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notes_19_2x2 - Branden Fitelson Philosophy 12A Notes 1...

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Branden Fitelson Philosophy 12A Notes 1 Announcements & Such Miles Davis (with Coltrane) : So What Administrative Stuff HW #3 resubs due today (usual drill). When you turn in resubmissions, make sure that you staple them to your original homework submission . – The Take-Home Mid-Term has been posted. It’s due next Friday, with resubmissions due the following Friday (just like a HW). Please submit any extra-credit solutions with (any one of) your homework assignment(s). There is no deadline for extra-credit. Today: Chapter 4 — Natural Deduction Proofs for LSL Our natural deduction system for LSL. * Learning the Intro. and Elim. rules for each connective. * Proof strategy — working backward and forward. MacLogic — a useful (free!) computer program for proofs. UCB Philosophy Chapter 4 10/10/08 Branden Fitelson Philosophy 12A Notes 2 An Example of a Natural Deduction Involving & and The following is a valid LSL argument form: A & B C & D (A & D) H H Here’s a (7-line) natural deduction proof of the sequent corresponding to this argument: A & B, C & D, (A & D) H H . 1 (1) A & B Premise 2 (2) C & D Premise 3 (3) (A & D) H Premise 1 (4) A 1 & E 2 (5) D 2 & E 1, 2 (6) A & D 4, 5 & I 1, 2, 3 (7) H 3, 6 E UCB Philosophy Chapter 4 10/10/08 Branden Fitelson Philosophy 12A Notes 3 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 We can see that our example proof begins, as it should, with the three premises of the argument-form, written as follows: 1 (1) A & B Premise 2 (2) C & D Premise 3 (3) (A & D) H Premise UCB Philosophy Chapter 4 10/10/08 Branden Fitelson Philosophy 12A Notes 4 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 We can see that our example deduction continues, in lines (4) and (5), with two correct applications of the &-Elimination Rule: 1 (4) A 1 & E 2 (5) D 2 & E UCB Philosophy Chapter 4 10/10/08
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Branden Fitelson Philosophy 12A Notes 5 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.
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