Thought for the Day: Nine Lines of Gemara, Three Powerful Lessons in Thinking, Learning, and Teaching

There is a standard line of reasoning in logic known as reductio ad absurdum. It is usually employed when a direct proof would be difficult/impossible and it works as follows: You want to prove that some proposition is true; say, for example, that the there is no smallest, positive, non-zero, rational number. The first step is to consider the opposite; in this case, suppose that there is, in fact, a  smallest, positive, non-zero, rational number; call it x. Next, we formulate some logical implications of our supposition; in our case, divide that smallest, positive, non-zero, rational number by two; call that y. We have posited that x is the smallest, positive, non-zero, rational number, which implies that any other positive, non-zero, rational number -- including y -- is bigger than x. So by assuming that there is, in fact, a smallest, positive, non-zero, rational number, we have shown by logical inference that would mean there is a number that is both half the size and larger than that smallest, positive, non-zero, rational number. That's absurd; which proves that they opposite or our original proposition is false, which means that that that statement we wanted to prove must be true. Simple.

I wanted to do a simple example in gory detail to note two critical factors upon which every proof by reductio ad absurdum stands (or not). First, the conclusion is truly absurd. Second, the original statement and the proposed to be false opposite are truly the only two possibilities. People like to use reductio ad absurdum because it is relatively easy and it lets the one using this technique formally and politely say that anyone who disagrees with him is just being absurd. Very, very, very often, though... neither are true; the conclusion is not actually absurd and the original statement and its proposed to be false opposite are not actually the only two possibilities.

How does one learn things like this? From the gemara, of course; one source: Bava Kamma, 72a.

Here's what you need to know before we start: (1) If someone steals and then slaughters an ox, then is caught and found guilty by Beis Din, then he owes five times the value to the owner. (Four times the value for a sheep.) However, that is only true if the ox still belongs to the owner; if something happened to abolish his ownership, then the thief would only owe the usual double. (2) It is forbidden to slaughter a non-consecrated animal in the courtyard if the Beis HaMikdash. Were one to commit such a heinous crime, then the animal remains are forbidden for any use whatsoever to everyone -- rendering it, for all intents and purposes, ownerless. (3) The mishna on a previous daf stated that a thief who slaughters a stolen (and non-consecrated) animal in the courtyard of the Beis HaMikdash would owe five (four for a sheep) times its value to the owner.

R' Chavivi from Chuzana said to R' Ashi, "Hey! I see a clear inference from the mishna that is true, because if it weren't true, then the animal would already be ownerless ans so the thief would only owe double!" (There are actually two versions of what is, but we don't need that for this discussion.) Rav Huna the son of Rava said, "Your premise is wrong." Rav Ashi then responds, "Don't dismiss question so fast; it is quite reasonable. However, we have no proof, as the mishna may very well be talking about a case where the slaughtering process started outside the courtyard and then finished inside the courtyard."

What just happened? R' Chavivi and Rav Huna were both jumping to a logical inference without really addressing whether or not such a leap was justified. R' Chavivi had a question on what he had learned and immediately began formulating wonderful new insights. Rav Huna agreed in principle, but was dismissive of the question.

Rav Ashi did a few things: (1) He demonstrated that the question was certainly reasonable and should not simply be dismissed. (2) He brought that question to the fore, demonstrating that appreciating the question is actually a part of the process of learning. (3) He demonstrated that the question could be addressed by broadening/deepening the context of the case stated by the mishna.

Rav Ashi at the very least demonstrated that there were more then the two possibilities ( and ) More than that, though, he was leading by example how to take apart a mishna. He was also teaching the need to respect your colleagues enough to take their questions seriously. He was also teaching not to jump to conclusions; spend time with the questions to deepen and broaden your understanding of the entire situation and case.

Nine lines of gemara...