Examples of Deduction
"Prove" axiom 1:
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- P 1. hypothesis
- Q 2. hypothesis
- P 3. reiteration of 1
- Q→P 4. deduction from 2 to 3
- P 1. hypothesis
- 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 2. hypothesis
- (P→Q)→(P→R) 8. deduction from 2 to 7
- P→(Q→R) 1. hypothesis
- (P→(Q→R))→((P→Q)→(P→R)) 9. deduction from 1 to 8 QED
Using axiom 1 to show ((P→(Q→P))→R)→R:
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- (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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