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Tuesday, July 24, 2007 10:26 AM The general form of the indirect derivation. The first form is to find a negation by assuming the nonnegated. SHOW: ~A ID A As SHOW: X DD ... ... ... X 1) 2) 3) 4) 5) 6) 7) 8) PQ Pr Q~P Pr SHOW:~P ID P As SHOW:X DD Q 1,4,O ~P 2,6,O X 4,7,XI 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) PQ Pr ~PQ Pr SHOW: Q DD SHOW:~~Q ID ~Q As SHOW:X DD ~P 1,5,O ~~P 2,5,O X 7,8,XI Q 4,DN 1) 2) 3) 4) 5) 6) 7) 8) ~(P&~Q) Pr SHOW: PQ CD P As SHOW:Q ID ~Q As SHOW:X DD P&~Q 3,5,&I X 1,7,XI The second form of ID is best used for atomic statements and disjunctions (A v B). 1) ~PQ Pr Intro to Logic Page 1 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) SHOW: P v Q ID ~(P v Q) As SHOW: X DD SHOW:~P ID P As SHOW:X DD P v Q 6,vI X 3,8,XI Q 1,5,O P v Q 10,vI X 3,11,XI New inference rule. ~vO ~(A v B) __________ ~A ~B 1) 2) 3) 4) 5) 6) 7) 8) ~PQ Pr SHOW: P v Q ID ~(P v Q) As SHOW: X DD ~P 3,~vO ~Q 3,~vO Q 1,5,O X 6,7,XI When dealing with derivations of disjunctions we use indirect derivation 2 and then in the SHOW:X section we use ~vO. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) (P v Q)(P&Q) Pr SHOW: (P&Q) v (~P&~Q) ID ~((P&Q) v (~P&~Q)) As SHOW:X DD ~(P&Q) 3,~vO ~(~P&~Q) 3,~vO ~(P v Q) 1,5,O ~P 7,~vO ~Q 7,~vO (~P&~Q) 8,9,&I X 6,10,XI Conjuctions and biconditionals SHOW:A&B SHOW:A Intro to Logic Page 2 ... ... SHOW:B ... ... A&B &I 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) SHOW:AB SHOW:AB ... ... SHOW:BA ... ... AB I (A v B) C Pr SHOW:(AC) & (BC) DD SHOW: (AC) CD A As SHOW:C DD A v B 4,vI C 1,6,O SHOW: BC CD B As SHOW: C DD A v B 9,vI C 1,11,O (AC) & (BC) 3,8,&I Conditionals Negations Disjunctions Conjunctions Biconditionals CD ~D vD &D D Atomic formulas ID Intro to Logic Page 3 ...
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 Spring '08
 Morgan
 Logic, ... ..., Disjunctions Conjunctions Biconditionals, DD ~P 1,5,O, ~Q 3,5

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