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.

For the care of baby Zero, the numbers decided to take turns. Each number would hold the baby beside it. As the days passed, the numbers realized a puzzling fact. Those numbers who gave care in the right manner saw a never-seen abundance at home; everything would increase tenfold. The numbers who passed their turns superficially did not see any difference in their living. Was it a sign from God that the numbers could not grasp?

Days passed, and the numbers were expecting to see some developments in the newborn. It was supposed to make facial expressions, start uttering some meaningful sounds, make purposeful movements with its arms, and so on. But zero did not exhibit any of the expected qualities. The numbers thought, “Maybe it’s a late-developer.” But soon they understood that this number was disabled in many respects. It would not give any known response to any of the stimuli. It would not make a difference when used in summation or subtraction games. In the multiplication game, it would give a deep silence as the result, no matter how big a number it played with. And in the division game, no matter what number divided Zero, the result would be a hopeless reluctance to change. The numbers could not reconcile in their minds the miraculous abundance upon the proper care for Zero with its extraordinary disability. What was the deal with this baby?

Years flew by like a stream, and Zero entered adolescence. Now its body was not only growing very fast but was also assuming a more rounded shape, leaving its primitive seed-like dot form. With the rebellious instincts of adolescence, Zero discovered its ability to roll, instead of the step-by-step walk of the others. And soon it discovered that it was able to move much faster than the others by rolling. This unexpected burst of motor skills from Zero made the numbers very happy. They did not have to put burden themselves anymore to move Zero. Once a totally disabled baby, Zero was now the new roller in the numbers kingdom that even the wealthiest numbers could not catch up with.

The downside of this was that Zero was not able to stop. It had to hit another number for that, just like in the multiplication game. Maybe part of the wisdom of the apparent disability of Zero at the beginning of its life was that it needed to gain wisdom to control its super ability to move. However, especially within the first months of the discovery of its new skill, Zero just rolled, rolled and rolled. Even some accidents did not scare Zero from its attempts to roll farther and faster. After all, it had a proper justification for it, considering its long years of immobility.

As Zero grew even more, and as the numbers watched it closely, they realized it had another interesting feature. It possessed an enthralling beauty due to its perfect symmetry. How was it possible that such unparalleled elegance emerged in a number that was initially thought to be disabled? The numbers decided to consult the Savant One. One was the oldest among the numbers, was held noble among all others, and knew history very well. As knowledgeable and noble as it was, One was humble as well, as it would bend its torso to talk with children, instead of distinguishing itself among everybody by raising its head straight up.

Savant One said, “I have been looking at the ancient books about the signs of this number. As told, it was born disabled, and it was raised by its relatives; it then surpassed everyone else by its speed of motion and its perfect rotational symmetry. More than that, as Zero grows, it is assuming the shape that we observe in all heavenly objects. This young member of our community has more to teach us. We need to wait and see.”

After the numbers had left the house of One, Prime Minister Two could not hold back its words and raised its voice: “Could it be that the old guy is mixing things up due to its age? I mean, how do we know that the things written in those books are authentic? Zero is not totally different from us; it is just another number, anyway. Why are we bothering ourselves with something that no one has seen before? Are we going to leave the greatest numbers for the sake of this fledgling egg?” And then, Two burst into laughter. Some of the numbers joined it in laughing, and some others said, “We’ll see!”

As the years passed, Zero started questioning its existence, and contemplating its character as a number. It had heard the words of both the Savant One and the Prime Minister Two, but it could not come to an exact conclusion about itself. Maybe Two was right. Even if the ancient books did actually talk about someone, who said that it was Zero that was expected, anyway? So, with this mindset, Zero preferred a midway between that of Savant One and that of Prime Minister Two.

As Zero was leaving adolescence, something started bothering Zero from inside again. It was as if somebody were periodically disturbing its comfortable life by triggering emotions that were not satisfied with the daily stuff that Zero did. So, Zero started reflecting again. This time, it considered the things it had experienced so far in its unusual life story: its initial disability in life and the consequent burst of motor skill, its initial negligible size and the unparalleled beauty it exhibited as it grew, the words of the Savant One and those of Prime Minister Two, and so on.

It was one of those times that Zero felt a sudden shiver with the coolness of an inspiration that penetrated its whole body. Zero felt that its life was full of opposites, so that, by pondering over the respective non-existence and existence of its qualities, it might understand infinity. These thoughts motivated Zero to investigate infinity by talking to other numbers.

The answer that best resonated with what Zero felt was from Savant One: “I have not seen infinity, but I do know that I was the first one to exist among numbers. I feel that I could not have come to be out of nothing. And seeing all these numbers that came after me, I better understand that what made me is infinity. And, oh my little one, if you ask what one step before me was, the ancient books say that it is you who is going to reveal that.” And the answer that Zero did not like much was from the Prime Minister Two: “There is no such thing as infinity. No matter how great a number you imagine, by adding one, you can always obtain a greater number.

And, this is all there is about it; there is no one who has actually seen infinity, anyway; it is no more than a figment of our imagination.”

When alone, Zero thought, “I can understand Two because when it came to the world, it saw One already there; so, Two cannot comprehend coming to existence without an ancestor. But what I feel inside is more like what One tells me. One can realize the existence of something that triggered its existence because when it came to the world, there was none other than One.” After a silence, Zero mumbled, “If I exist, then you exist. But where are you? If only I could see you…”

Life had been very turbulent for Zero. First, disabilities that burdened it, then a sudden explosion of skills and emotions, then the painful questioning it went through without satisfactory answers, and finally a conclusion that brought peace along with a wish for homecoming. Soon after, not being able to stand the harshness of such turbulence, Zero fell sick. It started losing weight to the point that it almost returned to its initial state at birth. With this final sickness at such a young age, Zero had had enough of it here. Although everybody was sad about what was going on, at the same time they thought that the best thing for this unique number was to die.

And Zero died. They buried it under the quotient line. Roses grew on its grave, as beautiful as Zero, as noble as One. Thus commemorated was the phrase “one over zero,” in order to refer to this beauty. It was then that the numbers understood what Zero felt inside. And the numbers visited Zero from time to time, whenever they needed a glimpse of beauty stretching from infinity. As powerless as it was, Zero helped others understand infinity not by its existence among them, but by lying below them.