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notes_21_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such • Peter Gabriel : In Your Eyes • Administrative Stuff – I have posted the sample in-class mid-term. Same structure as actual mid-term. The mid-term rules/TT handout is also posted. – The Take-Home is due today , with resubmissions due next 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. • Today: Chapter 4 — Natural Deduction Proofs for LSL – Our natural deduction system for LSL. * Proofs by contradiction ( reductio ad absurdum ), and the ∼ rules. * The ∨ rules. * Next: Sequent and Theorem Introduction (derived rules). – MacLogic — a useful (free!) computer program for proofs. UCB Philosophy Chapter 4 10/17/08 Branden Fitelson Philosophy 12A Notes 2 ' & Proofs by Contradiction and the Rules for ∼ • If assuming p leads us to a contradiction, then we may infer [ ∼ p . [Note: This was implicit in our “short” truth-table method.] • This style of proof is called proof by contradiction (or reductio ad absurdum ). It is a very powerful technique that we’ll see often. • In our natural deduction system, the introduction and elimination rules for negation ( ∼ I and ∼ E) allow us to perform reductio s. • We use the symbol ‘ ’ to indicate that a contradiction has been deduced ( i.e. , that both p and [ ∼ p have been deduced, for some p ). We call ‘ ’ the absurdity symbol . [added to the lexicon of LSL] • With these preliminaries out of the way, we’re ready to see what the negation rules look like, and how they work. . . UCB Philosophy Chapter 4 10/17/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % 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 10/17/08 Branden Fitelson Philosophy 12A Notes 4 ' & 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....
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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 Berkeley.

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notes_21_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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