notes_22_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such Phish : You Enjoy Myself Administrative Stuff I have posted the sample in-class mid-term. Same structure as actual mid-term. The mid-term rules handout is also posted. The Take-Home will be returned Wed. , with resubs due Friday. + Extra Credit Option: for each validity problem in (3), you must chose one way of proving it for both initial and re-submission. * You may use sequent/theorem introduction on the take-home. I have posted the solutions for HW #3. Branden will not have office hours on Friday. Today: Chapter 4 Natural Deduction Proofs for LSL Our natural deduction system for LSL. * The rules: I is simple, E is complex (but straightforward). * Next: Sequent and Theorem Introduction (derived rules). More on MacLogic for constructing and checking proofs. UCB Philosophy Chapter 4 10/20/08 Branden Fitelson Philosophy 12A Notes 2 ' & General Strategy Working in Both Directions Begin by writing the premises (if any) at the top of your scratch paper area, using the Rule of Assumptions. Then, write the conclusion (the main goal formula) youre trying to derive at the bottom of your scratch area. Next, determine what the main connective (if any) of your conclusion is, then apply the introduction rule for that connective. This will yield sub-goal formula(s). Write the sub-goal formula(s) directly above your conclusion. Then try to figure-out how to prove the sub-goal formula(s) from your premises. This will yield sub-sub-goal formula(s). And so on . . . Repeat this process until you have worked your way all the way back up to your premises/assumptions (if a formula is resisting proof, you might try to prove its double-negation using I with E). UCB Philosophy Chapter 4 10/20/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % Example Proof of a Theorem Using only the rules we have learned so far, we should be able to prove the following theorem : ` (A & A) . Lets do this one by hand first. Heres a simple proof, generated using MacLogic (Ill show how): Problem is: (A&A) 1 (1) A&AAssumption (!) 1 (2) A1 &E 1 (3) A1 &E 1 (4) 2,3 E (5) (A&A)1,4 I This proof makes use of no premises , and its final line has no numbers to its left indicating that we have succeeded in proving (A & A) from nothing at all . Its a theorem ( i.e. , a sequent with no premises)! UCB Philosophy Chapter 4 10/20/08 Branden Fitelson Philosophy 12A Notes 4 ' & The Introduction Rule for ( I) 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...
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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_22_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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