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 Todays 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. Youll be given 3 hours to do it. Ive 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, final, and Chapters 7 & 8 Intro. Well 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 Final & 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 Final & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 3 ' & $ % Proof of (8) Problem is: (x)(y)(Fx&Gy) (y)(x)(Fx&Gy) 1 (1) (x)(y)(Fx&Gy) Premise 1 (2) (y)(Fa&Gy) 1 E 3 (3) Fa&Gb Assumption 1 (4) (y)(Fc&Gy) 1 E 5 (5) Fc&Gd Assumption 5 (6) Fc 5 &E 1 (7) Fc 4,5,6 E 3 (8) Gb 3 &E 1,3 (9) Fc&Gb 7,8 &I 1,3 (10) (x)(Fx&Gb) 9 I 1,3 (11) (y)(x)(Fx&Gy) 10 I 1 (12) (y)(x)(Fx&Gy) 2,3,11 E UCB Philosophy Chapter 6 Final & Chapters 7/8 (L2PL) Intro. 06/29/10 Branden Fitelson Philosophy 12A Notes 4 ' & Proof of (9) Problem is: (y)(x)(Fx&Gy) (x)(y)(Fx&Gy) 1 (1) (y)(x)(Fx&Gy) Premise 2 (2) (x)(Fx&Gb) Ass umption 2 (3) Fa&Gb 2 E 2 (4) (y)(Fa&Gy) 3 I 1 (5) (y)(Fa&Gy) 1,2,4 E 1 (6) (x)(y)(Fx&Gy) 5 I UCB Philosophy Chapter 6 Final & 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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notes_14_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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