1 / 18

Advanced Proof Techniques Using the vOut Strategy in Logic

This article discusses the application of the vOut and vIn strategies in logical proofs, particularly focusing on establishing goals and managing missing conditionals through new goal setups. It outlines specific proof methods for demonstrating implications, considering various logical conditions and connecting premises effectively. The guide walks through step-by-step proofs using the framework of logical inequalities and conditional statements, ensuring a comprehensive understanding of how to structure arguments and reach valid conclusions.

duer
Télécharger la présentation

Advanced Proof Techniques Using the vOut Strategy in Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related