notes_23_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such Godspeed You Black Emperor : The Dead Flag Blues Administrative Stuff The in-class mid-term is on Friday. * Same structure as the sample mid-term. * Rules Handout will be provided at the exam (bring blue book). The Take-Home Mid-Term resubmission is due on Friday . I have posted the solutions for HW #3. I have posted HW #4 (due next Friday). Branden will not have office hours on Friday. Today: Chapter 4 Natural Deduction Proofs for LSL The rules, and lots of (solved) proof problems. Next: Sequent and Theorem Introduction (derived rules). More on MacLogic for constructing and checking proofs. UCB Philosophy Chapter 4 10/22/08 Branden Fitelson Philosophy 12A Notes 2 ' & The Disjunction Rules Rule of -Introduction : For any formula p , if p has been inferred at line j, then, for any formula q , either [ p q or [ q p may be inferred at line k, labeling the line j I and writing on its left the same premise and assumption numbers as appear on the left of j. a 1 ,. . . , a n (j) p a 1 ,. . . , a n (j) q . . . OR . . . a 1 ,. . . , a n (k) p q j I a 1 ,. . . , a n (k) p q j I The I rule is very simple an intuitive. Basically, it says that you may infer a disjunction from either of its disjuncts. The elimination rule ( E) for , on the other hand, is considerably more complex to state and apply. Its the hardest of our rules. UCB Philosophy Chapter 4 10/22/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % Rule of -Elimination : If a disjunction [ p q occurs at line g of a proof, p is assumed at line h, r is derived at line i, q is assumed at line j, and r is derived at line k, then at line m we may infer r , labeling the line g, h, i, j, k E and writing on its left every number on the left at line g, and at line i (except h), and at line k (except j). a 1 ,. . . , a n (g) p q . . . h (h) p Assumption . . . b 1 ,. . . , b u (i) r . . . j (j) q Assumption . . . c 1 ,. . . , c w (k) r . . . A (m) r g, h, i, j, k E where A is the set: {a 1 ,. . . , a n } {b 1 ,. . . , b u }/h {c 1 ,. . . , c w }/j. UCB Philosophy Chapter 4 10/22/08 Branden Fitelson Philosophy 12A Notes 4 ' & Another Example Involving I and Negation Heres a proof of the theorem : ` A A . Problem is: AA 1 (1) (AA)nobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspaceAssumption (I) 2 (2) AnobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspaceAssumption (I) 2 (3) AAnobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspace2 I 1,2 (4) nobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspacenobreakspace1,3...
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This note was uploaded on 09/23/2009 for the course PHIL 12A taught by Professor Fitelson during the Spring '08 term at University of California, Berkeley.

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notes_23_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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