Tuesday, August 12, 2003

Proof by contradiction in the thought of Chazal

I was thinking more about that yerushalmi I cited earlier, about the dispute between R Yochanan and Resh Lakish about whether mountains can function as walls to remove the status of reshut harabim from cities. I was discussing it at the Shabbos table as a guest at someone's home, and another guest's reaction was, "I hope you don't mind me saying this, but this all sounds absurd." I launched into appropriate apologetics, but it made me think more about what is going on, on a higher level, in this gemara and others, and how while the particulars seem extreme and farfetched, once you understand the system in which it is working, you can see a wonderful thought process going on.

There is a particular method of proving theorems in mathematics and computer science called proof by contradiction. Assuming you have some body of knowledge you are sure about, in the form of axioms and theorems which have already been proved. You want to prove some new theorem. To do so, you negate the theorem. That is, if I wished to to prove that it is raining outside, I would assume, that it is not raining outside. Then, I would derive, via some methods of proof, some statement that is false, and is known to be false because it contradicts some axiom or theorem already proved. Since we derive something false, some earlier assumption must be false. And, we only made one assumption, the negation. If the negation of the statement we want to prove is false, then the statement we want to prove is true.

Warning: heavy computer science terminology ahead
As an example, ATM is a language consisting of Turing Machines which halt and accept their input. We would like to know whether ATM is decidable - that is, whether we can construct a Turing Machine, that takes as input another Turing Machine B as well as B's input, and accepts the input if B would halt and accept B's input, and rejects the input if B would either loop indefinitely or would halt and reject B's input. This is a very useful type of Turing Machine to have. However, we prove via a method called diagonalization that such a Turing Machine cannot possibly exist.

ETM is another language, which consists of all Turing Machines that reject on all inputs. We want to show whether a decider for ETM exists. Such a deciding Turing Machine takes another Turing Machine B as input, and accepts if B will not accept any input, and rejects otherwise.

We prove that a decider for ETM cannot exist, not by analyzing ETM in any way to show mathematically that it cannot exist (which we can, by the way, do). Rather, we prove by contradiction.

We assume, for the purposes of contradiction, the negation of the statement we want to prove. So we assume that there is a Turing Machine C that decides ETM exists. Using C, we can construct a Turing Machine D which decides ATM. But, we know from before (we proved via diagonalization) that such a decider cannot exist. It must be that some statement or assumption we made earlier must be false, and the only assumption we made was the negation of the statement we wanted to prove. We assumed that a decider for ETM could exist, so it must be that a decider for ETM cannot exist.

And now back to the Torah

I think the same proof by contradiction occurs occasionally in the gemara. That is, we want to prove that a halacha, or halachic principle, is true. Rather than analyzing psukim, or mishnayot and braytos, or perhaps the halachic parameters of the question to determine truth or falsity, we demonstrate that assuming the opposite yields a falsehood or absurdity. Thus, proof by contradiction.

Axiom: Every law the Torah speaks about must exist in the real world.

Thus, as a made-up example, there is a mitzvah of sending away the mother bird before taking the eggs. If Chazal were to use hermeneutics to free from this obligation any birds whose species have feathers, all birds would be excluded, and then the Torah would contain a pasuk speaking about something that does not exist, which contradicts our axiom (which we know to be true), and so our assumption - that birds whose species have feathers are not within the commandment, would be false.

(This is NOT the same as "ain koach biyad chachamim laakor davar min hatorah," which refers to rabbinic enactments, rather than elaborations of Biblical law.)

Some examples of this in action occur on Bavli Chullin 10b-11a and on. There, we try to establish that first chazaka and then going after the majority are principles which are effective in the Biblical plane.

For the purposes of contradiction, they assume that chazaka does not exist. How then, asks R Shmuel bar Nachmeni, when the priest (Leviticus 14) seals up the house stricken with leprosy for 7 days. Perhaps from the time he observed it the leprosy had shrunk from the required size. The Torah speaks of the priest doing it, and the lack of a principle of chazaka would invalidate the entire process. It must be our assumption is false and chazaka does indeed exist.

R Acha bar Yaakov protests, perhaps the cohen walked out backwards, facing the leprosy. That is, by changing a detail not mentioned in the text, we can still assume the negation (the lack of chazaka) without achieving our desired contradiction.

Abaye answers two answers. Firstly, (the pasuk says "veyatza hakohen," the priest shall leave, and walking backwards is not called "yetziah," leaving. Thus, the cohen leaves facing forwards and so without chazaka the leprosy cannot be assumed to exist, so he cannot close up the house. Furthermore, what if the leprosy is on the back of the door? (This latter point I realize now is a sticking point for me, for Abaye is arguing here that in all cases, including the case with the back of the door.) The gemara continues arguing this case back and forth.

Similarly, by rov, majority, they try to prove from the head an olah offering, which cannot be broken apart, which presents a problem, for perhaps it is traif if the membrane of the brain were pierced, and then from a paschal offering, whose bones cannot be broken. They suggest that one might be obligated to burn through the skull with a coal, thus not breaking bones. The gemara continues. Thus, they try to show that rov must exist, for if we assume the negative, a mitzvah dictated in the Torah could not exist. This is pure proof by contradiction.

Now back to our yerushalmi (8:8) about a city surrounded by mountains:

"R Leizar asked R Yochanan, those cities surrounded by mountains, may one throw from it to reshus harabbim (public domain), or from reshus harabbim into it? He said to him, by your reasoning, there would never be any reshus harabim in the world!"

"Resh Lakish said, 'Indeed there is no reshus harabim until it is mefulash (open on both ends - this means, for example, open-ended on North and South, or alternatively, on East and West) from one end of the world until its (other) end."

"This seems the reverse of the opinion of Resh Lakish, who said 'There is no reshus harabim in this world, but there will be in the future to come, as it it said, (Yeshayahu 40:4), "kol gey yinasei," "every valley will be lifted up."

I said originally that there seems little contradiction within Resh Lakish. I now want to revisit this gemara.

R Yochanan rejected the idea of a city surrounded by mountains being a non-reshus harabbim (i.e. either a karmelis or a reshus hayachid). Why? Assume for the purposes of contradiction that the mountains would make it into a non-reshus harabbim. Then, we could say, as Resh Lakish actually does, that the only reshus haraabim would exist would be if it were mefulash (open) from the end of the world until its end. That is, no reshus harabim could exist in such a situation, says R Yochanan. However, we know there is a Biblical status of reshus harabbim, and a penalty of kares or chatas for advertently or inadvertently carrying from between a private and public domain, so reshus harabbim must exist. Thus, the assumption we made, that mountains prevent reshus harabbim's from, being reshus harabbims, must be false!

Note this discusses no feature of mountains (slope, being the product of nature rather than man, etc) that would invalidate them, but just the fact that we would arrive at an absurd conclusion.

The difficulty with using a reductio ad absurdum is that your opponent may take your absurdum and maintain it. This is in fact what Resh Lakish does. He claims the far reaching effects of mountains used by R Yochanan, and maintains that it is indeed so. However, his wording seems to suggest that there are some locations, though rare, which are indeed mefulash from one end of the world to the other, and in those grid coordinates we have reshus harabbim. Thus, we have no proof by contradiction, for the Torah's statement as to the existence of reshus harabbim is validated in reality.

(Alternatively, and I do not agree with this at all but am merely observing the possibility, one might say Resh Lakish was merely speaking out the reasoning for R Yochanan which was cited very tersely. At any rate, Resh Lakish's second statement surely argues with R Yochanan, but perhaps this is the meaning of "mechalfa shitetei deResh Lakish.")

Resh Laskish's second statement allows for the lack of the existence in reshus harabim in the present reality, since it will exist in the future. It is not clear that R Yochanan would disagree with such a statement, since this "future reality" argument does not seem to be juxtoposed to R Yochanan's statement.

Application to other situations.
A continental shelf surrounds all continents, and rises steeply for a sufficient height (certainly more than 10 tefachim). As such, this should make walls around all continents. Some suggest it, and people discuss the halachic applications of this, since this could theoretically take away the reshus harabbim status of certain places. Reshus Harabim is a major impediment to creating an eruv, for we can only make an eruv with tzuras hapesach etc. for a karmelis. (At least, that is what is commonly assumed. I think I can show this assumption to be false, but this is a topic for a later, more involved essay.)

I think there should be no question that the continental shelf cannot form walls to eliminate the status of reshus harabbim. Why? I make no appeals to the physical/halachic dimensions of a continental shelf. Rather, I point out, as R Yochanan did by mountains, that if we say this is the law then there would be no reshus harabbims in the world. Thus, via a proof by contradiction, they connot exist. We need not be concerned why the cannot exist. An explanation exists, but we need not know it, and it may not even be possible to know it (explanations of it might be rationalizations, and extrapolations to affect other halachos seems risky in terms of a strong possibility of determining a false halacha).

Further, we cannot even appeal to Resh Lakish. By mountains, he either maintains that reshus harabbim is rare but does exist, or does not exist now, but since all mountains will be flattened and all valleys lifted up, reshus harabim will exist. With continental shelves, no reshus harabim can exist now, and there is no verse telling us that continental shelves will be eliminated. So, Resh Lakish agrees to R Yochanan's principle, and so he would agree here that continental shelves are not efficacious, proving it by contradiction.

Postscript: bavli Eruvin 8a discusses a mavoi, one wall of which is formed by a steep rise from the sea or river. Such a wall is deemed invalid lest the sediment shift and change the slope to one that will not form an halachic wall. This seems to be rabbinic, so would not present a problem Biblically. Further, sediment shift could not reasonably eliminate the continental shelf. That is a limited discussion about the halachic parameters of a continental shelf. We still have, I maintain, the proof by contradiction.

{Updated Jan 18 2005 to allow comments and to correct some spelling errors.}

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