Given the emphasis by the Qur’an on logical reasoning, it is understood that its audience (all of mankind) is expected to have a basic understanding of logical principles. It is our purpose in this article to go over a few basic principles of logic and give examples of how they are used in the Qur’an.

**Logical principle 1: ** Proof by contradiction. An important aspect of logic concerns methods of proving or disproving statements. This is one of the basic principles of proving statements, and it is commonly used to prove mathematical statements. To show that a statement is true using this principle, suppose its negation is true, that is, that the statement itself is false. If you can then obtain a contradiction, something that is clearly false, this shows that the original statement is true. We shall illustrate this principle with an example. The reason we choose this example is that it is a classical result in mathematics, and the argument is beautiful. According to Hardy, this is an example of a first-class theorem about a significant mathematical fact (“a real mathematical theorem”) that is accessible to a general, educated audience (Hardy, 1967). There is a fundamental theorem in number theory which says that “there are infinitely many prime numbers.”1 There are many other proofs of this theorem but the most elementary and elegant proof is given by Euclid and it is by contradiction. Hardy claims that “both the statement and the proof can be mastered in an hour by an intelligent reader, however slender his mathematical equipment” (Hardy, 1967). Here is that proof: Suppose there is only a finite number of primes. Call them p1 , p2 , … pn. Now, consider the number N = p1 , p2 , … pn +1. Clearly, N is not equal to any of the primes in the list. Therefore, N itself is not a prime, and hence N must be divisible by one of the primes p1 , p2 , … pn (this is by “the fundamental theorem of arithmetic” which states that every integer greater than 1 can be written as a product of prime numbers). However, that is not the case: when N is divided by each one of pi there is a remainder of 1. This is a contradiction, and it arises from the assumption that there are only finitely many primes. Hence, the assumption cannot be true, and we conclude that there are infinitely many primes.

Proof by contradiction, or reductio ad absurdum in Latin, is one of the mathematician’s finest tools. There are many examples of theorems in mathematics where the simplest proof is by reductio ad absurdum. Another classical example is to show that √ is not a rational number (Hardy, 1967).

Now, we would like to give examples from the Qur’an where this logical principle is used. Consider the verse: “Had there been in the heavens and the earth any deities other than God, both would certainly fallen into ruin” (Anbiya 21:22). Here the argument is: suppose there were gods other than God. Then the earth and the heaven could not maintain their order (because the powers and wills of different gods would interfere with each other). Since this is clearly not the case (because we see perfect order in the universe), this contradicts the logical consequence of the initial assumption. Therefore, the initial assumption (i.e., there are more than one god) is wrong.

For another example, let us consider the verse: “Do they not contemplate the Qur’an? Had it been from any other than God, they would surely have found in it much inconsistency” (Nisa 4:82). Here the statement to be proven is “The Qur’an is the word of God, and it is from God.” To prove this statement by reductio ad absurdum, suppose for a moment the opposite statement is true, that is, that the Qur’an is not from God. A logical consequence of this assumption would be it is the word of a human being, but then there would be inconsistencies in it because the Qur’an talks about such great matters that if a human being had written it he would have made mistakes and inconsistent statements (for more on this point see Unal, 2007, page 228, and Nursi). However, we see no such statements in the Qur’an, thus contradicting the initial assumption. Therefore the Qur’an is the word of God.

The final example we would like to give related to this principle is expressed in the verse: “If you are in doubt about the Divine authorship of what We have sent down our servant (Muhammad) then produce just a sura (chapter) like it and call for help to all your supporters” (Baqara 2:23). Here the statement to be proven is the same as the one in the previous example. The argument goes like this: If it was possible for anybody other than God to produce the Qur’an, you should be able to make something like it with all of your supporters, if not an entire book, just a chapter of it. The Qur’an presents disbelievers with a challenge. However, they have not been able to meet the challenge. They chose to fight with the sword because fighting with words was not possible. Therefore the assumption that “the Qur’an could have been written by a human” is wrong.

** Logical principle 2: ** The next principle we would like to discuss is the equivalence of an implication to its contrapositive. A statement of the form “If P, then Q” is called an implication. In such an implication P is called the premise, or the hypothesis and Q is called the conclusion. Such a statement asserts that whenever P is true (or satisfied) then Q is guaranteed to be true. However, if/when P is not true, it does not say anything about the truth value of Q. An example of an implication from daily life is: “If it is rainy, then it is cloudy.” Here the premise is “it is rainy” and the conclusion is “it is cloudy.” Another example would be: “If you are in New York City, then you are in the US.” A mathematical example is: “if a number n is divisible by 6, then it is even.” An implication can also be read as “P implies Q,” or written “P => Q.” The contrapositive of an implication “if P then Q” is “if not Q then not P.” For example, the contrapositive of the first implication above is “if it is not cloudy then it is not rainy,” of the second one, “if you are not in the US, then you are not in NY city,” and of the third one, “if n is odd (not even) then n is not divisible by 6.”

Here are two examples from the Qur’an where we can apply this principle. The first one is the verse we discussed above: “If you are in doubt about the Divine authorship of what We have sent down our servant (Muhammad) then produce just a sura like it and call for help to all your supporters.” We can rephrase this statement as “if the Qur’an is not the word of God, then humans can produce a like of it,” and its contrapositive is “if humans cannot produce something similar to the Qur’an, then it is the word of God.” Remember that these two statements are logically equivalent. Now, we know that humans cannot produce anything comparable to the Qur’an despite much effort and motivation, therefore it follows that it is the word of God.

The second example we want to consider is the verse: “Say (to them, O Messenger) if you indeed love God, then follow me, so that God will love you and forgive your sins” (Al Imran 3:31). The contrapositive of this statement can be stated as (slightly simplified) “if you are not following his messenger, then you really do not love God.” So, we obtain a clear measure of whether one really loves God or not. Following the Prophet is a necessary condition to attain the love of God.

**Logical principle 3: ** The final principle we discuss is false implies anything. This is actually a special case of an implication of the form “P implies Q.” Recall that such a statement asserts that whenever P is true, Q is also true, so P promises Q. However, if P is not true then it does not claim anything about Q; it may or may not be true. For example, if it is not rainy then it may or may not be cloudy. An implication is false when P is true yet Q is false. Intuitively, we can think of this as breaking a promise. The question arises as to what the truth value of an implication should be when the premise P is false. It is accepted that the implication will be true in this case, because when the premise is false, the conclusion is totally irrelevant. This is called “false implies anything.” One can make any conclusion whatsoever based on a false premise, but it really does not mean anything. For example, the statements “if 2+2=5, then there are only finitely many primes,” and “if the world is flat, then there are no wars in the world” are both true statements. These statements are true but they really do not have any substance. They are true by a convention in logic. The important point one needs to pay attention to is this: Once you believe a premise that is not true, it is possible to make any conclusion you want and the whole argument looks right. However, such an argument is quite meaningless. So one needs to examine very carefully what is being assumed in a logical argument.

One example we want to consider from the Qur’an related to this principle is the verse: “Those who deny Our Revelations and turn arrogantly from them-for them, the gates of Heaven will not be opened and they will not enter Paradise unless a camel can pass through the eye of a needle” (A’raf 7:40). With a little simplification, we can rephrase this verse as “if a camel passes through the eye of a needle, then disbelievers will enter Paradise.” Since the premise is false (obviously, a camel cannot pass through the eye of a needle), this implication is true. However, this is not good news for disbelievers because the truth of this implication has nothing to do with the truth of its conclusion. This again shows the point: One needs to examine logical arguments carefully. What looks true or convincing formally and on the surface may not mean anything in reality if it is based on a false premise.

Nuh Aydin is an associate professor of Mathematics at Kenyon College, in Ohio, USA.

**Notes**

1. A positive integer n that is greater than 1 is called a prime if its only positive divisors are 1 and n. 1 is not considered to be a prime. Thus the sequence of prime numbers is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…

**References**

**Hardy, G. H. A Mathematician’s Apology, Cambridge University Press, 1967.**

**Nursi, Bediuzzaman Said. The Words (translation), The Twenty-Fifth Word, Available online at http://en.nurglobal.net/modules/mastop_publish/?tac=The_Twenty-Fifth_Word**

**Unal, Ali. The Qur’an with Annotated Interpretation in Modern English, The Light Inc., 2007. **

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