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Lecture #14 - Branden Fitelson Philosophy 12A Notes 1...

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Branden Fitelson Philosophy 12A Notes 1 Announcements & Such Townes Van Zandt Administrative Stuff – HW #3 resubmissions are due today. When you turn in resubmissions, make sure that you staple them to your original homework submission . – The Take-Home Mid-Term has been posted. [Discuss.] – A Sample In-Class Mid-Term has been posted. [Discuss.] The Actual In-Class Mid-term is next Thursday, 3/11. Today: Chapter 4 — Natural Deduction Proofs for LSL Continuing on with the LSL- natural deduction rules for . MacLogic — a useful computer program for natural deduction. * See http://fitelson.org/maclogic.htm . Natural deductions are the most challenging topic of the course. UCB Philosophy Chapter 4 03/04/10 Branden Fitelson Philosophy 12A Notes 2 The Rule of Assumptions (Preliminary Version) Rule of Assumptions (preliminary version): The premises of an argument-form are listed at the start of a proof in the order in which they are given, each labeled ‘Premise’ on the right and numbered with its own line number on the left. Schematically: j (j) p Premise UCB Philosophy Chapter 4 03/04/10 Branden Fitelson Philosophy 12A Notes 3 The Rule of & -Elimination ( & E) Rule of & -Elimination : If a conjunction p & q occurs at line j, then at any later line k one may infer either conjunct, labeling the line ‘j & E’ and writing on the left all the numbers which appear on the left of line j. Schematically: a 1 ,. . . , a n (j) p & q a 1 ,. . . , a n (j) p & q . . . OR . . . a 1 ,. . . , a n (k) p j & E a 1 ,. . . , a n (k) q j & E UCB Philosophy Chapter 4 03/04/10 Branden Fitelson Philosophy 12A Notes 4 The Rule of & -Introduction ( & I) Rule of & -Introduction : For any formulae p and q , if p occurs at line j and q occurs at line k then the formula p & q may be inferred at line m, labeling the line ‘j, k & I’ and writing on the left all numbers which appear on the left of line j and all which appear on the left of line k. [ Note: we may have j < k, j > k, or j = k. Why ? ] a 1 ,. . . , a n (j) p . . . b 1 ,. . . , b u (k) q . . . a 1 ,. . . , a n , b 1 ,. . . , b u (m) p & q j, k & I UCB Philosophy Chapter 4 03/04/10
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Branden Fitelson Philosophy 12A Notes 5 The Rule of -Elimination ( E) Rule of -Elimination : For any formulae p and q , if p q occurs at a line j and p occurs at a line k, then q may be inferred at line m, labeling the line ‘j, k E’ and writing on the left all numbers which appear on the left of line j and all numbers which appear on the left of line k. [ Note: We may have either j < k or j > k .] a 1 ,. . . , a n (j) p q . . . b 1 ,. . . , b u (k) p . . . a 1 ,. . . , a n , b 1 ,. . . , b u (m) q j, k E UCB Philosophy Chapter 4 03/04/10 Branden Fitelson Philosophy 12A Notes 6 Deduction #3 Using the Rules & E, & I, and E Let’s do a deduction of: A (B (C D)) C & (A & B) D 1 (1) A (B (C D)) Premise 2 (2) C & (A & B) Premise 2 (3) A & B 2 & E 2 (4) A 3 & E 1, 2 (5) B (C D) 1, 4 E 2 (6) B 3 & E 1, 2 (7) C D 5, 6 E 1 (8) C 2 & E 1, 2 (9) D 7, 8 E UCB Philosophy Chapter 4 03/04/10 Branden Fitelson Philosophy 12A Notes 7 Note on -E Rules — Avoiding a Common Error As with & E, E can only be applied to the main of a conditional — not to any other ’s which may be in a formula.
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