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 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:
-
- (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
Read more about this topic: Deduction Theorem
Famous quotes containing the words examples of and/or examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)