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ln2-page9 - 1 2 3 4 5 S(M N S(S P SHOW M N S S&P 7 2 Pr 3...

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1. 2. 3. 4. 5. 6. 7. 8. 9. S ! ( M _ N ) S ! ( S & P ) SHOW: M _ N ( M _ N ) SHOW: 6 S S & P S 6 Pr Pr ID Ass DD 1,4 ! O 2,6 ! O 8 &O 6,8 6 I We can do indirect derivations within conditional derivations (or, more rarely, vice-versa). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. P ! [ Q _ ( R & P )] SHOW: P ! Q P SHOW: Q Q SHOW: 6 Q _ ( R & P ) R & P P 6 Pr CD Ass ID Ass DD 1,3 ! O 5,7 _ O 8 &O 3,9 6 I 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. R ! ⇠ [ P ! ( P & Q )] Q SHOW: R R SHOW: 6 [ P ! ( P & Q )] SHOW: P ! ( P & Q ) P SHOW: P & Q P & Q 6 Pr Pr ID Ass DD 1,4 ! O CD Ass DD 2,8 &I 6,7 6 I (Remember we can introduce SHOW lines whenever we wish.) Often, it is helpful to do two indirect derivations in order to prove that a conjunction ( and -statment) is true. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. A ! ( B & B ) C _ [( D _ ⇠ C ) ! C ] SHOW: C & A SHOW: C C SHOW: 6 ( D _ ⇠ C ) ! C D _ ⇠ C C 6 SHOW: A A SHOW: 6 B & B B B 6 C & A Pr Pr DD ID Ass DD 2,5 _ O 5 _ I 7,8 ! O 5,8 6 I ID Ass DD 1,12 ! O 14 &O 14 &O 15,16 6 I 4,11 &I G. New Direct Derivation Rules: The Negation Rules Just using the rules we have already learned, we can
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