Paradox and anti-paradox in halakhic decision-making
The Spring – 5771
Rabbi Michael Avraham
outline:
On paradox and anti-paradox
Anti-paradox in Halacha ruling
A paradox in the ruling of halakhah
Summary
On paradox and anti-paradox
The 'liar paradox' is one of the oldest paradoxes. In its well-known formulation, it is presented as follows: A Cretan says: "All Cretans are liars." It is commonly thought that this is a paradox, because if indeed all Cretans are liars, then this resident is also a liar, and therefore this sentence is false, and in any case its content must be negated. In other words, the situation is that Cretans are not liars, and hence this sentence is true, and so on...
But this is a mistake. On a logical level, this sentence is not paradoxical at all, since the loop described here is based on an error in the operation of the negation operator. The negation of the sentence "All Cretans are liars" is not the sentence "All Cretans are truthful", but the sentence "There is at least one Cretan who is truthful". If we assume that there is indeed only one truthful person in Crete, and that this person is not the speaker himself but his friend, the loop stops immediately. The conclusion is that there is someone among the Cretans who is a liar, but it is not the speaker himself.[1].
In contrast, the following sentence is a truly paradoxical sentence:
Sentence A: Sentence A is false.
This is a sentence that deals with itself, and here the loop is truly unstoppable; if this sentence is false, then its content must be negated, and hence it is true. But if it is true, then its claim is true, meaning that it is false, and so on…
And what about the following sentence:
Sentence B: Sentence B is true.
Apparently there is no problem here, since the content of this sentence does not contradict its truth value. But it turns out that there is a logical problem here too, and this time the problem is the opposite (therefore it can be said that this is an 'anti-paradox').
Some define a 'claim' as a sentence that can take one of two truth values: true or false. A sentence 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 claim "The sun is rising now" is a true or false claim. If the sun is actually rising now, then this claim is true. And if the sun is actually not rising now, then this claim is false. For any ordinary claim, only one of these two possibilities is true. It can take only one of the two possible truth values, 'true' or 'false'.
This is precisely where the special nature of claim B appears. If we decide that it is true, then examining its content will reveal that it is indeed true, and therefore the truth value of the claim is 'true'. That is, the assertion that this claim is true is consistent with its content (= the state of affairs it describes) and therefore the truth value that corresponds to this sentence is 'true'. But even if we decide that it is false, this means that claim B is false, that is, that the content of claim B is not true. That is, this assertion also leads us to a consistent result (= correspondence between the truth value of the claim and the state of affairs it describes). Therefore, the truth value that corresponds to this verse is 'false'.
So, we have an anti-paradox here: in a paradox such as the 'liar paradox', i.e., proposition A above, there is no possible truth value. In contrast, in an anti-paradox such as proposition B, two truth values can be determined, and both will be consistent with the 'facts' (=the content of the claim).
Anti-paradox in Halacha ruling
We will 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 anti-paradox.
The Gemara in Eruvin 13 describes an ongoing dispute between B.S. and B.H., which ends with a decision according to Bat Kol, which is the law of B.H. 25. And in Toda here (Eruvin 6:2) they had difficulty in following Bat Kol, while in Achnai's Torah it was renewed that from the verse "It is not in the heavens" we learn that there is no mishgichin in Bat Kol? And the Toda wrote about this:
Here after a voice. And if you say, "Why did you hate that it did not last as long as the voice of Rabbi Eliezer of Zehav (B"M 55, 2)?" And it should be said that it did not come out except for his honor, as the word of the innocent. And even more, it was against many, and the Torah said to be after many, to be inclined, but here, rather, the House of Hillel was the majority, and they did not need a voice except because of Debith Shammai, who was a Harifi Tefi. And if you say, then why did Rabbi Yehoshua say, "There is no supervision over a voice," did not Rabbi Yehoshua say, "But the voice of Rabbi Yehoshua?" And it is said, "Without heaven," meaning that there is no supervision at all over any voice.
This means from the words of the Toss that beyond all the specific disagreements that existed between the B"S and the B", there was another broad meta-halachic disagreement between them: whether to follow the majority of wisdom or the majority of the minyan (similar to what the teaching brings in Mitzvah 88, and the Ramban and Rav Hai Gaon disagreed on this: whether in the B"D they follow the majority of wisdom or the majority of the dayanim, Aiyish and Minach).
The situation now is this: There is a dispute between the B"S and the B"H on the question of whether to follow the majority of wisdom or the majority of the minyan. The facts are that the B"S was the most prominent of the Tefi (meaning they were the majority of wisdom) and the B"H were more numerous. If so, if one follows the majority of wisdom, then the halakha of the B"S is that one must follow the majority of wisdom. In other words, this claim is consistent. But even if the halakha is to follow the majority of the minyan, the conclusion is that the halakha of the B"H is, and they believe that one must follow the majority of the minyan. In other words, this assertion is also consistent. This is a situation of anti-paradox, and therefore we have no logical way to decide this dispute. This is the explanation offered by the authors of the Tos' (which we cited above) for the need for a Bat Kol ruling here.
A paradox in the ruling of halakhah
In the BM, on page 1, Rav and Shmuel disagreed on the question of a gift for deceit: "Itamar, who says to his friend that you have no deceit against me, Rav said that he has deceit against him, and Shmuel said that he has no deceit against him."
How should the halakha be decided in this dispute? The general rule is that the halakha is as Samuel in its law and as Rabbi in its prohibitions (see Bekorot 44:2 and Nida 24:2), in disputes concerning foreigners the halakha is as Samuel and in disputes concerning prohibitions the halakha is as Rabbi. How should we define the scope of the dispute regarding the stipulation on what is written in the Torah? In Tud'a Bema (ibid., 2) they cited the words of Rabbi Chaim who ruled as Rabbi Chaim in his halakha as Rabbi in his prohibitions, and he criticized him: "It is strange, that he does not rule as a rabbi if it is permissible to do so, but rather as a rabbi if he must repay the fraud, and the halakha is as Samuel in his law"! Tosafots are perplexed by Rabbi Chaim's ruling as Rabbi, even though it is a question of merit. However, Rabbi Chaim himself apparently believed that it was a question of merit and not of merit: Is it possible to stipulate on what is written in the Torah?[2]But some ruled as Samuel, for example in the Maimonides glosses (Hebrew: הלהכתא פיעד האלה גאט ג) it is written that there are geonim who ruled as Samuel "דהלכתא קווטיה בידיני". Apparently, there is a place to hang the dispute regarding the ruling of the halakhah on the question of how to define the area of dispute, whether it is about the laws of money or about prohibition and permission.
And here we find later in the Gemara an analysis of this controversy, and its comparison to the controversy of the Tannaim (R"M" and R"I) regarding a gift for what is written in the Torah:
To Rav, who said as Rabbi Meir and Shmuel said as Rabbi Yehuda, it is said that one who says to a woman, "You are consecrated to me, so that you have no remaining covering on me," and she answers, "She is consecrated, and she has renounced." The words of Rabbi Meir are nullified. Rabbi Yehuda says that in the matter of wealth, the vow is valid.
And so, even though the Gemara presents that the disagreement between Rav and Shmuel is ostensibly like the disagreement between Ram and Ri regarding the stipulation of what is written in the Torah, immediately afterwards each of the Amoraim tries to explain why he goes as he does:
Rabbi said to you, please tell even Rabbi Yehuda, up to this point Rabbi Yehuda has not said anything but knowledge and not illness, but here who knows that he is sick? And Shmuel said, please tell even Rabbi Meir, up to this point Rabbi Meir has not said anything but knowledge and not barren, but here who would say that he is barren from me?
Rav explains that the conditionality of what is written in the Torah in a gift is a matter of forgiveness, and when a person does not know what he is being forgiven for, there is no forgiveness. And Shmuel explains that such a conditionality is fundamentally impossible, and in his opinion there is no difference between prohibitions and gifts, except that even in prohibitions, if there is no definite displacement of the words of the Torah, it is not considered a conditionality of what is written in the Torah.[3].
From these reasonings, it is quite clear that according to the Rav's method, the principle of stipulating what is written in the Torah regarding money belongs to the law of contracts, and he explains that according to Rabbi Yehuda, it is possible to stipulate because there is forgiveness here. Whereas according to Shmuel's method, it is a fundamental halachic question (in the law of conditions in general, and not necessarily in terms of money); in his opinion, the fact that according to Rabbi Yehuda, stipulating money is valid is not related to the law of forgiveness – but to the laws of the conditions.
It turns out that according to Rav, there is a dispute here in the law of contracts, which are part of the law of the covenant, and therefore the halakha should be as Samuel. Whereas according to Shmuel, this is a halakhic dispute of prohibition and permission, and therefore the halakha should be as Rabbi. This is the paradox of decision, which is parallel to the 'paradox of the liar': if Rav is right, then the halakha is as Samuel, and if Shmuel is right, then the halakha is as Rabbi.
Although this position of the dispute must be rejected, it certainly has a place (even in the Rishonim). It certainly demonstrates the possibility of a paradox of decision in halakhic law.
Summary
We presented two types of problems of decision in halakhah, and showed 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 accept the halakhic decision, is different: in the first case, that of the anti-paradox, we saw that a unanimous decision was made for us; in the second case, that of the paradoxical one, the poskim are the ones who decided, even if it is not clear whether they were aware of the paradoxes in 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 is a logically consistent meaning for this verse. According to what is called in logical interpretation the 'principle of grace', we attach a consistent meaning to every verse, even if it is not clear that the speaker intended it.
[2] Figure of evidence for the matter: In the issue of Gittin, page 11 of the book of Ra'b, they raised the possibility of tying the law of gifting to what is written in the Torah to the condition of doing a prohibition (such as "in order that you may eat pork"); and so on.
[3] Simply put, according to Shmuel, if a person makes a condition about what is written in the Torah in the prohibitions in a way that is not necessarily unreasonable, the condition also exists there. For example, if a person makes a condition that he is a Nazirite so that it is permissible to drink the contents of this bottle (and he does not know whether it is wine or water), there is no such condition in the Torah, and the condition exists. There is more to be said about this, Tuba, and so on.
[4] In my book 'Two Carts and a Hot Air Balloon' (second edition, Beit El 2007, chapter nine), I discussed the path to resolving paradoxes in halakhic law, and I linked this to 'Godel's theorem', which teaches us that in order to resolve a paradoxical loop, we must go outside the boundaries of the system in question and attach different weights to each of the principles involved in the process, and this may allow us to reach a decision (Chapter 1 of the aforementioned chapter describes the mechanism of going outside the system, and in the chapters that follow we discuss the connection to 'Godel's theorem' in mathematical logic). Indeed, it is possible to analyze the ways in which the various poskim reach decisions on the second issue (of conditions) from an awareness of the possibility of rationally resolving paradoxes, and to see what weights each of them gave to the principles involved in the decision; however, such an analysis is beyond the scope of this article.