notes_20_2x2 - Branden Fitelson ' Philosophy 12A Notes 1...

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Philosophy 12A Notes 1 ' $ % Ry Cooder : Chavez Ravine Administrative Stuff There will be no lecture on Wednesday (10/15). And, Branden will not have office hours on Wednesday either (I’ll be at Caltech). – The Take-Home Mid-Term has been posted. It’s due on Friday, with resubmissions due the following Friday (just like a HW). + “Extra Credit Option”: for each validity problem in (3), you must chose one way of proving it — for both initial and re-submission. Today: Chapter 4 — Natural Deduction Proofs for LSL Our natural deduction system for LSL. * How to deduce a conditional: the introduction rule for . * Proofs by contradiction ( reductio ad absurdum ), and the rules. * Proof strategy — working backward and forward. MacLogic — a useful (free!) computer program for proofs. UCB Philosophy Chapter 4 10/13/08 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 10/13/08 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 10/13/08 Branden Fitelson Philosophy 12A Notes 4 ' $ % Using The I Rule: An Example Let’s do a deduction of: A (B C) (A B) (A C) 1 (1) A (B C) Premise 2 (2) A B Assumption 3 (3) A Assumption 2, 3 (4) B 2, 3 E 1, 3 (5) B C 1, 3 E 1, 2, 3 (6) C 4, 5 E 1, 2 (7) A C 3, 6 I 1 (8) (A B) (A C) 2, 7 I F UCB Philosophy
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notes_20_2x2 - Branden Fitelson ' Philosophy 12A Notes 1...

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