Survey of the Qur'an and Mathematical Logic

Survey of the Qur'an and Logic
by Butrus Rasool, Waleed Quwan, and Faddaan Sinnawr

We, the kuffaar, are constantly reminded that Islam is a religion for those who think. Muslims do not follow Islam out of blind faith; rather they do so after seeing evidence that is beyond dispute. If you are a rational and logical individual, you will embrace Islam.

This article will offer an introduction to some of the logical and mathematical anomalies in the Qur'an. It is our point to note that precisely these sorts of abberations are to be expected if the text is of a human origin, and we will adduce precisely that after treating these peculiarities as corroborating evidence for such a conclusion.

Some readers may wonder why a survey of logic would cover mathematics. While it is not clear to us whether mathematics is a branch of logic or logic is a branch of mathematics, we are certain of two things: (1) logic is at least a bridge between mathematics and philosophy, and (2) the Qur'an has problems with both categories. For introductions to logical approaches to mathematics and mathematical approaches to logic, consider Bertrand Russell's Principia Mathematica, or Willard Van Orman Quine's Mathematical Logic.

An Erroneous Challenge

In the Qur'an there is a verse that many Muslims cite as a challenge to find an error. They claim that the Qur'an is laying down the gauntlet, and it is now incumbent upon the nonbeliever to find a problem with the Qur'an. The verse reads:

Afalaa yatadabbaroona al-Quraana wa law kana min indi ghayri Allaahi lawajadoo feehi ikhtilaafan katheeran

Do they not consider the Qur'an? If it were from other than Allaah, they would have found many errors.

Soorat an-Nisaa'

Shaykh Gary Miller, in his article The Amazing Qur'an, argued that this verse makes Islam the first religion to present itself as being falsifiable. The Qur'an does not ask you to accept it on blind faith; rather it gives you a method to prove it false. Now all that is left, the Muslims claim, is for one brave soul to present an error or contradiction in the text.

The great irony is that if one were to search for an error in the Qur'an, they would not have to look further than this very verse! The proposition being put forth in the text is "if the Qur'an is not from Allaah, it would have errors." A quick intro to basic logic will explain how Shaykh Gary Miller and many other Muslims have badly missed a couple of key points about the logical structure of this verse.

First we would like to introduce the "conditional" connective of formal logic (in this text it will be represented by an arrow, '-->'). So, if we let 'A' stand for "the Qur'an is from Allaah," let 'E' stand for "the Qur'an has errors," and let '~' represent the symbol for negation, the proposition can be translated into sentential logic as:

~A --> E

In other words, if the Qur'an is not from God, it would have errors. The first question one might ask is why we should accept this as true. Is it necessarily the case that a text that is not from God has errors? Certainly not. This problem is further complicated when we invoke the logical rule known as "contraposition," and state that if the above is true, the following is also true:

~E --> A

To explain why this is the case, let us give an example. Suppose we say that if you are in Jerusalem, you are in Israel (J --> I). If this proposition is true, then it follows that if you are not in Israel, you are not in Jerusalem (~I --> ~J). The logical structure of one implies the other.

So, this means that the verse is putting forth a proposition that is logically equivalent to saying "if the Qur'an does not have any errors, it is from Allaah." That is a statement that is demonstrably false. I could easily present a text that is both free of error and not of a divine origin (exempli gratia: a phone directory). The Muslims might argue that we are not discussing phone directories, and put forth some special pleading on behalf of the Qur'an, but that too is fallacious.

So, it seems we have pointed out a logical error in the Qur'an. Does this allow us to meet the challenge? The shrewd logician would respond with the question "what challenge?" Were we not discussing the challenge mentioned in the verse above? Well, we were discussing the verse, and we noted that Muslims treat it as a challenge, but the reality is that there is no challenge being put forth.

To explain why, suppose we found a bunch of errors and contradictions in the Qur'an. If we were to use the proposition put forth in the verse from the Qur'an, our argument would go as follows:

If the Qur'an is not from Allaah, it would have errors.
The Qur'an has errors.
Therefore, the Qur'an is not from Allaah.

We have already rejected the first premise as untrue, so automatically the argument is unsound. More important, the above argument is INVALID!!! To use an argument like the above is to commit the fallacy known as "affirming the consequent," yet this is precisely what the followers of "the most logical religion" are asking us to do.

Let us explain the logical structure of a conditional proposition (so-called "if-then statement"). Suppose I say "if Santa Claus came last night, you will have presents under the tree." This statement could very well be true. However, if you woke up on Christmas morning and found presents under your tree, this does not allow you to conclude that Santa came the night before. To do such would be to affirm the consequent. It is wholly possible that your loved ones put those presents under the tree.

So, the challenge does not exist. We have no doubt that the author of the of this Qur'anic passage thought he was putting forth a challenge of some sort. However, the uncareful (or sloppy) nature of the way the language was structured implies it was from a human hand, as this is exactly the sort of error a person unfamiliar with formal logic would make.

The only way out would be to strain the text and argue that it should be rendered as a biconditional proposition, where a text from Allaah is treated as being equivalent to a text that is free of error. While this would allow the Muslim to avoid the fallacy of affirming the consequent and the paradox of the non-existent challenge, we would sink right back into the previous problem: it is false to claim that a text that is free of error is also divine.

Not Errors, But Ikhtilaafs

Amazingly, there is one other approach to the problem rooted in the logical structure verse 4:82. In the article On Claims, Contradictions, Context & Internal Relationships, Dr. Muhammad Sayfullaah and other members of the Islamic-Awareness team argue that the verse can be seen in a different way.

They point out that the words "many errors" (or "much discrepancy") are translated from the last two words of the Arabic, ikhtilaafan katheeran. They further note that you can leave the word

ikhtilaafan untranslated. From here they present the following argument:

Now, what a clever logician would do is to simply treat ikhtilafan as a word and not consider its meaning and start counting number of its occurrences in the Qur'an. This would simply enable him to check whether the author of the Qur'an understands the difference between use and mention of the word. The clever logician would find that the word ikhtilafan occurs only once in the Qur'an, i.e., in the above verse.

So, this is presented as yet another miracle of the Qur'an. How ironic that the Qur'an states that if the text were not from God, you would find many "ikhtilaafan," and (SURPRISE!) the word appears only once (at least according to the Islamic-Awareness team).

A number of objections can be raised. First, how many instances of the word ikhtilaafan would be sufficient? We suppose it doesn't really matter. The word ikhtilaafan is nothing more than

ikhtilaaf in the accusative. The word ikhtilaaf appears in the Qur'an at least seven times.

Of course, we can't use this to meet the challenge, as that would again be a case of affirming the consequent, so there was really no point to this word game. Even if we pay attention to the alif (the difference between ikhtilaafan and ikhtilaaf), this gimmick has not escaped the fact that it is false to claim a text having only one instance of this word is divine. Almost every book in my house lacks instances of the word ikhtilaafan, and surely none of them were written by Allaah. So, even when taking the approach offered by the Islamic-Awareness team, the Qur'an still contains a false proposition.

The Errors Of Inheritance

The Qur'an lays down a number of complicated rules with regard to inheritance. Interestingly, the first person on the net to point out the mathematical errors in the passages putting forth these rules was Jochen Katz, creator of the Answering Islam site, and at the time a student at Georgia Tech's School of Mathematics.

Whatever the Muslims may think of Katz or any of his other articles (one is reminded of the line of anti-Trinitarian ridicule that goes roughly "I'm surprised that a math major doesn't know that one plus one plus one equals three, not one"), none have been able to dispute his piece Who is going to pay the bill for the "promise in overdraft?" with any thoroughness.

In his general outline of internal contradictions, Katz sums up the problems of Qur'anic inheritance as follows:

And it just doesn't add up: Sura 4:11-12 and 4:176 state the Qur'anic inheritance law. When a man dies, and is leaving behind three daughters, his two parents and his wife, they will receive the respective shares of 2/3 for the 3 daughters together, 1/3 for the parents together [both according to verse 4:11] and 1/8 for the wife [4:12] which adds up to more than the available estate. A second example: A man leaves only his mother, his wife and two sisters, then they receive 1/3 [mother, 4:11], 1/4 [wife, 4:12] and 2/3 [the two sisters, 4:176], which again adds up to 15/12 of the available property.

This is not a hidden error that Katz stumbled upon by himself; on the contrary, many Muslims were aware of these problems as well. While certain individuals within the more hateful spheres of the proverbial anti-Islamic camp deny such, the fact is that many of history's greatest mathematicians were Muslim. Many subjects within mathematics received their foundation from medieval Muslims. With all these brilliant Muslim mathematicians, some of them were certainly trying to make sense of the problematic verses on inheritance:

The Koranic rules for the distribution of estates to various relatives are complicated, and their application calls for some skill in arithmetic and first-order algebraic equations. The Prophet Muhammad is reported to have said that the laws of inheritance comprise one-half of all useful knowledge; this may explain why al-Khwarizmi devoted half of his book on algebra to problems of inheritance. Very few of the mathematical writings on this subject have ever been studied.

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

The fact that great mathematical thinkers like Muhammad bin Musa al-Khurizimee had to devote large amounts of their mental energy to making sense of the Qur'anic laws of inheritance should be treated as a hint that there was a problem. Al-Khurizimee was trying to correct this problem, or at least find a mathematically sound approach. His solutions, however, came under heavy criticism. For example, in his commentary on the section on inheritance in al-Khurizimee's work on algebra, Frederic Rosen wrote:

The solutions which the author has given of the remaining problems of this treatise are, mathematically considered, for the most part incorrect. It is not that the problems, when once reduced into equations, are incorrectly worked out; but that in reducing them to equations, arbitrary assumptions are made, which are foreign or contradictory to the data first enounced, for the purpose, it should seem, of forcing the solutions to accord with the established rules of iheritance as expounded by Arabian lawyers.

The Algebra of Mohammed ben Musa

In other words, al-Khurizimee could only make the laws of Qur'anic inheritance work if he employed rules that are found not in the text of the Holy Book, but in the solutions of others who have commented on these passages. Moritz Cantor raised a similar objection:

The established rules of the law of inheritance, as contained in the Quran, are not only very complicated but also confused, because their directions pretty often contradict each other. These contradictions pretty often necessitated decisions and solutions that deviated, symmetrically and proportionally, from both rules of the law and the rules of computation, because it seemed inopportune, or inexpedient, to violate the one in favor of the other.

Osiris: Commentationes de Scientiarum et Eruditionis Historia Rationeque

Interestingly, Solomon Gandz, who set out to defend al-Khurizimee from the attack of Rosen and Cantor, falls right into the hands of his subject's critics. Gandz offers an example given by al-Khurizimee himself (which Rosen treated with disdain): a woman dies and leaves behind her husband, her mother, and two daughters. The Qur'an rules that each dauther gets a third, the husband gets a quarter, and the mother gets a sixth. The breakdown and combine total would be as follows:

Daughter₁	1/3	4/12
Daughter₂	1/3	4/12
Husband	1/4	3/12
Mother	1/6	2/12
Total		13/12

Obviously there is a problem here. Katz gave similar examples, with totals reaching 17/12, 7/6, 9/8, et cetera. With the example above (and by extension the examples given by Katz, Rosen, Cantor, et al.), Gandz notes the following:

In such and similar cases a later amendment rules that the denominator is to be made equal to the numerator, instead of 13/12 we have to take 13/13, and in the legal shares of the heirs the numerators remain unchanged with the new denominator. In our case, the two daughters get 8/13, the husband 3/13, and the mother 2/13... The technical term for this amendment is al-'aul "the deviation" from the original law.

opere citato

Unfortunately, Gandz played right into the hands of Rosen and Cantor. This "deviation" from the law is precisely what they were making reference to, where unjustified rules are added to prop up the failing Qur'anic laws of inheritance. As Katz himself concluded, all the solutions subconsciously admit that the Qur'an by itself is logically inconsistent; they then import data in an attempt to amend this problem. We were not arguing about one's ability to fix an inconsistent equation by arbitrarily changing the numbers; rather we were arguing that the Qur'an is in a state of logical discordance.

Concluding Remarks On Errors

The errors listed above are to be expected in a text written by human hands. Uncareful wording can cause a human to think he is offering a challenge when really he is only inviting others to commit a fallacy. An unfamiliarity with logic can lead to a human to fumble with the theological axiom that divine texts are infallible and accidentally state that texts that are free of error are divine. A human can certainly put forth inconsistent propositions when discussing complex mathematical divisions. Based on this, we feel justified in reasoning via abduction that it is probable that the Qur'an is of a wholly human origin. From there, we can allow Ockham's razor to slice away the claims of celestial and heavenly beings authoring the text.

Post Script

For another foray into the realm of Islam and mathematical logic, see the article Logical Inconsistency of Islamic Dogma, written by a logician. The argument is very interesting, and is certainly valid. However, we wonder about its soundness in light of a possible lack of justification for the first axiom... we reserve judgement until further elucidation is offered by the author.

As for the topics covered in this article, there is some discussion about them available on the net. For debate on the logical structure of Soorat an-Nisaa' 4:82, see Logic of Allaah, a usenet thread including contributions from Dr. Muhammad Sayfallaah, Mete Gulenoglu and others. MENJ of the Bismika Allaahuma site has written a rather potent response. MENJ has also taken to defending his attack on this article in a mildly technical (and quite fascinating) discussion in Ali Sina's FFI forum. Finally, for a defense of the laws of inheritance that employs an interpretation that is quite different from al-Khurizimee's (and thus creates a new paradigm which is immune to the above listed criticisms), see Moiz Amjad's 1997 response to Jochen Katz, Understanding the Law of Inheritance of the Qur'an.

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