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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such • Graham Nash • Administrative Stuff – HW #3 solutions have been posted. – The TakeHome MidTerm has been posted. – A Sample InClass MidTerm has been posted. + The Actual InClass Midterm is on Thursday. * Bring bluebooks and something to write with. * The exam is closedbook, closednotes. * It shouldn’t take you the full 75minutes. • 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 03/09/10 Branden Fitelson Philosophy 12A Notes 2 ' & $ % How to Deduce a Conditional: I • To deduce a conditional, we assume its antecedent and try to deduce its consequent from this assumption. If we are able to deduce the consequent from our assumption of the antecedent, then we discharge our assumption, and infer the conditional. • To implement the → I rule, we will first need a refined Rule of Assumptions that will allow us to assume arbitrary formulas “for the sake of argument”, later to be discharged after making desired deductions. Here’s the refined rule of Assumptions: • Rule of Assumptions (final version): At any line j in a proof, any formula p may be entered and labeled as an assumption (or premise, where appropriate). The number j should then be written on the left. Schematically: j (j) p Assumption (or: Premise) UCB Philosophy Chapter 4 03/09/10 Branden Fitelson Philosophy 12A Notes 3 $ % How to Deduce a Conditional: II — The → I Rule • Now, we need a formal Introduction Rule for the → , which captures the intuitive idea sketched above ( i.e. , assuming the antecedent, etc .): • Rule of →Introduction : For any formulae p and q , if q has been inferred at a line k in a proof and p is an assumption or premise occurring at line j, then at line m we may infer [ p → q , labeling the line ‘j, k → I’ and writing on the left the same assumption numbers which appear on the left of line k, except that we delete j if it is one of these numbers. Note: we may have j < k, j > k, or j = k ( why ?). Schematically: j (j) p Assumption (or: Premise) . . . a 1 ,..., a n (k) q . . . {a 1 ,..., a n }/j (m) p → q j, k → I UCB Philosophy Chapter 4 03/09/10 Branden Fitelson Philosophy 12A Notes 4 ' & $ % Examples Involving & E, & I, → E, and → I • Can you deduce the following, using &E, &I, → E, and → I? (a) A → B A → C ∴ A → (B & C) (b) (A & B) → C ∴ A → (B → C) (c) B & C ∴ (A → B) & (A → C) (d) A → B ∴ (A & C) → (B & C) (e) A & (B & C) ∴ A → (B → C) (f) A → B ∴ A → (C → B) UCB Philosophy Chapter 4 03/09/10 Branden Fitelson Philosophy 12A Notes 5 ' & $ % Important Tips For Using the → I Rule • Use → I only when you wish to derive a conditional...
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This note was uploaded on 04/20/2010 for the course PHIL 63170 taught by Professor Fitelson during the Spring '10 term at Berkeley.
 Spring '10
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