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# notes_10_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & \$ % Announcements and Such • Today’s Music: Pink Floyd • HW #4 is due on Thursday @ 4pm, usual drill (chapter 4 — proofs). • I’ve posted my solutions for HW #2 and HW #3. • Grade Curve (so far). Take the average of: (1) your average HW score (all on 100-point scale), and (2) your mid-term score. – The approximate “curve” for the course is as follows: A-ish ( ≥ 90), B-ish (80–90), C-ish (70–80), D-ish (60-70). • This should be a reasonably good (but not perfect) guide to where the (overall) grade curve will end-up for the entire course. • Today: Chapter 4, Finalé + Chapter 5, Intro. + You should be doing as many proofs as you can. UCB Philosophy Chapter 4 (Finalé) & Chapter 5 (Intro) 06/16/10 Branden Fitelson Philosophy 12A Notes 2 ' & The Rule of Definition for the Biconditional Rule of Definition for ↔ (Df): If [ (p → q) & (q → p) occurs as the entire formula at line j, then at line k we may write [ p ↔ q , labeling the line ‘j Df’ and writing on its left the same numbers as are on the left of j. Conversely, if [ p ↔ q occurs as the entire formula at a line j, then at line k we may write [ (p → q) & (q → p) , labeling the line ‘j Df’ and writing on its left the same numbers as are on the left of j. a 1 ,. . . , a n (j) (p → q) & (q → p) . . . a 1 ,. . . , a n (k) p ↔ q j Df OR a 1 ,. . . , a n (j) p ↔ q . . . a 1 ,. . . , a n (k) (p → q) & (q → p) j Df UCB Philosophy Chapter 4 (Finalé) & Chapter 5 (Intro) 06/16/10 Branden Fitelson Philosophy 12A Notes 3 ' & \$ % Using ↔ in MacLogic • Using the Definition strategy of MacLogic (accessed via the ... button), we can implement our Df. rule for ↔ . Do not use ↔ I or ↔ E! • Using MacLogic ’s Definition strategy is much simpler than using its Tautology strategy (I did that last time, which was cumbersome). + To get to Definition , first: then . • Here is a non-trivial example: A ↔ ∼ B ` ∼ (A ↔ B) . Let’s try to tackle this one, using MacLogic ’s Definition strategy for our Df. • The shortest proof I’ve been able to find is 18 steps (next slide). Forbes gives a 20-stepper in his discussion of this example (p . 118). UCB Philosophy Chapter 4 (Finalé) & Chapter 5 (Intro) 06/16/10 Branden Fitelson Philosophy 12A Notes 4 ' & Problem is : AÍÒB Ê Ò(AÍB) 1 (1) AÍÒB Ass 2 (2) AÍB Ass 1 (3) (AÁÒB)&(ÒBÁA) 1 Defn. 1 (4) AÁÒB 3 &E 1 (5) ÒBÁA 3 &E 6 (6) B Ass 2 (7) (AÁB)&(BÁA) 2 Defn. 2 (8) BÁA 7 &E 2,6 (9) A 8,6 ÁE 1,2,6 (10) ÒB 4,9 ÁE 1,2,6 (11) Ï 10,6 ÒE 1,2 (12) ÒB 6,11 ÒI 1,2 (13) A 5,12 ÁE 1,2 (14) ÒB 4,13 ÁE 2 (15) AÁB 7 &E 1,2 (16) B 15,13 ÁE 1,2 (17) Ï 14,16 ÒE 1 (18) Ò(AÍB) 2,17 ÒI UCB Philosophy Chapter 4 (Finalé) & Chapter 5 (Intro) 06/16/10 Branden Fitelson Philosophy 12A Notes 5 ' & \$ % Sequent and Theorem Introduction: I • You may have noticed that certain important sequents or theorems tend to get proven over and over again in different problems.tend to get proven over and over again in different problems....
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notes_10_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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