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Derivation Rules and Justification

This text presents the derivation rules and their justification, including the simple rules and complex rules. It explains how to derive conclusions based on given premises using logical reasoning.

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Derivation Rules and Justification

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  1. Derivation Rules and Justification

  2. m A n A m R The Simple Rules m A n B o A & B n, m &I m A & B n A n &E o B n &E m A B n A o B m,n E m AB n B o A m,n E p B m,o E m A n AB m I o B p AB o I

  3. Reiteration m A n A m R S, AA

  4. Reiteration m A n A m R S, AA S, A S, AA T T T T T T F T F F

  5. Reiteration m A n A m R S, AA S, A S, AA T T T T T T F T F F

  6. &E m A & B n A n &E o B n &E S, A & BA S, A & BB

  7. &E m A & B n A n &E o B n &E S, A & BA S, A & BB S, A, B S, A & BA T T T T T T T T F T F T T F T T F F T F F T F F S, A, B S, A & BB T T T T T T T T F T F F T F T T F T T F F T F F

  8. &E m A & B n A n &E o B n &E S, A & BA S, A & BB S, A, B S, A & BA T T T T T T T T F T F T T F T T F F T F F T F F S, A, B S, A & BB T T T T T T T T F T F F T F T T F T T F F T F F

  9. &I m A n B o A & B n, m &I S, A, BA & B

  10. &I m A n B o A & B n, m &I S, A, BA & B S, A, B S, A, BA & B T T T T T T T T T F T T F F T F T T F T F T F F T F F F

  11. &I m A n B o A & B n, m &I S, A, BA & B S, A, B S, A, BA & B T T T T T T T T T F T T F F T F T T F T F T F F T F F F

  12. I S, AAB S, BAB m A n AB m I o B p AB o I

  13. I S, AAB S, BAB m A n AB m I o B p AB o I S, A, B S, AAB T T T T T T T T F T T T T F T T F F T F F T F F S, A, B S, BAB T T T T T T T T F T F T T F T T T T T F F T F F

  14. I S, AAB S, BAB m A n AB m I o B p AB o I S, A, B S, AAB T T T T T T T T F T T T T F T T F F T F F T F F S, A, B S, BAB T T T T T T T T F T F T T F T T T T T F F T F F

  15. E m A B n A o B m,n E S, A  B, AB

  16. E m A B n A o B m,n E S, A  B, AB S, A, B S, A  B, AB T T T T T T T T T F T F T F T F T T T F T T F F TTF F

  17. E m A B n A o B m,n E S, A  B, AB S, A, B S, A  B, AB T T T T T T T T T F T F T F T F T T T F T T F F TTF F

  18. E m AB n B o A m,n E p B m,o E S, A  B, AB S, B  A, AB

  19. E m AB n B o A m,n E p B m,o E S, A  B, AB S, B  A, AB S, A, B S, A  B, AB T T T T T T T T T F T F T F T F T T F F T T F F T T F F S, A, B S, A  B, BA T T T T T T T T T F T F F T T F T T F T F T F F T T F F

  20. E m AB n B o A m,n E p B m,o E S, A  B, AB S, B  A, AB S, A, B S, A  B, AB T T T T T T T T T F T F T F T F T T F F T T F F T T F F S, A, B S, A  B, BA T T T T T T T T T F T F F T T F T T F T F T F F T T F F

  21. m A Assumption n B o AB m-n I The Complex Rules m A Assumption n B o B p A q AB m-n, o-p I m AB n A Assumption o C p B Assumption q C r C m, n-o, p-q I m A Assumption n B o B p A m-o I m A Assumption n B o B p A m-o E

  22. I m A Assumption n B o AB m-n I If S, A B, then SA  B

  23. I m A Assumption n B o AB m-n I If S, A B, then SA  B S, A, B If S, A B, then SA  B T T T T T T T T T T F T T F T F T F T T F T T T T F F TFF T T

  24. I m A Assumption n B o AB m-n I If S, A B, then SA  B S, A, B If S, A B, then SA  B T T T T T T T T T T F T T F T F T F T T F T T T T F F TFF T T

  25. I m A Assumption n B o B p A q AB m-n, o-p I If S, A B, and S, B A, then SA  B

  26. I m A Assumption n B o B p A q AB m-n, o-p I If S, A B, and S, B A, then SA  B S, A, B If S, A B, and S, B A, then SA  B T T T T T T T T T T T T T F T T F T F T T F T F T T F T T T F T F T F F TFF T F F T T

  27. I m A Assumption n B o B p A q AB m-n, o-p I If S, A B, and S, B A, then SA  B S, A, B If S, A B, and S, B A, then SA  B T T T T T T T T T T T T T F T T F T F T T F T F T T F T T T F T F T F F TFF T F F T T

  28. m AB n A Assumption o C p B Assumption q C r C m, n-o, p-q I E If S, AB, A C, and S, AB, B C, then S, ABC

  29. m AB n A Assumption o C p B Assumption q C r C m, n-o, p-q I E If S, AB, A C, and S, AB, B C, then S, ABC S, A, B, C If S, AB, A C, and S, AB, B C, then S, ABC T T T T T T T T T T T T T T T T T T F T T T F T T T F T T F T T F T T T T T T T F T T T T T T F F T T T F T T F F T T F T F T T T T F T T T T T T T T T F T F T T F F T T T F T T F T F F T T F F T T F F T T F T T F F F T F F F T F F F T F F

  30. m AB n A Assumption o C p B Assumption q C r C m, n-o, p-q I E If S, AB, A C, and S, AB, B C, then S, ABC S, A, B, C If S, AB, A C, and S, AB, B C, then S, ABC T T T T T T T T T T T T T T T T T T F T T T F T T T F T T F T T F T T T T T T T F T T T T T T F F T T T F T T F F T T F T F T T T T F T T T T T T T T T F T F T T F F T T T F T T F T F F T T F F T T F F T T F T T F F F T F F F T F F F T F F

  31. ¬I m A Assumption n B o B p A m-o I If S, A B and S, A B, then SA

  32. ¬I m A Assumption n B o B p A m-o I If S, A B and S, A B, then SA S, A, B If S, A B and S, A B, then SA T T T T T T T T F T F T T F T T F T T T T F T F T T F T T F F T T T F F TFF T F T T T

  33. ¬I m A Assumption n B o B p A m-o I If S, A B and S, A B, then SA S, A, B If S, A B and S, A B, then SA T T T T T T T T F T F T T F T T F T T T T F T F T T F T T F F T T T F F T F F T F T T T

  34. ¬E m A Assumption n B o B p A m-o E If S, A B and S, A B, then SA

  35. ¬E m A Assumption n B o B p A m-o E If S, A B and S, A B, then SA S, A, B If S, A B and S, A B, then SA T T T T T T T T F T F T T F T T F T T T T F T F T T F T T F F T T T F F TFF T F T T T

  36. ¬E m A Assumption n B o B p A m-o E If S, A B and S, A B, then SA S, A, B If S, A B and S, A B, then SA T T T T T T T T F T F T T F T T F T T T T F T F T T F T T F F T T T F F TFF T F T T T

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