# Negation elimination rule in natural deduction

1. May 22, 2008

### drguildo

Can somebody please explain the negation elimination rule in natural deduction to me? I've read a few explanations and none of them make any sense whatsoever. Nor do I understand how you can (or why you would want to) be able to infer anything from a contradiction.

Thanks.

2. May 22, 2008

### Kittel Knight

Suppose you have a set of preestablished facts.
If a new assumption leads you to a contradiction, then it is false.

3. May 23, 2008

### drguildo

Then surely that would just reinforce the original set of facts? Why should you be able to deduce anything from that contradiction?

4. May 23, 2008

### CRGreathouse

I take it you've never played sudoku?

5. May 23, 2008

### Kittel Knight

Consider the pertinent preestablished facts , also known as axioms or theorems :
Axiom 1 : "If a guy knows Logic then he understands my first post."
Axiom 2 : "Peter doesn't understand my first post"

Let's make the assumption Peter knows Logic.

So, since Peter is a guy and we are assuming he knows Logic, then, according to the Axiom 1, Peter understands my first post.

However, according to the Axiom 2, Peter doesn't understand my first post - which is a contradiction!

So, our assumption is false, and the final conclusion is that Peter doesn't know Logic.

Last edited: May 23, 2008
6. May 23, 2008

### drguildo

I have played Sudoku, I don't see the relationship though. Especially not when it comes to being able to deduce anything from a contradiction.

7. May 23, 2008

### drguildo

Thanks; that's a good intuitive example of how contradiction can be a useful tool in reasoning, but I think I'm being misunderstood when I say you can conclude "anything" from a contradiction. I literally mean anything. For example, the book I'm reading (Proof and Disproof in Formal Logic) states in regard to contradiction:

8. May 23, 2008

### exk

Sudoku provides with a set of numbers that whose position is predefined. If you try to place a number in a manner that breaks the rules of sudoku you receive a contradiction, which leads to the conclusion that placing the number in that manner is impossible.

9. May 23, 2008

### drguildo

Indeed, if a square has to be one of X, Y or Z, and you know it isn't X or Z, then it must be Y (queue Sherlock Holmes quote). But I don't understand how this relates to the formal rule of negation elimination and it's relationship with contradiction which states that should you encounter a contradiction, then you can conclude whatever you want.

10. May 23, 2008

### exk

The definition you posted above states that in an "impossible" situation you can do that. Consider a situation with three 5s in one square in sudoku. If you encounter that situation you can conclude that the placement of the numbers was incorrect to tidy it up, or you can conclude that this is some special type of sudoku, or that the moon is made of cheese (if that somehow helps clear up the situation).

11. May 23, 2008

### LukeD

Let's say you have a few statements
P,Q,R
all of which you know as a fact are true.

Then $$P \Rightarrow Q$$ is a true statement. However,$$\neg P \Rightarrow Q$$ is also a true statement. In fact, $$\neg P \Rightarrow \neg Q$$ is also a true statement.

This is because in classical logic, you take it as an axiom that if you have statement of the form $$P \Rightarrow Q$$ and P is false, then the statement is true no matter what Q is. This is just basically taken as true so that the logic is consistent and you cover all of your bases as far as what sort of statements you can conclude from a false statement.

Let's say you have the statement $$1=2$$. Using the usual definitions of 1 and 2, this is a false statement. However, from this statement, you can conclude that $$2=1$$, and using both of these together with the transitive property, you can conclude that $$1 = 1$$, so you were able to conclude 1 false thing as well as 1 true thing.

In principle, however, if you are using "correct logic", then it is not possible to conclude "everything" from a false statement. It is just that in classical logic, concluding anything from a false statement is considered valid. There are other logical systems though where this is rejected. Try looking up paralogical systems in Wikipedia.

Last edited: May 23, 2008
12. May 23, 2008

### drguildo

What I don't get is how it's used to validate certain inferences. My understanding is that any path of reasoning that leads to a contradiction is fallacious and should be abandoned with nothing inferred from it except the opposite case (at least in classical logic which has the law of excluded middle).

For example arguing by cases in or-elimination. I don't know how to write it out in TeX and I would probably do a bad job of explaining it so I've attached a small snippet in the form of an image. I hope this is OK.

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13. May 23, 2008

### LukeD

Sorry, I meant paraconsistent logic

14. May 23, 2008

### Kittel Knight

Well, there is nothing wrong in saying "If Marilyn Monroe kisses me tonight, then I will fly for 2 hours".
This is a true statement, isn't?
Of course, since we know MM is dead, I can say any thing as "the consequence" of being kissed by her.
In fact, I could say "then the Sun will disappear instantaneously", and the statement would be true, again.
Agreed?

15. May 23, 2008

### drguildo

So we're not really inferring anything from Marilyn Monroe kiss scenario, it's just that we can state whatever we like because it's impossible (cf. contradiction)?

16. May 23, 2008

### CRGreathouse

Now hang on. I'm responding to your question "Why should you be able to deduce anything from that contradiction?" (1). That anything can be deduced from a contradiction (2) is a different principle:

1. Given A, suppose B. A and B imply not A. Thus, not B.
2. Given A and B. A and B imply not A. Thus, ___.

In Sudoku, there are many situations where I try a number and find that it forces a contradiction. This allows me to conclude that the number I chose is wrong.

17. May 24, 2008

### Kittel Knight

Perfectly !

18. May 24, 2008

### drguildo

Okay, but this raises the question what is the point in deducing something from a situation that is never going to be able to arise in order to lead to it? I also still don't understand how it can be used to justify or-elimination.

19. May 25, 2008

### Kittel Knight

Now,suppose MM really kisses me tonight... off course 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!

20. May 25, 2008

### drguildo

Yes, we've already been over that. What I want to know is what's the point.