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Lecture #16

# Lecture #16 - Branden Fitelson Philosophy 12A Notes 1& \$...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & \$ % Announcements & Such • Fleet Foxes • Administrative Stuff – Take-Home Mid-Term re-subs 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 mid-term grades are in (both the take-home and the in-class). – Branden will not be holding office hours this week. • Today: Chapter 4 — Natural Deduction Proofs for LSL – We’ll 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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Lecture #16 - Branden Fitelson Philosophy 12A Notes 1& \$...

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