• Issue 97 / January - February 2014

    Nasraddin Hodja's Pot and Reductio Ad Absurdum

    Nathan Ruler

    From a children's tale to Socrates and the Qur'an, the need for logic and sound reasoning is universal.

    Nasraddin Hodja is a well-known folktale character in the Middle East. He is famous for his wit, which is apparent in the wisdom-filled stories that have been ascribed to him for eight centuries. Although Hodja was not a philosopher, many of his tales contain elements that demonstrate philosophical concepts and principles. As an example, the story of Hodja's pot neatly exercises a fundamental philosophical method, i.e. the Socratic method or reductio ad absurdum. This story, though apparently simple and funny, has a deep philosophical dimension and teaches us the Socratic method in a very accessible way. The story goes as follows:

    One day, Nasraddin Hodja borrowed a pot from his neighbor. After he had finished using it, he took it back to the neighbor with a smaller pot put inside it.When the neighbor saw the smaller pot, he was surprised.

    "What is that?" he asked.
    "Well, said Hodja, when I borrowed your pot it was pregnant and it gave birth."
    The man smiled and accepted both pots.
    A few days later, Hodja borrowed the pot again but this time he did not return it.
The neighbor was rather cross. He went to Hodja and asked, "What about my pot?"

    "I am very sorry," said Hodja, "but it died."
    "Don't make jokes with me," replied the neighbor. "How can a pot die?"
    "If you believe that it brought a child into the world," said Hodja, "why can't you believe that it died?"

    In this story, Nasraddin Hodja suggests that his neighbor's behavior of accepting the smaller pot is improper, by indicating an absurd consequence of his behavior, namely that a pot can die if it can give birth. This method is commonly used in philosophical argumentation since the Ancient Greeks. We can see similar examples in Plato's dialogues. For example, in the Republic, Cephalus defines justice as speaking the truth and paying the debt. However, Socrates shows that this definition logically leads to some absurd consequences. Let us assume that somebody lends us arms when his mind is right, but wants them back later when he lost his mind. According to Cephalus' definition of justice, we should give the arms to that insane and crazy person since they are not ours but his. Yet, no one would approve such an act. Once Cephalus encounters this counterexample, he understands that his initial definition has some problems; he approves Socrates' point and tries to modify his idea of justice. Thus, the dialogue continues. Different definitions are introduced and checked by this method.

    We can display the logical structure of the Socratic method in the following way. In a conversation, somebody holds an assumption, and makes a definition or claims something. The other person shows that this assumption, definition, or claim leads to an absurdity. For this reason, this method is also called reductio ad absurdum, i.e. reducing to absurdity. Since the initial assumption leads to something unacceptable, it is unacceptable as well by logical inference. Then the person who held that assumption is forced to abandon or modify it. See Table 1 for a comparison of these examples.

    Table 1
    Socratic Method Hodja's example of pot Plato's example of justice
    X holds an assumption. The neighbor accepts that a pot can give birth to another pot. Cephalus defines justice as speaking the truth and paying the debt.

    Y shows that X leads to an absurdity. Hodja shows that a pot can die on the basis of this assumption. Socrates shows a counterexample to this definition, which is absurd.

    Absurdities cannot be accepted. The neighbor admits the absurdity of the death of a pot. Counterexample: returning the arms to an insane person since they belong to him.

    Thus, assumptions that lead to absurdities are unacceptable. Hodja points out then that the neighbor's initial claim is also absurd. Cephalus admits that this conclusion is unacceptable and abandons his initial definition.

    The Socratic method is one of the basic methods that we use in reasoning. We use it in daily life, in sciences, and in any area where we engage in rational thinking. The strictest kind of absurdity is a contradiction, i.e. accepting and denying the exactly same thing under the same conditions. Because of this, logicians and mathematicians call the Socratic method, "the indirect proof," or "proof by contradiction." See the appendix for an example of the proof by contradiction in mathematics.

    Interestingly enough, we also see this method in the holy books. For example, the Qur'an frequently suggests people use their rationality and carefully think about the universe. It is remarkable that, in the Qur'an, we find arguments that rely on the reductio ad absurdum method. For example, let us consider the following verse: "But the fact is that had there been in the heavens and the earth any deities other than God, both (of those realms) would certainly have fallen into ruin. All-Glorified God is, the Lord of the Supreme Throne, in that He is absolutely above all that they attribute to Him" (21:22). This verse proposes that one must have observed disorder in the universe if that person associated partners with God's activity in the universe. Since this is not what we observe, the initial assumption is incorrect. This example shows that the Qur'an addresses the rationality of human beings, and suggests actively using that faculty.

    As we have seen, deriving absurd consequences from an assumption and denying it on the basis of those absurdities is a common and fundamental way of reasoning that is exercised in many different areas and aspects of human life. It is called by different names over history, namely "the Socratic method," "reductio ad absurdum," or "proof by contradiction." Yet, the basic idea behind these technical terms is the same. Most of us probably exercise this method without thinking about it.

    Theorem: The number that is equal to itself, when added to itself, is zero.
    Prove that for any x, x is a number; if x +x = x, then x=0.
    1. x+x=x (assumption for conditional/direct proof)
    2. x is not equal to 0. (Assumption for indirect proof/reductio ad absurdum)
    3. x+x=2x (by addition)
    4. x=2x (since "x+x" is common for both the 1st and 3rd steps, by the principle of transitivity)
    5. 1=2 (cancel x's) –Contradiction
    6. x=0. (The assumption that leads to the contradiction in the 5th step must be false. Therefore, it is false that x is not equal to 0. Thus, x=0 is true.) Indirect proof is complete.
    7. If x+x=x, then x=0. (Conditional proof is complete)


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