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Logic and the Law of non-Contradiction

INTRODUCTION

Can two opposing opinions be simultaneously correct? The opinion that is correct can only be one

– Imam Malik (d.179 AH)

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, known intuitively and presented directly to our consciousness.
  2. In order to blindly defend their religious doctrine and/ or preconceived notions which they know to be contradictory.

In what follows, God willing, we will show how preposterous it is for one to deny and negate this law.

ISLAM AND THE LAW OF NON-CONTRADICTION

(P and ¬P) false, where ¬ stands for “not” or “negation of”.

This law adjudges as false any meaningful proposition P and its denial, ¬P, as 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 uttered and hold as true 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 illAllaah – 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 are for none but Allah alone. Hence, in Islam, it is impossible to accept the worship of other than Allah as true precisely 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 – Malik ibn Anas (aka Imam Malik d.179 AH) – who ruled on what a person should do when faced with two opposite opinions:

Ashab, one of Imam Malik’s students said: “Malik 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. Imam Malik replied: ‘No! By Allah! Not unless it is correctthe truth can only be one. Can two opposing opinions be simultaneously correct? The opinion that is correct can only be one.'” [1]

We may also conclude from the above that the understanding of the truth “only be one”, was not something invented by Aristotle, as some ignoramuses assert. It maybe true that the earliest historical documentation of said law comes to us from Aristotle; but to infer that this could not, therefore, be approved of and applied to the source of all truth, is beyond preposterous.

BELIEF IN THE IMPOSSIBLE – ALLAH DOES NOT REVEAL CONTRADICTIONS

Logic is a priori – Immanuel Kant

The denial of this law entails that opposite things can be true at the same time and in the same respect. Hence, any proposition can be true and false at the same time, e.g. an answer is both right and wrong at the same time, which, it goes without saying, is impossible!

What would shake this self-evident law of logic? The answer is: an impossible universe!

Let’s entertain for the moment the inconceivable notion of the essential nature of the universe changing entirely every second. Only in such an impossible universe could it be said that the laws of logic would not be applicable. Fortunately, we do not live in such a universe; we could not, for there would be no ‘we’. This type of universe is not possible and cannot be given meaning precisely because it contradicts the only reality we can make any real sense of things.

In fact, we might go on to propose something even more profound, and that is:

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

THE A PRIORI LAWS OF BI-VALUED LOGIC [2]

Western German philosopher, Immanuel Kant, was correct when he said that “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 that are known intuitively; that is to say, they are self-evident and directly presented to our consciousness, without which nothing would make any sense.

If one were ever to assert:

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

Our 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

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 may 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, i.e. 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 propositions (i.e. that the law of non-contradiction is true and false) is impossible, thus confirming the principle of non-contradiction.

What is more, if we do admit to just one contradiction being true, then we must also accept that all contradictions can be true. The result of this would be that truth itself would lose all meaning and value. The principle or law of non-Contradiction is inescapable and incorrigible if we intend to state anything meaningful about reality.

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

[1] Ibn ‘Abdil-Barr, Jaami’ Bayaanul ‘Ilm, (Cairo: al-Muneerah Press, vol.2), p. 32.
[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.

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