By Abu Adeeba & Abu 'Abdur Rahman

(Original Source: http://www.islam-sikhism.info/rebut/non_con/non_con02.htm)

INTRODUCTION

Those who deny the Law of non-Contradiction do so at the cost of their own intellectual well-being.

There are two main reasons why one may end up denying this law: 

1. Ignorance of the law itself and of the fact that it is self-evident, that is to say, it is known intuitively and directly presented to our consciousness.
2. In order to blindly defend their religious doctrine which they know to be contradictory.

In what follows, insha'Allah (God Willing), we will show how preposterous it is for one to deny and negate this law.

ISLAM & THE LAW OF NON-CONTRADICTION 

(P and not-P) is false. 

This law judges as false any meaningful proposition P and its denial, proposition not-P, true at the same time and in the same respect. In the words of Aristotle:

"One cannot say of something that it is and that it is not in the same respect and at the same time."

The first pillar of Islam, the declaration of faith (shahaadah), cannot be made and hold any meaning without firstly affirming this law and the laws of bi-valued logic (see below).

The first part of the declaration of faith is:

Laa ilaaha illAllah - None has the right to be worshipped in truth except Allah

The first pillar of Islam cannot be realised except through negation (nafiyun) and affirmation (ithbaatun). Laa ilaaha from the Arabic language means: a complete negation of worship towards all deities, followed by the exception to the rule (illa istithinaa), which is to affirm that all forms of prescribed worship is for none but Allah alone. Hence, in Islam, it is impossible to accept that worshipping other than Allah is true simply because it violates the declaration of faith. 

In the Qur'an Allah establishes that something and its opposite cannot both be true at the same time:

"And such is your Lord in truth. Then what is there after Truth except falsehood. How have they then turned away?" (Qur'an 10:32)

This universal law was articulated by the great jurist and practitioner of Islam - Maalik ibn Anas (aka Imaam Maalik d.179AH) - who ruled on what a person should do when faced with two opposite opinions:

Ashab, one of Imaam Maalik's students said: "Maalik was once asked whether one was safe to follow a ruling related to him by reliable narrators who had heard it from companions of the Prophet. Imaam Maalik replied: 'No! By Allah! Not unless it is correct - the truth can only be one. Can two opposing opinions be simultaneously correct? The opinion that is correct can only be one.'" [1]

From this we say that the understanding that 'truth can only be one' was not something invented by Aristotle as some ignoramuses assert. Although Aristotle famously formulated these laws, it is evident that he could not have been the first to invent them.
For a more thorough read including proofs from the Islamic sources please read the following two pages:

BELIEF IN THE IMPOSSIBLE

ALLAH DOES NOT REVEAL CONTRADICTIONS 

The denial of this law entails that opposite things can be true at the same time and in the same respect, which is impossible. Hence, any proposition can be true and false at the same time, or that an answer is both right and wrong at the same time!

What would shake this self-evident law of logic? An impossible universe; if the nature of the universe were such that it could totally change every second, if this is even conceivable, the laws of logic would not be applicable. Fortunately, we do not live in such a universe; we could not, there would be no 'we'. This type of universe is not possible and cannot be given meaning precisely because it contradicts reality and thus logic.

In fact, something even more profound occurs:  

Those who deny this principle (or law) of contradiction contradict themselves.

BI-VALUED LOGIC IS A PRIORI [2]

For once, Immanuel Kant was correct when he said, "logic is a priori". Before anything can be postulated, bi-valued logic is required to make sense of it. Hence, for any meaningful proposition P, at a given time, in a given respect, there are three related laws: the Law of Non-Contradiction, the Law of Bivalence and the Law of the Excluded Middle. [3] These three are universal givens known intuitively; that is to say, they are self-evident and directly presented to our consciousness, without which nothing would make any sense.

If someone asserts:  

It is merely your opinion that logic is a priori and a universal given 

The response would be:  

You will have to make recourse to the laws of logic in order to prove that it is merely opinion. Hence, the above statement requires logic to prove it is true. This proves that logic is indeed a priori and a universal given.

CONCLUSION

As we said before, those who deny this principle (or law) of contradiction end up contradicting themselves.

Reductio ad Absurdum

Reductio ad absurdum (from Latin 'reduced to an absurdity') is a type of logical argument where one: a) assumes a claim for the sake of argument b) arrives at an absurd result, and then c) Concludes that the original assumption must have been wrong, since it resulted in an absurdity. This is also known as 'proof by contradiction'. [4] Now, consider the following statement "The law of non-contradiction is false". We outline said argument as follows:

• The law of non-contradiction is false
• Hence, that "the law of non-contradiction is true" must be reckoned false (since it contradicts the above)
• Hence, the law of non-contradiction is actually true.

If one asserts that the law of non-contradiction is false, then the opposite of this claim (that the law of non-contradiction is true) must be false; otherwise two opposing things could and would be true at the same time. But, the claim that the law of non-contradiction is false, as opposed to true and false, implies that there is one absolute truth, which is essentially what the law of non-contradiction establishes. By this, the denier actually concedes that the affirmation of two mutually exclusive opposites (i.e. that the law of non-contradiction is true and false) is impossible, thus confirming the principle of non-contradiction. If we admit one contradiction as true we can accept all contradictions as being true. The result is that truth itself loses all meaning and value. The principle or law of Non-Contradiction is inescapable, incorrigible if we intend to state anything meaningful about reality.  

Hence, it is a self-evident truth - a priori.  

[1] Jaami' Bayaanul 'Ilm, Ibn 'Abdil-Barr (Cairo: al-Muneerah Press, vol.2), p32.
[2] A Priori: known to be true independently of any particular kind of experience or empirical information.
[3] Law of Bivalence: Any proposition, either P is true or P is false, e.g. the man is dead, which is either true or false.

 Law of the Excluded Middle: (P or not-P) is true, e.g. the man is either dead or not dead, which is true.
[4] For a simple example, consider the proposition "there is no smallest rational number greater than 0". In a reductio ad absurdum argument, we would start by assuming the opposite: that there is a smallest rational number, say, r0.

 It is not uncommon to use this type of argument with propositions such as the one above, concerning the non-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist.