Timeline for Reductio ad absurdum or the contrapositive?
Current License: CC BY-SA 2.5
18 events
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| Jan 20, 2010 at 16:05 | comment | added | compguy | The original question mentions using contraposition to infer (P → Q) from (¬Q → ¬P)--this clearly relies on classical logic/PEM. To me, this would "turn negative statements into positive statements", as would a proof by reductio of (P → Q). But perhaps you meant something different here. | |
| Jan 20, 2010 at 15:07 | comment | added | Charles Stewart | @compguy: Equivalent, yes, with some room needed for equivocation. The point of PEM/RAA is normally to turn negative statements into positive statements. The normal axiom of contraposition can't do that. It's the "trivially" I was objecting to; I would not have objected to "they're basically the same, if you look at them this way". | |
| Jan 20, 2010 at 13:27 | history | edited | compguy | CC BY-SA 2.5 |
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| Jan 20, 2010 at 13:24 | comment | added | Joel David Hamkins | I agree with your edit, about pushing the contradictory context to the end. When successful, this method essentially turns the proof into a direct proof, with the advantages that I mention for the intermediary statements, and an additional proof by contradiction at the end. | |
| Jan 20, 2010 at 13:11 | history | edited | compguy | CC BY-SA 2.5 |
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| Jan 20, 2010 at 12:43 | comment | added | compguy | No, they are still equivalent. In the absence of the EM, both proof methods only apply to negative statements: positive statements cannot be proven either by contradiction or by a contrapositive proof. | |
| Jan 20, 2010 at 9:42 | comment | added | Charles Stewart | -1: Misleading: they are equivalent only in the presence of the excluded middle, which is to say that they are not trivially equivalent. | |
| Jan 20, 2010 at 1:32 | history | edited | compguy | CC BY-SA 2.5 |
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| Jan 20, 2010 at 1:22 | vote | accept | Harry Gindi | ||
| Jan 20, 2010 at 9:58 | |||||
| Jan 20, 2010 at 0:54 | comment | added | Harry Gindi | Yeah, I just posted a similar idea for the transformation in the comments in the answer below. | |
| Jan 20, 2010 at 0:47 | history | edited | compguy | CC BY-SA 2.5 |
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| Jan 20, 2010 at 0:39 | comment | added | Harry Gindi | I'm reserving judgement on that argument until all of the facts come out. | |
| Jan 20, 2010 at 0:36 | comment | added | Anweshi | @Harry. Yes that is true. But there are genuine examples of ad absurdum, like the Cantor proof. | |
| Jan 20, 2010 at 0:12 | comment | added | Harry Gindi | It's just that 80-90% of the proofs I've done by contradiction have the contrapositive proof stuck inside somewhere. | |
| Jan 20, 2010 at 0:10 | history | edited | compguy | CC BY-SA 2.5 |
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| Jan 19, 2010 at 23:55 | comment | added | Anweshi | Ah, F G Dorais has already written it below. I had not seen it yet when I was typing the above comment. Sorry. | |
| Jan 19, 2010 at 23:52 | comment | added | Anweshi | This works only for "P implies Q" types of statements. What do you have to say about "P is not true" type of statements? Here specifically I have the Cantor's proof that a set and its power set are not equivalent. | |
| Jan 19, 2010 at 23:37 | history | answered | compguy | CC BY-SA 2.5 |