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.
~: 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
The result is that all logical statements are evaluated as True. Even a statement and its contradiction became both true.
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!
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 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:
= 2 * 3
= 1 * 3 ; premise
^: The power operator
= 4 ^ 2
= 4 ^ 1 ; premise
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)