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Proofs Using vOut

Proofs Using vOut. A v B A > G B > G G. A v B. G. 1. V v -V A 2. V > -B A 3. -V > -N A -Bv-N GOAL. Set up the vOut Strategy. Proofs Using vOut. A v B A > G B > G G. A v B. G. 1. V v -V A 2. V > -B A 3. -V > -N A -Bv-N GOAL.

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Proofs Using vOut

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  1. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A -Bv-N GOAL Set up the vOut Strategy

  2. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A -Bv-N GOAL ... by making the missing conditionals new goals.

  3. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A V>(-Bv-N) GOAL -V>(-Bv-N) GOAL -Bv-N 1,?,? vO Prove the top goal using the >I Strategy

  4. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA -Bv-N GOAL V>(-Bv-N) 4-? >I -V>(-Bv-N) GOAL -Bv-N 1,?,? vO Apply >O whenever you can.

  5. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA -B 2,4 >O -Bv-N GOAL V>(-Bv-N) 4-? >I -V>(-Bv-N) GOAL -Bv-N 1,?,? vO ... and your goal comes by vIn

  6. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA 5. -B 2,4 >O 6. -Bv-N 5 vI 7. V>(-Bv-N) 4-6 >I -V>(-Bv-N) GOAL -Bv-N 1,7,? vO Now use >In to obtain your second goal.

  7. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA 5. -B 2,4 >O 6. -Bv-N 5 vI 7. V>(-Bv-N) 4-6 >I 8. -V PA -Bv-N GOAL -V>(-Bv-N) 8-? >I -Bv-N 1,7,? vO Apply >O whenever you can.

  8. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA 5. -B 2,4 >O 6. -Bv-N 5 vI 7. V>(-Bv-N) 4-6 >I 8. -V PA -N 3,8 >O -Bv-N GOAL -V>(-Bv-N) 8-? >I -Bv-N 1,7,? vO ... and your goal comes by vIn

  9. Proofs Using vOut A v B A>G B>G G A v B G 1. V v -V A 2. V>-B A 3. -V>-N A 4. V PA 5. -B 2,4 >O 6. -Bv-N 5 vI 7. V>(-Bv-N) 4-6 >I 8. -V PA 9. -N 3,8 >O 10. -Bv-N 9 vI 11. -V>(-Bv-N) 8-10>I 12. -Bv-N 1,7,11 vO The proof is now complete.

  10. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A (A&B)v(A&C) GOAL Do &Out whenever you can

  11. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O (A&B)v(A&C) GOAL Set up the vOut Strategy

  12. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O (A&B)v(A&C) 3,?,? vO Set the missing conditionals as new goals.

  13. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O B>[(A&B)v(A&C)] GOAL C>[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,?,? vO Use >In to prove the first goal.

  14. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O 4. B PA (A&B)v(A&C) GOAL B>[(A&B)v(A&C)] 4-? >I C>[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,7,? vO Prove A&B so that you can use vIn to obtain the goal.

  15. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O 4. B PA 5. A&B 2,4 &I (A&B)v(A&C) GOAL B>[(A&B)v(A&C)] 4-? >I C>[(A&B)v(A&C)] GOAL (A&B)v(A&C) Now the subproof can be completed with vIn.

  16. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O 4. B PA 5. A&B 2,4 &I 6. (A&B)v(A&C) 5 vI 7. B>[(A&B)v(A&C)] 4-6 >I C >[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,7,? vO Now set up the >In Strategy to prove the second goal.

  17. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O 4. B PA 5. A&B 2,4 &I 6. (A&B)v(A&C) 5 vI 7. B>[(A&B)v(A&C)] 4-6 >I 8. C PA (A&B)v(A&C) GOAL C>[(A&B)v(A&C)] 8-? >I (A&B)v(A&C) 3,7,? vO This subproof can be completed in the same way.

  18. Proofs Using vOut A v B A>G B>G G A v B G 1. A&(BvC) A 2. A 1 &O 3. B v C 1 &O 4. B PA 5. A&B 2,4 &I 6. (A&B)v(A&C) 5 vI 7. B >[(A&B)v(A&C)] 4-6 >I 8. C PA 9. A&C 2,8 &I 10. (A&B)v(A&C) 9 vI 11. C >[(A&B)v(A&C)] 8-10 >I 12. (A&B)v(A&C) 3,7,11 vO The proof is now complete.

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