Thursday, February 28, 2008

The Law Of Non-Contradiction

In logic, the law of non-contradiction is an important principle. This law states that no instance in the same respect can be in two or more contradicting states. Contradicting states can be defined as mutually exclusive states, or disjoint states.

In boolean logic, the not operator, -also called the negation operator- is defined as the operator which evaluates true as false, and false as true, where true and false are by definition -in the context of boolean logic- two contradicting states.

The law of non-contradiction is a concept that holds several types of logic from dissolving into trivial systems. Even higher forms of logical systems like set theory and mathematics would dissolve into trivial systems if the law of non-contradiction is not taken as part of its axioms.

Now I will try to demonstrate why the law of non-contradiction is important. First in logic. And for people who are not familiar with logic notation, I will then demonstrate it using mathematics. And finally, the importance of the law of non-contradiction for language and communication of ideas in general.

In Logic:
~: The logical negation
^: Logical AND
v: Logical OR

The law of non-contradiction states:
p: any statement
p ^ ~p = False

Denying the law of non-contradiction:
j: is a statement that defies the law of non-contradiction
j ^ ~j = True

The law of non-contradiction prohibits the existence of such statement as j. Lets see what happens if a statement like j actually existed:
= p v False
= p v ~True
= p v ~(j ^ ~j) ; Denying the law of non-contradiction
= p v (~j v j) ; By DeMorgan's rule
= j v True ; Tautology
= True ; Tautology

= ~p v False
= ~p v ~True
= ~p v ~(j ^ ~j)
= ~p v (~j OR j)
= ~p v True
= True

The result is that all logical statements are evaluated as True. Even a statement and its contradiction became both true.

In Mathematics:
The law of non-contradiction would say that two different numbers are different.
IF a is always not equal to b THEN a is never equal to b

Lets say that there exists only one exception to this rule. Lets say: "1=2". Denying the law of non-contradiction would mean that a statement like "1=2" can be true. The law of non-contradiction prohibits such statements.

Lets now deny the law of non-contradiction and say: "1=2".... Now we can simply prove that any number is equal to any other number!
Consider this:
1=2 ; premise

= 2 + 3
= 1 + 3 ; premise
= 4 ; Result 1 (5=4)

= 5 + 2 + 13
= 4 + 2 + 13 ; Result 1 (5=4)
= 4 + 1 + 13 ; premise
= 18 ; Result 2 (20=18)
= 2 + 16
= 1 + 16 ; premise
= 17 ; Result 3 (20=17) and (18=17)

= 1 - 1
= 2 - 1
= 1 ; Result 4 (0=1)

Practically, by having ONLY ONE such an exception to the law of non-contradiction, as "1=2"... We can show that ALL other numbers are equal to one another! Not only for integer, but also for any other number. Observe:
For rational numbers:
= 10 / 20

We can prove that 10=200, and that 20=13

So we get:
= 10 / 20
= 200 / 13

For irrational numbers:
SQR: Square root
SQR 20
= SQR 50
= SQR 901

Now someone might say, that the operator of addition (+) is the problem, and that saying "20 = 1 + 19" is the problem. Fine, if we can't do it with addition, we do it with some other operators. Observe:
Using multiplication:
= 2 * 3
= 1 * 3 ; premise
= 3

Using power:
^: The power operator
= 4 ^ 2
= 4 ^ 1 ; premise
= 4

The end result, that to protect mathematics from dissolving by one such instance where the law of non-contradiction does not hold, you must deny numerous other operators that can be used to dissolve the system, including -among others-: Addition, subtraction, multiplication, division, powers, logarithms... etc.

In short, the whole concept of algebraic manipulation needs to be abandoned to hold the system from dissolving into a trivial system.

This would mean heaven for lazy students who hate math and science. Because this means that any answer they put in their exams is necessarily correct!! All students now get 20 out of 20. And its no problem if you get 0 out of 20, because 20 equals 0! So if you are a lazy student, I suggest you start a campaign with the slogan "Say NO to the law of non-contradiction!"...

Communication and Language:
Denying the law of non-contradiction makes communication of ideas very difficult. To deny the law of non-contradiction is to say that: "To say something, and to negate that same something are two compatible views"...

Funny dialogs can result from this assertion. Consider this dialog, between two people: A and B... Hegel is a philosopher who -is said to have- denied the law of non-contradiction, and this fictional dialog was assembled to demonstrate how problematic this denial is.

A: Are you still a follower of Hegel?

B: Of course! I believe everything he wrote. Since he denied the law of noncontradiction, I deny this too. On my view, P is entirely compatible with not-P.

A: I'm a fan of Hegel myself. But he didn't deny the law of noncontradiction! You read the wrong commentators!

B: You're wrong, he did deny this! Let me get my copy of The Science of Logic.

A: Don't get so upset! You said that he did deny the law, and I said that he didn't. Aren't these compatible on your view? After all, you think that P is compatible with not-P.

B: Yes, I guess they're compatible.

A: No they aren't!

B: Yes they are!

A: Don't get so upset! You said that they are compatible, and I said that they aren't. Aren't these two compatible on your view? Recall that you think that P is compatible with not-P.

B: Yes, I guess they're compatible. I'm getting confused.

A: And you're also not getting confused, right?

source: Philosophy, et cetera Blog (link)


No_Angel said...

simply that stems from misunderstanding what hegel said ...
read that and i hope you do get it

The Observer said...

hehehe, you and no_angel are head to head!

I like the post. Simple and clear explanation for the law of non-contradiction.

I personally feel like looking in a blank page if this law fails because nothing would have a meaning then!

Devil's Mind said...

No_Angel, as a side note, it is preferred if you use the "a" tag for links. I noticed this here and in the comments at your blog, where links are truncated and unreadable.

The link that no angel provided is this.

Several points arise... First, Hegel did NOT actually deny the law of non-contradiction.

Look at the definition of the law of non-contradiction: This law states that no instance in the same respect can be in two or more contradicting states.

The definition says: In the same respect.

According to the link you provided Hegel says that something can be in one or more contradicting states in different respects. Honestly, he isn't saying anything new....

I have discussed this idea before here. Nothing new.. Nothing contradicting!

bambam said...

hey thanks but i do it at my own leisure ... sometimes am lazy.
lol ... u were making fun of hegel earlier and i was pointing it out that u misunderstand him ... not saying it is a contradiction just a lack of understanding.

still u missed something ... he is saying that in the state of becoming all possible states are innate and reside within an object ... having all those contradictory states before any specific state gets actualized irrelevant of time since the perception of time is that of change so it will be placing a limit of the potential of becoming :D
and you wonder why ppl misunderstand hegel

Devil's Mind said...
This comment has been removed by the author.
Devil's Mind said...

No offense, but it seems to me that its you who most misunderstands Hegel. Hegel is probably a smart philosopher, but it seems to me that you put in his mouth words (or meanings) that he didn't say!

You have been arguing that Hegel denied the law of non-contradiction, but you now say: "i was pointing it out that u misunderstand him ... not saying it is a contradiction just a lack of understanding." - There is a world difference between the two! Dude, stop contradicting yourself!

In short: I don't know what Hegel said... And I don't care... I have a mind of my own, and a judgment of my own... If he was stupid, I don't have to be like him, and if he was smart, then good for him! Either way, it doesn't affect me!

Devil's Mind said...

My point is, don't be like Mr. "B" in the funny dialog I quoted above. Standing for something you don't understand, and ending up confused by your own statements!!

Devil's Mind said...

Follow up:
If you are interested in further discussion about this topic, follow this link. Comments on that post discuss criticism to the logic demonstration provided.

Jarvis said...

Hi there. Can you show that admitting that there can be one contradiction in reality would make all of reality contradictory, and so unknowable, a la the way you showed that admitting the possibility of one mathematical contradiction results in all numbers being the same?

Devil's Mind said...

Yes Jarvis, if we admit a contradiction in reality, then we admit reality is unknowable.

This simply follows from realizing that reaching such a conclusion requires use of physics and mathematics. Once mathematics is now involved, and all numbers are the same, then reality is unknowable.

But this is sort of admission is unreliable, because there when we admit a contradiction in reality, such a contradiction exists only in our understanding and modeling of reality, not necessarily reality itself!