Paradox and Anti-Paradox in Halakhic Decision-Making

HaMa'ayan – 5771

Rabbi Michael Abraham

Outline:
On paradox and anti-paradox
Anti-paradox in halakhic decision-making
Paradox in halakhic decision-making
Summary

On Paradox and Anti-Paradox

The 'liar paradox' is one of the oldest paradoxes. In its well-known formulation it is presented thus: a resident of Crete says: "All the inhabitants of Crete are liars." It is commonly thought that this is a paradox, because if indeed all the inhabitants of Crete are liars, then this resident too is a liar, and therefore this statement is false, so its content must be denied. That is, the situation is that the inhabitants of Crete are not liars, from which it follows that this statement is true, and so on…

But this is a mistake. On the logical plane this statement is not paradoxical at all, since the loop described here is based on an error in applying the negation operator. The negation of the statement "All the inhabitants of Crete are liars" is not the statement "All the inhabitants of Crete are truth-tellers," but rather the proposition "There is at least one inhabitant of Crete who tells the truth." If we assume that in Crete there is indeed only one truth-teller, and that this inhabitant is not the speaker himself but rather his friend, the loop stops immediately. The conclusion is that there is someone among the inhabitants of Crete who is a liar, but it is not the speaker himself[1].

By contrast, the following statement is genuinely paradoxical:

Statement A: Statement A is false.

This is a statement that refers to itself, and here the loop truly cannot be stopped; if this statement is false, then its content must be denied, from which it follows that it is true. But if it is true, then its claim is correct — namely, that it is false — and so on…

And what about the following statement:

Statement B: Statement B is true.

At first glance there is no problem here at all, since the content of this statement does not contradict its truth-value. But it turns out that here too there is a logical problem, and this time the problem is the reverse (therefore one may call it an 'anti-paradox').

Some define a 'proposition' as a sentence that can receive one of two truth-values: true or false. A statement is true if its content corresponds to the facts (= the state of affairs in the world), and it is false if it does not correspond to the facts. The proposition "The sun is now shining" is either true or false. If, as a matter of fact, the sun is now shining, then that proposition is true. And if, as a matter of fact, the sun is not now shining, then that proposition is false. For every ordinary proposition, only one of these two possibilities is correct. It can receive one and only one of the two possible truth-values, 'true' or 'false'.

It is precisely here that the special character of Statement B appears. If we decide that it is true, then examining its content will show that it is indeed true, and therefore the truth-value of the proposition is 'true'. That is, the determination that this proposition is true is consistent with its content (= the state of affairs it describes), and therefore the truth-value appropriate to this statement is 'true'. But if we decide that it is false, this means that Statement B is false, that is, that the content of Statement B is not correct. Here too, that determination leads us to a consistent result (= correspondence between the truth-value of the proposition and the state of affairs it describes). Therefore the truth-value appropriate to this proposition is 'false'.

So we have here an anti-paradox: in a paradox of the type of the 'liar paradox,' that is, Statement A above, there is no possible truth-value at all. By contrast, in an anti-paradox of the type of Statement B, two truth-values may be assigned, and both will be consistent with the 'facts' (= the content of the proposition).

Anti-Paradox in Halakhic Decision-Making

Let us now move on to describe two examples of problems in halakhic rulings, one of the type of the liar paradox and the other of the type of an anti-paradox.

The Talmud in Eruvin 13 describes a prolonged dispute between Beit Shammai and Beit Hillel, which ends with a ruling on the basis of a heavenly voice that the Jewish law follows Beit Hillel. And in Tosafot here (Eruvin 6b, s.v. here) they ask how one can follow a heavenly voice, when in the case of the Oven of Akhnai it was established that from the verse "It is not in heaven" ("It is not in heaven") we learn that one does not heed a heavenly voice. Tosafot write:

Here, however, it is after the heavenly voice. And if you ask: what is different such that we do not rule in accordance with the heavenly voice in the case of Rabbi Eliezer in "Gold" (Bava Metzia 59b)? One may say that there it emerged only in his honor, as is evident there. Furthermore, there it was against the majority, and the Torah said, "follow the majority." But here, on the contrary, the House of Hillel were the majority, and they needed the heavenly voice only because the House of Shammai were sharper. And if you ask: if so, what does it mean when it says, "It is Rabbi Joshua, who said: we do not pay attention to a heavenly voice"? After all, Rabbi Joshua said this only regarding the heavenly voice in Rabbi Eliezer's case. One may say that "It is not in heaven" implies that one should not pay attention at all to any heavenly voice. (Here it was after the heavenly voice. And if you ask: what is different here, such that we do not rule in accordance with Rabbi Eliezer's heavenly voice in the chapter "Gold" [Bava Metzia 59b]? One may say that there it came forth only in his honor, as is evident there. Furthermore, there it was against the majority, and the Torah said follow the majority ("follow the majority"), whereas here, on the contrary, Beit Hillel were the majority, and the heavenly voice was needed only because Beit Shammai were sharper. And if you ask: if so, what does it mean when it says that this follows Rabbi Yehoshua, who said that one does not heed a heavenly voice? Did Rabbi Yehoshua say this only regarding Rabbi Eliezer's heavenly voice? One may answer that It is not in heaven implies that one should not heed any heavenly voice at all.)

 

It follows from Tosafot's words that beyond all the specific disputes between Beit Shammai and Beit Hillel, there was between them another broad meta-halakhic dispute: whether one follows the majority in wisdom or the numerical majority (somewhat along the lines of what Sefer HaChinukh brings in commandment 78, and Nachmanides and Rav Hai Gaon disagree there about whether in a court one follows the majority in wisdom or the majority of judges; see there and in Minchat Chinukh).

The situation now is as follows: there is a dispute between Beit Shammai and Beit Hillel over whether one follows the majority in wisdom or the numerical majority. The facts are that Beit Shammai were sharper (that is, they were the majority in wisdom), whereas Beit Hillel were more numerous. Therefore, if one follows the majority in wisdom, then the Jewish law follows Beit Shammai, who hold that one should follow the majority in wisdom. That is, this proposition is consistent. But if, in practice, one should follow the numerical majority, the conclusion is that the Jewish law follows Beit Hillel, and they hold that one should follow the numerical majority. That is, this determination too is consistent. This is a case of anti-paradox, and therefore we have no logical way to decide this dispute. This is the explanation proposed by the Tosafists (cited above) for why a heavenly voice was required here.

Paradox in Halakhic Decision-Making

In Bava Metzia 51a, Rav and Shmuel disputed the question of a stipulation regarding overreaching (ona'ah): It was stated: If one says to another, "On condition that you have no claim against me for overcharging," Rav said: he does have a claim against him for overcharging, and Shmuel said: he does not have a claim against him for overcharging. ("It was stated: if one says to his fellow, 'On condition that you have no claim of overreaching against me,' Rav said: he does have against him a claim of overreaching, and Shmuel said: he does not have against him a claim of overreaching.").

How is the Jewish law to be decided in this dispute? The rule in our hands is that the Jewish law follows Shmuel in monetary law and Rav in prohibitions (see Bekhorot 49b and Niddah 24b); in disputes that concern civil law, the Jewish law follows Shmuel, and in disputes that concern prohibitions it follows Rav. How should we define the area of this dispute regarding stipulating away what is written in the Torah? In Tosafot, s.v. what (ad loc., 51b), they cite the words of Rabbenu Hananel, who ruled like Rav because for the Jewish law follows Rav in matters of prohibition ("the Jewish law follows Rav in prohibitions"), and they object to him: This is puzzling, for they do not disagree about whether it is permitted to do so; rather, they disagree about whether he is obligated to return the overcharge, and the Jewish law follows Shmuel in monetary law. ("This is puzzling, for they do not dispute whether it is permitted to do so; they dispute whether he is obligated to return the overcharge, and the Jewish law follows Shmuel in monetary matters!"). Tosafot wonder about Rabbenu Hananel: how does he rule like Rav even though this is a monetary question? But Rabbenu Hananel himself apparently held that this is a question of prohibition rather than money: whether it is possible to stipulate away what is written in the Torah[2]. But some ruled like Shmuel. For example, Hagahot Maimoniyot (Laws of Sale 13:3, letter 3) writes that there were Geonim who ruled like Shmuel, for the Jewish law follows him in monetary law ("for the Jewish law follows him in monetary matters"). At first glance, there is room to make the dispute about the halakhic ruling depend on the question of how to define the scope of the dispute: is this a matter of monetary law or of prohibition and permission?

Now, later in the Talmud there, we find an analysis of this dispute and its comparison to the tannaitic dispute (Rabbi Meir and Rabbi Yehuda) concerning stipulating away what is written in the Torah:

Shall we say that Rav speaks in accordance with Rabbi Meir, and Shmuel speaks in accordance with Rabbi Yehuda? For it was taught: If one says to a woman, "Behold, you are betrothed to me on condition that you have no claim upon me for sustenance, clothing, and conjugal rights," she is betrothed, but his condition is void—these are the words of Rabbi Meir. Rabbi Yehuda says: in a monetary matter, his condition stands. (Shall we say that Rav speaks in accordance with Rabbi Meir and Shmuel in accordance with Rabbi Yehuda? For it was taught: if one says to a woman, "Behold, you are betrothed to me on condition that you have against me no claim of support, clothing, or conjugal rights," she is betrothed and his condition is void — these are the words of Rabbi Meir. Rabbi Yehuda says: in a monetary matter, his condition stands.)

Now, although the Talmud presents the dispute of Rav and Shmuel as seemingly parallel to the dispute of Rabbi Meir and Rabbi Yehuda regarding stipulating away what is written in the Torah, immediately afterward each of the amoraim nonetheless tries to explain why he accords with everyone:

Rav could say to you: I state my view even according to Rabbi Yehuda. Rabbi Yehuda says so there only because she knew and waived it; but here, does he know that he is waiving it? And Shmuel says: I state my view even according to Rabbi Meir. Rabbi Meir says so there only because he certainly uproots the condition; but here, who says that he uproots anything? (Rav would say to you: I say my view even according to Rabbi Yehuda; Rabbi Yehuda said so there only because she knew and waived her claim, but here who knows what he is waiving? And Shmuel said: I say my view even according to Rabbi Meir; Rabbi Meir said so there only because he is certainly uprooting something, but here who can say that he is uprooting anything?)

Rav explains that stipulating away what is written in the Torah in monetary matters is a matter of waiver, and when a person does not know what he is waiving there is no waiver. Shmuel explains that such a stipulation is impossible in an essential sense, and in his view there is no difference between prohibitions and monetary matters; rather, even in prohibitions, if there is no definite uprooting of Torah law, this is not considered stipulating away what is written in the Torah[3].

From these rationales it emerges fairly clearly that according to Rav's approach, the principle of stipulating away what is written in the Torah in monetary matters belongs to contract law, and he explains that according to Rabbi Yehuda one may stipulate because there is waiver here. But according to Shmuel, this is a principled halakhic question (about the laws of conditions in general, and not specifically about monetary conditions); in his view, the fact that according to Rabbi Yehuda the stipulation in monetary matters takes effect is not connected to the laws of waiver, but rather to the law of conditions.

It follows that according to Rav there is here a dispute in contract law, which is part of civil law, and therefore the Jewish law should be like Shmuel. But according to Shmuel this is a halakhic dispute of prohibition and permission, and therefore the Jewish law should be like Rav. This is a paradox of decision, parallel to the 'liar paradox': if Rav is correct, then the Jewish law is like Shmuel; and if Shmuel is correct, then the Jewish law is like Rav.

To be sure, this presentation of the dispute can be rejected, but it certainly has a place (even among the medieval authorities). It certainly suffices to illustrate the possibility of a paradox of decision in Jewish law.

Summary

We have presented two types of problems of decision in Jewish law, and shown their correspondence to two types of problems on the general logical plane: paradox and anti-paradox. The question of what to do in the two cases, that is, how to arrive at the halakhic ruling, is different: in the first case, that of the anti-paradox, we saw that a heavenly voice emerged and decided for us; in the second case, the paradoxical one, it was the halakhic decisors who decided, even if it is not clear whether they were aware of the paradoxical character of the problem before them[4].

[1] I do not mean to say that this was necessarily the speaker's intention, but only to argue that there exists a logically consistent interpretation of this proposition. According to what is called in logical interpretation the 'principle of charity,' we attach to every proposition a consistent interpretation, even if it is not clear that the speaker intended this.

[2] There is some support for this: in the Talmudic passage in Gittin 84b they raised the possibility of linking the law of stipulating away what is written in the Torah to a stipulation to commit a prohibition (such as "On condition that you eat pork", "on condition that you eat pork"); but this is not the place to elaborate.

[3] Simply speaking, according to Shmuel, if a person stipulates away what is written in the Torah in the realm of prohibitions in a manner where it is not certain that he has uprooted the law, there too the condition stands. For example, if a person stipulates I am hereby a nazirite on condition that I be permitted to drink the contents of this bottle ("I am hereby a nazirite on condition that I be permitted to drink the contents of this bottle"), and he does not know whether it is wine or water, this is not a case of stipulating away what is written in the Torah, and his condition stands. Much more could be discussed about this, but this is not the place.

[4] In my book Two Wagons and a Hot-Air Balloon (second edition, Beit El, 2007, ninth section), I discussed the way to resolve paradoxes in Jewish law, and connected this to 'Gödel's theorem,' which teaches us that in order to resolve a paradoxical loop we must step outside the boundaries of the system under discussion and assign different weights to each of the principles involved in the process, and this may enable us to reach a decision (in chapter 1 of that section, the mechanism of stepping outside the system is described, and in the chapters that follow the connection to 'Gödel's theorem' in mathematical logic is discussed). Indeed, one can analyze the methods of decision of the various halakhic decisors in the second passage (that of conditions) from awareness of the possibility of rational decision of paradoxes, and see what weights each of them assigned to the principles involved in the decision; however, such an analysis lies outside the scope of this article.