notes_9_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: Neil Young • 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, Continued – More on ∼ and ∨ rules. Sequent & Theorem Introduction (SI/TI). + You should be doing as many proofs as you can. UCB Philosophy Chapter 4 (Cont’d) 06/15/08 Branden Fitelson Philosophy 12A Notes 2 ' & The Three Negation Rules Negation Elimination ( ∼ E) a 1 ,. . . , a n (j) ∼ q . . . b 1 ,. . . , b u (k) q . . . a 1 ,. . . , a n , b 1 ,. . . , b u (m) j, k ∼ E Negation Introduction ( ∼ I) j (j) p Assumption . . . a 1 ,. . . , a n (k) . . . {a 1 ,. . . , a n }/j (m) ∼ p j, k ∼ I Double Negation (DN) a 1 ,. . . , a n (j) ∼∼ p a 1 ,. . . , a n (k) p j DN UCB Philosophy Chapter 4 (Cont’d) 06/15/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % The Two Disjunction Rules a 1 ,. . . , a n (j) p a 1 ,. . . , a n (j) q . . . OR . . . a 1 ,. . . , a n (k) p ∨ q j ∨ I a 1 ,. . . , a n (k) p ∨ q j ∨ I a 1 ,. . . , a n (g) p ∨ q . . . h (h) p Assumption . . . b 1 ,. . . , b u (i) r . . . j (j) q Assumption . . . c 1 ,. . . , c w (k) r . . . A (m) r g, h, i, j, k ∨ E [ A = {a 1 ,. . . , a n } ∪ {b 1 ,. . . , b u }/h ∪ {c 1 ,. . . , c w }/j] UCB Philosophy Chapter 4 (Cont’d) 06/15/08 Branden Fitelson Philosophy 12A Notes 4 ' & A Simple Example Involving ∨ I and ∨ E • Here’s a proof of the sequent: A ∨ B ` B ∨ A . Problem is: AÎB Ê BÎA 1 (1) AÎB„„„ÊÊÊÊÊÊÊÊÊÊPremise 2 (2) A„„„„ÊÊÊÊÊÊÊÊÊÊÊÊAssumption (ÎE) 2 (3) BÎA„„„ÊÊÊÊÊÊÊÊÊÊ2 ÎI 4 (4) BÊÊÊÊÊÊÊÊÊÊÊÊÊAssumption (ÎE) 4 (5) BÎA„„„ÊÊÊÊÊÊÊÊÊÊ4 ÎI 1 (6) BÎA„„„ÊÊÊÊÊÊÊÊÊÊ1,2,3,4,5 ÎE UCB Philosophy Chapter 4 (Cont’d) 06/15/08 Branden Fitelson Philosophy 12A Notes 5 ' & $ % An Example Involving ∨ E and DN • Here’s a proof of the sequent: A ∨ B, ∼ B ` A . Problem is: AÎB, ÒB Ê A 1 (1) AÎB„„ÊÊÊÊÊÊÊÊÊÊPremise 2 (2) ÒB„„ÊÊÊÊÊÊÊÊÊÊÊPremise 3 (3) ÒA„„ÊÊÊÊÊÊÊÊÊÊÊAssumption (for ÒI) 4 (4) A„„ÊÊÊÊÊÊÊÊÊÊÊÊAssumption (for ÎE) 3,4 (5) ÏÊÊÊÊÊÊÊÊÊÊÊÊ3,4 ÒE 6 (6) B„„„ÊÊÊÊÊÊÊÊÊÊÊÊAssumption (for ÎE) 2,6 (7) ÏÊÊÊÊÊÊÊÊÊÊÊÊ2,6 ÒE 1,2,3 (8) ÏÊÊÊÊÊÊÊÊÊÊÊÊ1,4,5,6,7 ÎE 1,2 (9) ÒÒA„ÊÊÊÊÊÊÊÊÊÊ3,8 ÒI 1,2 (10) A„„ÊÊÊÊÊÊÊÊÊÊÊÊ9 DN UCB Philosophy Chapter 4 (Cont’d) 06/15/08 Branden Fitelson...
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