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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such Fleet Foxes Administrative Stuff TakeHome MidTerm resubs are due Thursday. + When you turn in resubmissions, make sure that you staple them to your original homework submission . I will be discussing the grade curve for the course as soon as all of the midterm grades are in (both the takehome and the inclass). Branden will not be holding office hours this week. Today: Chapter 4 Natural Deduction Proofs for LSL Well be done with the LSL natural deduction rules for ` this week. 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 03/16/10 Branden Fitelson Philosophy 12A Notes 2 ' & $ % The Elimination Rule for Rule of Elimination : For any formula q , if [ q has been inferred at a line j in a proof and q at line k (j < k or j > k) then we may infer at line m, labeling the line j, k E and writing on its left the numbers on the left at j and on the left at k. Schematically (with j < k): a 1 ,..., a n (j) q . . . b 1 ,..., b u (k) q . . . a 1 ,..., a n , b 1 ,..., b u (m) j, k E Note: we have added the symbol to the language of LSL. It is treated as if it were an atomic sentence of LSL. We can now use it in compound sentences ( e.g. , A , , etc .). UCB Philosophy Chapter 4 03/16/10 Branden Fitelson Philosophy 12A Notes 3 $ % The Introduction Rule for Rule of Introduction : If has been inferred at line k in a proof and {a 1 ,..., a n } are the assumption and premise numbers depends upon, then if p is an assumption (or premise) at line j, [ p may be inferred at line m, labeling the line j, k I and writing on its left the numbers in the set {a 1 ,..., a n }/j. j (j) p Assumption . . . a 1 ,..., a n (k) . . . {a 1 ,..., a n }/j (m) p j, k I I is used (typically with E) to deduce [ p via reductio ad absurdum , by ( i ) assuming p , ( ii ) deducing , and ( iii ) discharging the assumption. UCB Philosophy Chapter 4 03/16/10 Branden Fitelson Philosophy 12A Notes 4 ' & $ % The Rule of Double Negation (DN) Negation is an odd connective in our system. It not only has an introduction rule and an elimination rule, but it also has an additional rule called the double negation (DN) rule. The DN rule says that we may infer p from [ p . Without this DN rule, we would not be able to prove certain valid LSL argument forms e.g. , (A & B) (A B) . Rule of Double Negation : For any formula p , if [ p has been inferred at a line j in a proof, then at line k we may infer p , labeling the line j and writing on its left the numbers to the left of j....
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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 University of California, Berkeley.
 Spring '10
 FITELSON

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