Negation elimination rule in natural deduction

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Discussion Overview

The discussion revolves around the negation elimination rule in natural deduction, exploring its implications and the reasoning behind inferring conclusions from contradictions. Participants express confusion about the rule's application and its relevance in logical reasoning, particularly in relation to classical logic and examples like Sudoku.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants seek clarification on the negation elimination rule and express difficulty in understanding its implications.
  • One participant suggests that if a new assumption leads to a contradiction, it indicates that the assumption is false, reinforcing preestablished facts.
  • Another participant questions the validity of deducing anything from a contradiction, seeking examples to illustrate the point.
  • A participant provides an example involving axioms about understanding logic, leading to a contradiction that suggests a conclusion about knowledge of logic.
  • Some participants draw analogies to Sudoku, discussing how contradictions in the game lead to conclusions about incorrect placements.
  • There is mention of classical logic allowing conclusions from false statements, while other logical systems may reject this notion.
  • One participant expresses skepticism about the utility of deducing conclusions from impossible scenarios, questioning its relevance in logical reasoning.
  • Several participants agree that while one can state anything from a contradiction, it does not imply meaningful inference.

Areas of Agreement / Disagreement

Participants express a mix of agreement and disagreement regarding the implications of contradictions in logical reasoning. Some acknowledge the ability to conclude anything from a contradiction, while others question the practical utility and validity of such conclusions.

Contextual Notes

Participants highlight the limitations of their examples and reasoning, noting that the discussion involves various interpretations of logical principles and the application of negation elimination in different logical systems.

  • #31
FernBarc said:
You should read post #19 again...:cry:

(In fact, you should read all this thread again and again, and again...)

Why? Is the explanation I pasted incorrect?
 
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  • #32
drguildo, you asked what the benefit was to the principle of explosion. I told you it was because proofs by contradiction necessarily show the principle of explosion. Then you post a proof of the principle of explosion (by disjunctive introduction and quantifier elimination) and somehow this explains it for you? I thought you were after a reason, not a proof -- and haven't people posted proofs already?

FernBarc said:
You should read post #19 again...:cry:

(In fact, you should read all this thread again and again, and again...)

Actually, I don't think there's any special benefit to reading post #19. That essentially explains what explosion is, not why it's true or why we care.
 
  • #33
drguildo said:
Why? Is the explanation I pasted incorrect?
The both explanations are the samething.
 
  • #34
CRGreathouse said:
drguildo, you asked what the benefit was to the principle of explosion. I told you it was because proofs by contradiction necessarily show the principle of explosion. Then you post a proof of the principle of explosion (by disjunctive introduction and quantifier elimination) and somehow this explains it for you? I thought you were after a reason, not a proof -- and haven't people posted proofs already?

The explanation in your post was fine and answered my question, but only in the context of a proof of the principle of explosion which you said was contained in a previous post of yours, but which I couldn't (and still can't) "see". Once I read the proof I cited then it clicked. Not all explanations are equal and just because one (or two, or three) makes perfect sense to people who understand the concepts already, doesn't mean it will to somebody who doesn't.

I'm obviously still learning so it's probable that I have used certain terms in the wrong context and in an imprecise manor which has lead to wires getting crossed. I'm also searching around and making small bits of progress here and there and so things that didn't make sense earlier make sense now.
 
  • #35
CRGreathouse said:
Actually, I don't think there's any special benefit to reading post #19. That essentially explains what explosion is, not why it's true or why we care.

Yes, there is nothing new in post #19. But it does more than explaining explosion: it proves it is true.
I've mentioned post #19 cause it was the last post explaining (or "proving", once more) the same as post #26.
And, if it is clearly proved, there is no need to "explain in other words".
 
  • #36
FernBarc said:
The both explanations are the samething.
Hehehe, I figured you held this fundamentally incorrect view. I suppose it's what leads you to post comments (unless I misinterpreted you) such as #28. That is, if somebody does not understand one (or two, three etc.) explanation of something, then they cannot understand any explanation of it and should give up.

I shan't bother explaining why this isn't the case as if you haven't realized it by now then, according to you, you never will.
 
  • #37
Post #19:
Kittel Knight said:
...this is a contradiction! And thanks to that, it is true I will fly for 2 hours.
As you see, if there is a contradiction, then we can conclude anything we want!


Post #29:
drguildo said:
I found this nice explanation on Everything2:
...
We conclude that if I am and am not the Pope, then roses are blue.


Post #33:
FernBarc said:
The both explanations are the samething.
Agreed!


Post #36:
drguildo said:
I figured you held this fundamentally incorrect view.
Oops! Those explanations really are equivalents!
I hope some day you understand this...
 
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  • #38
Kittel Knight said:
Oops! Those explanations really are equivalents!
I hope some day you understand this...

Even if the intended meanings are equivalent, then thanks in part to the ambiguity of natural language and differences in interpretation, the end results certainly aren't. I hate to break it to you but your command of English isn't that great and I found the explanation I pasted much easier to parse.

Nice try removing the bulk of the second quote, though. You know, the bit that actually explained why.
 
  • #39
drguildo said:
Even if the intended meanings are equivalent, then thanks in part to the ambiguity of natural language and differences in interpretation, the end results certainly aren't. I hate to break it to you but your command of English isn't that great and I found the explanation I pasted much easier to parse.

Nice try removing the bulk of the second quote, though. You know, the bit that actually explained why.

Yes, I am very sorry you have difficulties in natural language interpretation.

That's why I've removed the bulk of the first and second quotes. You know, the bit where trivial logic is used to prove those "mysteries".

Anyway, don't believe FernBarc: even if it seems too hard to understand this kind of things, keep reading your book again, and again...and again!

Don't give up: some day you will finally learn logic, and this "chimera" will be a distant past...

:smile:
 
  • #40
Removed post.. late night... bad posting time...
 
Last edited:

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