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.
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- 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 3. hypothesis
- (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 2. hypothesis
- ((P→Q)→Z)→Z 12. deduction from 2 to 11
- P→Z 1. hypothesis
- (P→Z)→((P→Q)→Z)→Z) 13. deduction from 1 to 12 QED
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