Peirce's Law - Using Peirce's Law With The Deduction Theorem

Using Peirce's Law With The Deduction Theorem

Peirce's law allows one to enhance the technique of using the deduction theorem to prove theorems. Suppose one is given a set of premises Γ and one wants to deduce a proposition Z from them. With Peirce's law, one can add (at no cost) additional premises of the form Z→P to Γ. For example, suppose we are given P→Z and (P→Q)→Z and we wish to deduce Z so that we can use the deduction theorem to conclude that (P→Z)→(((P→Q)→Z)→Z) is a theorem. Then we can add another premise Z→Q. From that and P→Z, we get P→Q. Then we apply modus ponens with (P→Q)→Z as the major premise to get Z. Applying the deduction theorem, we get that (Z→Q)→Z follows from the original premises. Then we use Peirce's law in the form ((Z→Q)→Z)→Z and modus ponens to derive Z from the original premises. Then we can finish off proving the theorem as we originally intended.

    • P→Z 1. hypothesis
      • (P→Q)→Z 2. hypothesis
        • Z→Q 3. hypothesis
          • P 4. hypothesis
          • Z 5. modus ponens using steps 4 and 1
          • Q 6. modus ponens using steps 5 and 3
        • P→Q 7. deduction from 4 to 6
        • Z 8. modus ponens using steps 7 and 2
      • (Z→Q)→Z 9. deduction from 3 to 8
      • ((Z→Q)→Z)→Z 10. Peirce's law
      • Z 11. modus ponens using steps 9 and 10
    • ((P→Q)→Z)→Z 12. deduction from 2 to 11
  • (P→Z)→((P→Q)→Z)→Z) 13. deduction from 1 to 12 QED

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