊÊÊÊÊÊ 7,8 &I 1,3 (10) (Ëx)(Fx&Gb)„„„ÊÊÊÊÊÊÊ 9 ËI 1,3 (11) (‰y)(Ëx)(Fx&Gy)„„„ÊÊÊÊ 10 ‰I 1 (12) (‰y)(Ëx)(Fx&Gy)„„„ÊÊÊÊ 2,3,11 ‰E UCB Philosophy Chapter 6 Finalé & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 4 ' & Proof of (9) Problem is: (‰y)(Ëx)(Fx&Gy) Ê (Ëx)(‰y)(Fx&Gy) 1 (1) (‰y)(Ëx)(Fx&Gy) Premise 2 (2) (Ëx)(Fx&Gb) Ass umption 2 (3) Fa&Gb 2 ËE 2 (4) (‰y)(Fa&Gy) 3 ‰I 1 (5) (‰y)(Fa&Gy) 1,2,4 ‰E 1 (6) (Ëx)(‰y)(Fx&Gy) 5 ËI UCB Philosophy Chapter 6 Finalé & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 5 ' & \$ % Two LMPL Extensions of Sequent Introduction • Here are two additions to our list of SI sequents: (QS) One can infer [ ( ∀ x) ∼...
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## This note was uploaded on 11/26/2011 for the course PHILOSOPHY 101 taught by Professor Buechner during the Fall '06 term at Rutgers.

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notes_14_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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