Deduction Theorem - Examples of Deduction

Examples of Deduction

"Prove" axiom 1:

    • P 1. hypothesis
      • Q 2. hypothesis
      • P 3. reiteration of 1
    • Q→P 4. deduction from 2 to 3
  • P→(Q→P) 5. deduction from 1 to 4 QED

"Prove" axiom 2:

    • P→(Q→R) 1. hypothesis
      • P→Q 2. hypothesis
        • P 3. hypothesis
        • Q 4. modus ponens 3,2
        • Q→R 5. modus ponens 3,1
        • R 6. modus ponens 4,5
      • P→R 7. deduction from 3 to 6
    • (P→Q)→(P→R) 8. deduction from 2 to 7
  • (P→(Q→R))→((P→Q)→(P→R)) 9. deduction from 1 to 8 QED

Using axiom 1 to show ((P→(Q→P))→R)→R:

    • (P→(Q→P))→R 1. hypothesis
    • P→(Q→P) 2. axiom 1
    • R 3. modus ponens 2,1
  • ((P→(Q→P))→R)→R 4. deduction from 1 to 3 QED

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