Logical Loops and Self-Reference (Column 406)
Why Russell’s theory of types is not a solution but an artificial blockage
The column begins again with the liar paradox: a sentence that refers to itself and generates a rolling contradiction. From there it turns to Russell’s proposal, which arranges statements in a hierarchy and forbids a statement from referring to what is above it or on its own level, thereby preventing self-reference. But the rabbi argues that this is ad hoc: the rule has no independent justification beyond the fact that it prevents paradoxes, and it also bans many legitimate formulations. Moreover, forbidding the formulation of a problem does not solve it; it only builds a poorer language in which one cannot talk about it.
Why it matters to recognize legitimate self-reference
The critique of Russell becomes stronger because there are many cases of self-reference that are not paradoxical at all. So it is unreasonable to disqualify the whole phenomenon merely because it sometimes produces contradiction. The column stresses that the cost of a sweeping ban is too high: it erases valid self-reference as well, and even references to other statements on the same level, where there is no difficulty at all.
Gödel, Turing, and proofs by contradiction rely on self-reference
The central mathematical example is Gödel’s second incompleteness theorem, which relies on the constructive building of a self-referential statement. If such statements are illegitimate, the proof itself collapses. The same is true of Turing’s theorem on the halting problem, which uses a similar mechanism of a machine that checks itself. The conclusion is that self-reference is not treated here as a flaw, but as a legitimate tool for proof by contradiction.
The debate over “These and those” shows how pluralism collapses when applied to itself
The column recalls the dispute between halakhic monism, according to which there is one halakhic truth, and pluralism, according to which several halakhic positions can be true. The argument against pluralism is a self-referential one: if pluralism is applied to this dispute itself, it follows that the monist is also right; but the monist says the pluralist is wrong, so the self-application of pluralism yields a contradiction. One can of course exempt this particular dispute, but the rabbi sees that as a move similar to the theory of types: an artificial exception without sufficient justification.
Migo and lifnei iver: not every loop is a paradox
In migo, the Illui of Meitshet challenges the rule by saying that it undermines itself: the defendant may choose the weaker claim precisely because he knows that migo will strengthen it. This indeed has a loop-like smell, but the rabbi stresses that this is not necessarily self-reference in the strong sense; it is a logical challenge to grounding the rule by means of its own logic. Likewise, in the example of lifnei iver cited in the name of the rabbi of Ponevezh, there appears to be an infinite chain of mutual stumbling blocks, but infinity as such is still not a paradox. This yields an important distinction: there are loops that create no logical problem, so the mere existence of a loop or of self-application is not enough to disqualify it.
The interim conclusion: self-reference is not invalid, and only sometimes exposes a contradiction
After these examples, the column pauses and sharpens the main point: self-reference as such is not illegitimate. Sometimes it is used precisely to show that a certain assumption leads to paradox, and thus to refute that assumption. The very possibility of proving things this way shows that self-reference is considered a valid move; the problem begins only when one gets a loop with no stable outcome.
The red heifer and the yoke: why discomfort created by the disqualification cannot validate
From here the column moves to halakhic examples in which a genuine loop does arise. In the sugya of “a yoke was placed upon it,” the disqualification applies only when the owner is pleased with what happened. Tosafot ask: if so, an owner would never want an expensive red heifer to become disqualified, so why should it ever be invalid? Their answer is that discomfort produced by the very disqualification is irrelevant; what matters is whether, from the standpoint of the act itself, he wanted it. The rabbi notes that this initially sounds like a technical Russell-like exclusion, but suggests a rationale: the law must derive from the facts, not retroactively create its own factual conditions.
The Ran’s two answers formulate the ban on a law depending on itself
The Ran offers two more precise formulations. According to the first, the heifer’s disqualification cannot be the cause of its validation; the law cannot operate inside the causal framework that generates it. According to the second, if you say that the disqualification creates discomfort and therefore validates it, then once it is validated the owner is again pleased, and it becomes disqualified again, and so on. In both answers the shared principle is that a law is not made to depend on itself, but the second answer also exposes a full paradox. Even so, a difficulty remains: even if there is a loop, why stop it specifically at disqualification rather than at validation?
Psik reisha de-lo nicha lei reproduces the same loop
The same structure appears in the Arukh’s position on psik reisha de-lo nicha lei: even when a forbidden result will certainly occur, if the result is unwelcome to the person he is exempt. But then the objection arises: nobody wants to become liable to a sin-offering or stoning, so on that logic every psik reisha should disappear. Here too one can answer along the two lines already seen: either one measures convenience without taking the halakhic consequence into account, or one says that if we exempt because of the discomfort we immediately generate the opposite loop, in which the result again becomes welcome and liability returns. And again the same question remains open: why choose to stop the loop דווקא on one side.
The open conclusion: the issue is not only that there is a loop, but the criterion for stopping it
The column closes by noting that in the last three examples — migo, the red heifer, and psik reisha de-lo nicha lei — the commentators use the existence of a loop to justify a specific halakhic position. But merely pointing to a loop still does not explain why halakha is decided this way rather than the opposite. So the real question, which will be addressed in the next column, is what non-arbitrary criterion can stop halakhic loops.
What Is a Loop
In columns 195–196 (see also column 319) I discussed self-referential paradoxes in halakhah and beyond. This is a special kind of paradox based on a statement’s reference to itself. Their archetype is the Liar Paradox, which is built as a sentence that refers to itself. We also saw instances of self-reference that do not lead to paradox (beyond what I called there an “anti-paradox,” there are self-references that pose no problem at all).
Those columns presented several halakhic examples of self-reference. Here I will again begin with the Liar Paradox and demonstrate legitimate and illegitimate uses of self-reference. I will then bring two further halakhic examples in which self-reference is used, and finally explain a logical mechanism proposed by R. Shimon Shkop for stopping loops in halakhah.
A Further Look at the Liar Paradox and the Theory of Types
As noted, the Liar Paradox is formulated as a claim that refers to itself:
(A) Statement (A) is false.
If it is true, then its content is true—namely, that statement (A) (i.e., itself) is false. But if it is false, then its content is not correct—meaning that statement (A) is true—and so on in an endless loop.
I already mentioned that Bertrand Russell, in the introduction to his monumental work (with Whitehead) Principia Mathematica, proposed to solve the paradox by a theory he called “the theory of types.” He there defines a hierarchy of statements (different “types” of statements, one above another), and proposes adopting a rule that a statement cannot refer to statements higher than it or equal to it in the hierarchy. It is very easy to see that this rule prevents self-reference (since within it a statement cannot refer to itself).
The problem is that this rule smells ad hoc. What justifies such a rule, other than the fact that it blocks self-referential paradoxes? Anyone can see that they can be prevented in other ways as well—the simplest and most natural being to directly forbid self-reference. So why go the roundabout way when one can do it directly? One can do something even more limited: simply forbid paradoxes, i.e., stipulate that a paradoxical sentence is not admissible in the language, period. You understand that these are ad hoc solutions suffering from at least one of two main problems: (a) they lack justification in themselves; (b) together with paradoxical claims they also block many additional legitimate, non-problematic claims. As noted, there are claims that include self-reference yet create no problem—so why forbid their expression in the language?!
Beyond all that, there is a more fundamental issue: when we forbid the expression of something, that does not solve anything. This is exactly the method Stalin used to “solve” problems that arose in the Soviet Union: he simply forbade their expression and chopped off the heads of those who violated this friendly and self-evident rule. What Russell proposes—even if less violent and extreme—is essentially very similar. He does not solve the paradoxes, but constructs an artificial language in which it is forbidden or impossible to express them. But as I argued above, forbidding the expression of a problem is not a solution. In our language we can express these claims, and therefore the problem within each of them certainly exists. The fact that the problem cannot be translated into Russell’s language is at most an expression of the limitations of his language, but certainly not a solution to the problem.
I will say more. The fact that Russell’s rule prevents the emergence of paradoxes does not justify adopting that rule. A true solution to such paradoxes can only be a rule that is justified on its own terms (beyond the fact that it prevents paradoxes). There are many formal and arbitrary ways to prevent such paradoxes, and therefore it is hard to see the fact that his rule blocks paradoxes as a proof by contrapositive that establishes the rule.
Legitimate Self-Reference
As noted, one of the problems with Russell’s type of solution is that the rule he proposes forbids many statements that seem entirely legitimate. Note that he forbids any self-reference, not only paradoxical self-reference. Beyond that, he forbids even reference to other statements of the same type (even though there is not only no paradox here—there isn’t even self-reference at all). Furthermore, he forbids reference to statements of higher types, not only of the same type. This is an “expensive” and utterly unjustified price, certainly not justified by the fact that it prevents the expression of paradoxes.
The best way to illustrate this is to bring examples of statements that include self-reference without any problem, and therefore it is unreasonable to forbid them. A fortiori for statements that include no self-reference at all (statements that refer to other statements of the same type or to statements of a higher type).
Mathematical Examples
The first example is Gödel’s second incompleteness theorem. Without entering its content, I will only say that Gödel proved his theorem constructively—namely, by systematically building a self-referential statement—showing that it is necessarily true but unprovable within the system. If one may not construct self-referential statements, then Gödel’s proof collapses, since it relies on constructing an “illegal” statement.
It is interesting to note that Gödel’s theorem itself constitutes a frontal assault on Russell’s entire project in the above-mentioned book (Russell and Whitehead attempted to ground all of mathematics in set theory). But the theorem itself is built on an assumption that Russell’s framework rejects from the outset.[1] I will add that Turing’s theorem regarding the halting problem in computability theory (the theorem states that the halting problem is undecidable) is formulated and proven in a very similar fashion to Gödel’s theorem (mathematicians have shown equivalences between these theorems). There too, within the proof, one constructs a Turing machine that examines itself—that is, one uses self-reference that leads to a paradox to prove the claim by contradiction.
“These and Those”
This technique of proof by contradiction via self-application is used in other places and contexts as well. For example, there is a dispute over the meaning of the dictum “These and those are the words of the living God.” Monists hold that there is only one halakhic truth and the other opinion is legitimate but not true. Pluralists, by contrast, hold that there are multiple halakhic truths (i.e., there is no single halakhic truth). There is also a position known as “harmonism,” but I will not enter it here (it is a sophisticated monism). In my article “Is Halakhah Pluralistic?” I presented a self-applicative argument for monism. I asked there: according to the pluralist, how should we relate to this very dispute? On his view, it seems that the monist is also correct; but the monist claims that the pluralist is mistaken. That is, applying pluralism to the dispute over pluralism yields a contradiction. This is a proof by contradiction in favor of monism. To complete the picture, I add that monism is consistent, and of course can be applied coherently to this very dispute as well.
One could try to answer that the pluralist rule applies to all disputes except this one itself. This solution is very similar to the theory of types, as it carves out the problematic claim from the rule and thus saves it. But as I noted above, this is an artificial carve-out and does not seem justified. The principle underlying the pluralist approach in halakhah (as distinct from pluralism in general) is that a halakhic sage never errs (he has ruach ha-kodesh, special heavenly assistance, etc.). But if in this very dispute you admit there is an error by a halakhic sage, it follows that in principle he can err. So why should we not assume this also about other statements of halakhic sages? This is an example of a proof by contradiction through self-reference: I apply the meta-halakhic approach under discussion to the dispute about that very approach.[2]
Migo
In the laws of evidence in halakhah there is a principle called, in short, “migo” (“since/inasmuch as”). When a person advances a weak claim in a situation where he could have advanced a stronger claim that would have won him the case, this itself serves as evidence that he is not lying with the weaker claim. For example: someone is sued by his fellow who claims he lent him money a month ago and it was not repaid (the loan was for a week), and the defendant claims he repaid “within the time” (e.g., after two days). This is a weak claim, since there is a presumption that a person does not repay “within the time.” But in such a case the defendant has a migo—he could have claimed that he paid after the time (say, after two weeks). The logic of migo says that if he wanted to lie, he would surely have chosen the better claim (that he paid after the time). If he advances the weaker claim, it is presumably the truth. This is evidence in favor of the weaker claim.
The Illui of Meitshit challenges the principle of migo and argues that it self-destructs. The defendant knows that if he utters the weaker claim he will be believed because of the logic of migo—and perhaps precisely for that reason he chooses to utter the weaker claim. In other words, the migo principle turns the weaker claim into a stronger one, thereby pulling the rug out from under migo itself. I will not enter into answers to this difficulty (see my pamphlet on migo, and more pointedly the thread here).
Despite the whiff this move gives off, this in itself is not self-reference, since there is no statement referring to itself—just a regular logical difficulty. Still, there is a reference of the migo principle to a situation in which the migo principle exists, and in that sense there is something perhaps contrary to the theory of types. In different words: one cannot ground the migo principle on itself, but there is no bar to challenging it by virtue of its own logic.
Some sought to resolve this problem by claiming that it leads to a paradox: if we were to abolish migo because of this difficulty, it would re-emerge, since if migo gives no advantage, the weaker claim remains weak, and consequently he again has a migo if he advances it. I note in advance that this is a problematic solution, since it is still not clear why to stop the loop specifically by ruling that there is migo, and not at the state of “no migo.” That is, why should this argument explain the halakhah that does recognize migo as evidence? We will see this difficulty below and address it more generally.
“Before a Blind Person”
I once heard in the name of the Ponovezh Rav[3] that in a situation where I cause my fellow to stumble in the prohibition of “lifnei iver” (“before a blind person”) vis-à-vis me, an infinite chain of stumblings—and therefore of transgressions—is created. For example: I am a nazir who asks my friend for a cup of wine and he hands it to me from the other side of the river so he can pass it back to me. He causes me to violate the nazirite prohibition of drinking wine and has transgressed “lifnei iver,” and therefore I have caused him to transgress “lifnei iver.” But in that case I have transgressed “lifnei iver” (“lifnei de-lifnei”) in which he caused me to stumble, and therefore he has himself committed an additional “lifnei iver.” But I caused him to transgress that, too, and so I have myself transgressed yet another “lifnei iver,” and so on ad infinitum. Here there is apparently self-reference, and the Ponovezh Rav does not see a flaw in it.
However, regarding “lifnei de-lifnei” (causing a person to violate “lifnei iver” with respect to a third person), the commentators dispute its status (see briefly, for example, here and here). The Ponovezh Rav’s words were, of course, only according to the view that there is a prohibition of “lifnei de-lifnei,” and he applies this to the very person himself (where the “third person” is the original instigator himself). But the very discussion of “lifnei de-lifnei” indicates that there is a kind of potentially problematic self-reference here; otherwise, why exclude the prohibition of “lifnei iver” from the list of prohibitions in which causing a transgression is itself prohibited?
One could argue that such a situation does not touch the problem of self-reference, because this is a loop that does not create a logical problem. True, an infinite chain of prohibitions can be generated, but that in itself is not problematic. On the other hand, the very fact that even in such a situation some exclude “lifnei iver” perhaps hints that there is some problematic nature to self-reference even without a paradoxical structure. I note that all the explanations I saw that try to justify this exclusion rely on the specific parameters of the prohibition of “lifnei iver” itself—that is, they do not attribute the exclusion to the fact that this is self-reference. It seems they saw no problem in it.
Interim Summary
All these were examples of self-reference that does not create a paradox, unlike the Liar Paradox. Sometimes self-reference is used to prove a claim by pointing out that a given assumption generates a paradox (proof by contradiction), but it is important to understand that this does not mean that self-reference is paradoxical or inadmissible. On the contrary, as I noted, using such a proof technique shows that self-reference itself is not seen as problematic. When one applies it, a paradox arises and this refutes some assumption that led to it. If self-reference were inadmissible, there would be no place for such proofs.
We will now see two examples of loops that do create paradoxes.
First Example: “A Male Mounted Her—It Is Invalid”
The Gemara, Bava Metzia 30a, cites an aphorism of Rav Pappa regarding the eglah arufah:
For Rav Pappa said: If it were written “works” (we read “works”), I would say even if it happened on its own; and if it were written “worked” (we read “worked”), I would say only when he himself worked with it. Now that it is written “worked” but we read “works,” we require “working” analogous to “worked”: just as “worked” implies to his satisfaction, so too “working” must be to his satisfaction.
When a corpse is found and the killer is unknown, an eglah arufah is brought. The calf must be one upon which no yoke has ever been placed. The Gemara says that the disqualifying yoke is only one placed upon it with the owner’s consent (or at least where he has satisfaction from it). From the passage there we see that the same law applies to the red heifer (parah adumah) (see the Ran’s novellae there, who discusses whether the whole sugya concerns only the red heifer).
Tosafot there, s.v. “Af oved,” ask:
And if a male mounted her—why is she invalid? Surely he is certainly not pleased to invalidate a cow whose value is so great because of a minor act.
That is, the condition that the yoke must be placed upon her with the owner’s satisfaction empties the law of content, since an owner will never be pleased that his cow loses all its value (a valid red heifer is extremely valuable).
Tosafot answer:
One can say that if she were valid, he would be pleased; therefore we cannot validate [her].
The owner’s displeasure stems from the very fact that the cow is invalid, and such displeasure does not prevent disqualification. The “displeasure” the Gemara speaks of (which would validate the cow) is an objective displeasure—i.e., independent of the very law that the cow becomes invalid.
At first glance this sounds like a purely technical answer, and its logic is unclear. After all, in practice the owner truly is not pleased—so why should it matter that the displeasure is due to the disqualification itself? It is hard to ignore the resemblance between this solution and Russell’s solution to self-referential paradoxes (the theory of types): he proposes an arbitrary rule that artificially carves out problematic cases from the language’s admissible framework, without any intrinsic justification.
Precisely in a halakhic context, one could perhaps say that the requirement of the owner’s satisfaction is a requirement that defines the very act of placing the yoke. The Gemara defines that only actions that, in themselves, are desired by the owner disqualify the cow. Actions that, in themselves, are done for other purposes and are not desired by the owner do not disqualify it. Therefore, displeasure at the outcome—the cow’s becoming invalid—is irrelevant, as it pertains not to the definition of the act itself but only to its result.
The Ran there also found this difficult and offered two answers:
“If a male mounted her—she is invalid.” One could ask: why? Since you have invalidated her, he is not pleased; for a small benefit one does not incur a great loss, as said in “One may not maintain [animals].” And one can say: since the act, in itself, is to his satisfaction, we should not validate her on account of her invalidity—for if so, the invalidity would become the cause of validation—and that is impossible. And furthermore: if you validate her, he will certainly be pleased.
The first answer seems similar to that of Tosafot. But the Ran formulates it a bit differently, adding a rationale that explains this seemingly arbitrary carve-out. He argues that it is inconceivable that the cow’s invalidity would be the cause of its validation. That is, even if the fact that we invalidate the cow causes the owner’s displeasure, it still cannot be that because of that she reverts to being valid. There seems to be an assumption that, by the nature of halakhah, the factual circumstances are always the cause and the law is the effect. Therefore, the law cannot operate on the plane of causes and generate or neutralize itself. The proper order in halakhic determination is to first address the factual circumstances (whether there is or is not satisfaction) and only then, and from that, to determine the law. It is unreasonable to consider the law as part of the web of its own causes. Here we already have a rationale; in that sense there is progress beyond Russell’s theory of types, which employs a carve-out rule without intrinsic justification.
The Ran’s second answer is a logical “hack”: let us suppose that indeed the cow reverts to validity because the owner is not pleased by her invalidation. But if we now rule on that basis that she is valid, then placing the yoke is indeed to the owner’s satisfaction—and she is again to be invalidated because it was to his satisfaction.
There is a common foundation to the Ran’s two answers: the claim is that one cannot hang a law upon itself—that is, there is self-application here. The first answer suffices with the assertion that there is self-reference—i.e., a mixing of planes—even without the self-application creating a paradox. The second answer takes a further step and explains that such a situation creates a loop whose tail is in its mouth. It yields no clear result, and therefore it is an inadmissible claim.
However, the Ran’s second answer raises a difficulty. Suppose we follow him in that further step and there is indeed no justification to validate the animal—still, it is not clear why he stops the loop specifically at invalidity. He argues, by appeal to the looping argument, that when there is owner satisfaction the cow is invalid, as the Gemara rules. But to the same extent he could have stopped the loop one step later—at the cow’s validity. As I noted, the looping form of thought teaches that there is no clear law here; it is therefore not clear why this explains the halakhah that the cow is invalid. I raised a similar challenge above to the solution to the Illui of Meitshit’s difficulty regarding migo.
Second Example: Psik Reisha that Is Not to His Liking
A similar example can be found regarding R. Natan of Rome, author of the Arukh. The Tannaim dispute the law of davar she-eino mitkaven (an act not intended). For example, one who drags a bench from place to place and in doing so creates a furrow (the melakhah of plowing or building on Shabbat): is he liable for the furrow (so holds R. Yehudah), or is he exempt because his intention was only to move the bench and not to create a furrow (R. Shimon)? In practice we rule like R. Shimon that a non-intentional act is permitted.
But the Gemara reports a limitation to this exemption in a case of psik reisha (see on this in column 325). That is, if the prohibited outcome necessarily follows from the permitted act,[4] then even R. Shimon agrees he is liable. For example, when the ground is so soft that it is clear in advance that dragging the bench will create a furrow. In such a situation even R. Shimon agrees that even if he does not intend it, he is liable. The author of the Arukh adds a third layer to this structure. Based on difficulties in several sugyot, he argues that if the person is not pleased by the prohibited result, he is not liable even in a case of psik reisha. For example, if the furrow does not suit his plans (it ruins the ground), then even in a case where the creation of a furrow is certainly inevitable—he is exempt.
I have seen that some challenge the Arukh in a manner similar to the above challenges in Bava Metzia. Seemingly no one is pleased to incur stoning or a sin-offering, and therefore it is clear that in every case of psik reisha the person is displeased and should be exempt. The law of psik reisha would then be emptied of content, since a person who does not intend would always be exempt even in psik reisha.
Here, too, one can answer in the two ways we saw above. One can say, like Tosafot and like the Ran’s first answer, that the result that obligates is only one that is to his liking if we do not take the halakhic outcome into account. One can also answer like the Ran’s second answer: if we continue the loop and exempt the person in such a situation, we will again reach a state where it is to his liking and he is liable, and so on. Of course, my earlier comment on the Ran—that it is unclear why the conclusion is specifically this one and not the other (i.e., how one decides where to stop the loop)—applies here as well.
A Criterion for Stopping Loops
We are left to examine the three examples we have seen (migo, red heifer, and psik reisha that is not to his liking), in all of which commentators use the existence of a loop to defend a certain position, and seemingly choose arbitrarily one of the two possible outcomes at which to stop. In all these cases there is the difficulty of why they chose to stop the loop specifically at one of the two states and not the other. That will be the subject of the next column.
[1] In principle, one could defend Russell by arguing that Gödel’s theorem is not true because its proof uses an illegitimate statement (a self-referential statement, which is illegal under the theory of types). On its face this seems to me a possible defense; nonetheless, as far as I know, mathematicians and philosophers do not use it. For our purposes it suffices to note that mathematicians today accept Gödel’s theorem—i.e., they assume there is no principled bar to the existence of self-referential statements. In doing so they implicitly reject the theory of types. The reason is that that rule lacks intrinsic justification, and for them it is not enough that it prevents self-referential paradoxes.
[2] One could distinguish and say that the dictum “these and those” concerns halakhic positions and therefore should not be applied to meta-halakhic claims. The dispute about the meaning of the dictum is meta-halakhic and therefore should not be subject to the dictum. This is an independent justification for the carve-out and can therefore resolve the difficulty. But as I showed in the above-mentioned article, this dispute also has halakhic ramifications (regarding causing someone whose position differs from mine to commit, by his view but not by mine, a transgression), and therefore it is in fact a halakhic dispute.
[3] I have now found this briefly in his article “On the Matter of Lifnei Iver,” cited in the book Ohel Mordechai, p. 191; these words appear there on p. 196.
[4] The term psik reisha comes from the case of one who cuts off a rooster’s head to make it a toy for his child, but in doing so—surprisingly—the rooster does not survive and dies. On this the Gemara says: “Psik reisha—and it won’t die?!” meaning that death is a necessary result of cutting off the head; therefore he is liable for taking a life on Shabbat.
Discussion
The links in this column throw me back to two points that remained murky for me. And one more from the past. And another one.
A. Type theory as a solution to paradoxes.
Russell, in that introduction, does not say this. He is looking for a set of axioms and rules of inference from which one can formally derive everything we want to derive, while not allowing false things to be derived. If the method allows us to derive all the mathematical claims already known to us and does not allow us to derive any error, then it is a good method that mathematicians can conveniently and confidently use. The whole book, full of proofs of mathematical claims, is intended only to show that the set is adequate and complete, where “complete” means everything that is “interesting.” The set of axioms does not need to hit upon some truth, but to be a good working tool. That is what Russell says there, and not a general proposal to ignore paradoxes. And in relation to this, all the objections you wrote in the column are irrelevant. And type theory is an excellent mathematical tool. And see further here https://mikyab.net/posts/70517#comment-47457
B. Halakhic pluralism.
I said various things about this here https://mikyab.net/posts/71518#comment-50537 and I will repeat some of them, because I was not satisfied.
B1. The argument attacks pluralism on the strength of the fact that there exists a sage, deaf to others and subtle in understanding, who held monism. Who is this sage who held monism?
B2. Explain to me how you understand halakhic pluralism regarding some question of your choice, which you will choose to illustrate it with, and then I will explain pluralism also regarding the question of this very dispute itself. Even on a third reading of the article there, it did not become clear to me what is special about this dispute more than any other dispute in which pluralism is required.
B3. After all, there is no problem excluding a certain position from the rule if there are good considerations in favor of that exclusion. If indeed your argument proves that such an exclusion is necessary in order to sustain the pluralist position, which is the correct position, then fine—we exclude it. Obviously sages can err in an explicit Mishnah, and if the court ruled that the sun had set and in the end it had risen, that is not a ruling but an error. But if there are no decisive reasons against the position, then halakhic pluralism assumes it is correct.
C. “Do not place a stumbling block before the blind” of “before the blind.”
C1. Here https://mikyab.net/posts/66820#comment-35769 I proposed a simple solution to the loop of “before the blind.” It is that the one who stumbles causes the one who caused him to stumble only to violate “before the blind” retroactively, and a retroactive violation, according to a proposal you raised elsewhere, is not forbidden (like the Turei Even, who held that one may ask to annul a consecration of a sacrifice even though it would retroactively become ordinary meat slaughtered in the Temple courtyard. This aside from vows and oaths, where “he ate it in violation of its prohibition and then ate it in fulfillment of its condition—he is lashed”).
C2. And one may further say very simply that if in every case of “before the blind” there are infinitely many “before the blind” violations, then this prohibition itself includes that whole infinity, and it is one prohibition, because that is how it is defined: that in such an act (one that contains infinitely many ‘first-level before-the-blind’ cases) one violates the prohibition of “before the blind.” And there is no reason to think that one really violates infinitely many prohibitions.
C3. It is still not clear what practical difference it makes whether one violates “before the blind” once or infinitely many times. One is not lashed for this prohibition because it is a general prohibition. Is there a case in which one must choose between an act containing five prohibitions and an act containing four prohibitions, and one chooses the act with four? I am not aware (and that proves nothing) that there is a difference between two prohibitions and one prohibition—not in the case of two acts of one prohibition versus one act of one prohibition (like slaughtering instead of feeding carrion), but where both are one act: here there are ten prohibitions and there eight. And not in the case of a prohibition and a positive commandment, which is more severe than a prohibition alone.
So in human law there is no practical difference. In terms of repentance, one repents for the decision and not for its consequences. In terms of higher worlds, for those interested in their construction and destruction, it should be said that there is here a convergent geometric series of destruction (something like the nice joke that appears in the comment I linked above), and the spiritual destruction was designed so that all the infinitely many “before the blind” violations together create only a finite amount of destruction.
D. Type theory in halakhah (the red heifer and pesik reisha).
As I recall, I think I once saw in Rabbi Ovadia Yosef’s Taharat HaBayit that he brings another example. With immersion, if most of the body is covered by an intervening substance and one is particular about it, it is a Torah-level interposition. If it is most of the body and one is not particular, or a minority and one is particular, it is a rabbinic interposition. The question is about most of the body and one is not particular: since it is an interposition rabbinically, now he is particular to remove it so that he can immerse, and if so it should also be a Torah-level interposition. Rabbi Ovadia brings in connection with this the red heifer (that if one is particular because of the laws of the red heifer and their implications, that is not a kind of particularity that invalidates) and the Arukh (that if it is undesirable to him because of the laws of Shabbat and their implications, that is not the kind of consequence that invalidates), in order to propose—perhaps in the name of later authorities—also regarding interposition, that if one is particular because of the laws of immersion and their implications, that is not considered “being particular.”
But in the case of interposition there is only the hierarchical component, and no paradox is created, upon examination.
Or perhaps my memory is simply betraying me, and this example of interposition was brought in a note in Shtei Agalot and not in Taharat HaBayit [and here it was not brought together with its companions precisely because it does not create a paradox]
Me too. I referred to sources where I discussed this. But that is not the topic here.
A. I agree. He acted as a mathematician. But there are many who see this as a solution. Beyond that, even as a method for preventing paradoxes, in my opinion it is not correct, because it does not allow one to formulate legitimate claims, as I wrote. In this context, Gödel’s theorem is an interesting example, like any self-reference in mathematics.
B1. I do, for example. And if I am too insignificant in your eyes (: then go to all the sources I mentioned there. All halakhic discourse is monistic, at least in its offhand mode (talk about error in ruling, fear of issuing rulings, and so on).
B2. Pluralism is the view that there is no halakhic error. Every position of a halakhic sage is acceptable/valid.
B3. Your sources are evidence to the contrary. That is exactly what shows the monism underlying halakhic discourse. I explained in the column that exceptions need to be justified in their own right, and it is not enough that they prevent the problem.
C1. I do not know what there is to solve here. There is a loop whose result is infinitely many transgressions for each person. Even if there is some case in which this does not happen, so what? Why is that important?
C2. I did not claim this about every case of “before the blind.” Though perhaps one could also extend it there.
C3. The practical differences are the difficulties Rabbi Kahaneman came to solve. There are several such difficulties.
D. Your memory is not betraying you (unless mine is). I do not know this example. It is indeed a nice example of an application of self-reference that stops and does not continue forever.
A. Do you mean that in your opinion it is not correct as a method for preventing paradoxes in mathematics? (Or not correct as a general method for preventing paradoxes?)
B2. An example, please. Suppose that one who intended to misappropriate a deposit is liable for accidents: Beit Shammai obligate and Beit Hillel exempt. Could you formulate for me halakhic pluralism regarding this dispute in such a way that one could not formulate it regarding the dispute over methods of halakhic ruling?
B3. If one assumes pluralism is correct, then monism is an error, like a court that ruled that the sun had set and in the end it rose, and therefore we exclude it. Your argument is an indication that there is an error, because it contradicts a correct position (pluralism). And what is the error? The error is that the Holy One, blessed be He, decided that there would not be monism—and that is His prerogative; we merely discover what He decided.
C1. Perhaps it is important. If it is preferable to slaughter once rather than eat two olive-bulk measures of carrion, then perhaps it is also preferable to commit an act in which one violates one prohibition rather than an act in which one violates infinitely many prohibitions. And one might say that even two acts, each involving one prohibition, are preferable to one act involving infinitely many prohibitions. This opens the door to problems. And if indeed a retroactive transgression is nothing, then there is only one prohibition and everything is comfortable.
C2. I truly did not understand why you did not claim this about every case of “before the blind.” And I assumed an extension. In the comment I linked, you did present it regarding every case of “before the blind.”
D. It is indeed in Taharat HaBayit and brought here https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%A8%D7%95%D7%91-%D7%95%D7%90%D7%99%D7%A0%D7%95-%D7%9E%D7%A7%D7%A4%D7%99%D7%93. According to what is written there, it is in Taharat HaBayit, vol. 3, p. 8.
There is no problem with migo? After all, this is a formula for success in court. A swindler who would not take advantage of it is a stupid swindler.
And one who pursues justice must take into account that not all swindlers are stupid, and therefore all these rules of thumb like migo and the like for administering justice run contrary to the pursuit of justice and should be ignored.
And if you say that migo applies only in a case where we know the claimant is not a swindler… the problem is that in such a case one does not need migo at all. For if he is not a swindler, then we should believe whatever he says.
In the liar paradox the relation is direct.
In the examples you brought, the relation is not direct but includes another mediating factor along the way, and the examples are also very similar to Catch-22.
Perhaps the title should have been Catch-22 in Halakhah.
In mathematics everything is a matter of definition. If you define and remain consistent, that is legitimate. The question is whether this solves some real problem, philosophical or mathematical. In my opinion, it does not.
B2. I do not understand the question. Pluralism says that both are correct. You can rule however you like. One can perhaps issue a practical ruling, but it is arbitrary (not related to truth).
B3. If it is an error, then whoever said it erred. If so, there is no basis for the assumption that halakhic sages do not err.
C2. In an ordinary case of “before the blind,” it is not clear to what extent one can say that my transgression counts as causing the other to stumble. I cause his act to become a transgression, but I do not cause him to stumble. Beyond that, in the ordinary case of “before the blind,” your reasoning exists—that the Torah defined the infinity of transgressions together as the transgression of “before the blind.” But that is not true with respect to the other’s causing me to stumble through “before the blind.”
B2. “You can rule according to whomever you want.” If so, then why in the monism-pluralism dispute can the pluralist not say that one can rule according to whomever one wants? Exempt is opposed to liable just as monism is opposed to pluralism. My claim is that the meta-dispute is not special and does not move the pluralist.
As for the rest regarding pluralism, I have still not been persuaded, but I will leave it for a more relevant column.
He cannot rule like the monist, because this is not a halakhic question but a meta-halakhic one. Monism does not say what to do but what to think. Here there is a truth, and its opposite cannot go together with it. This has halakhic implications (in causing someone who thinks differently to stumble), but those are derivatives of the meta-halakhic question.
Something similar applies in disputes of belief: there too, in most cases, one cannot accept all positions together (because these are factual claims). So one can say there is no truth (which is nonsense, but some say so), but one cannot say that there is and is not truth simultaneously.
And in the question of liability versus exemption, is the action not also a derivative of the view regarding the true halakhah? I did not understand why there one can allow all the implications without committing oneself to some monistic truths.
The true halakhah is not a fact. Regarding that, a person can adopt a pluralistic stance. I am a monist and think there is halakhic truth, and still there is an obligation to act autonomously. Such conduct cannot be applied to thought. What I think, I think.
It doesn’t compile.
C2. Also in causing the other person to stumble via “before the blind” against me (a nazirite who handed a cup to someone handing it back), the infinity is only “I cause his act to become a transgression,” as in every case of “before the blind.” When the one handing it back hands it back, he causes the nazirite to violate his naziriteship and turns the nazirite’s handing over into a transgression of “before the blind,” and therefore he also causes the nazirite to stumble in “before the blind.” Therefore the nazirite caused the one handing it back to commit two “before the blind” transgressions, and he himself violates “before the blind” twice. Therefore the one handing it back caused three transgressions and violates “before the blind” three times. And so on.
So too in every case of “before the blind,” the blind person’s transgressive act turns the act of causing him to stumble into a transgression of “before the blind,” and therefore there is a “before the blind” of “before the blind.” Thus the blind person stumbles in the transgression itself and in the transgression of “before the blind,” so the one who caused him to stumble has caused two transgressions and transgressed twice, and therefore the blind person stumbles in the transgression and in “before the blind” twice. Therefore the one who caused him to stumble has caused three transgressions and transgressed three times. And so on. What is the difference?
But on second thought, I do not understand at all why there is an infinity. A person violates “before the blind” even if the blind person does not stumble. Therefore the blind person’s transgression changes nothing. When the nazirite handed it over, he violated “before the blind,” period, because he enabled the other person to commit a transgression (the transgression of “before the blind”). When the other person hands it back, he violated “before the blind” because he enabled the nazirite to commit a transgression (drinking wine). It is not the nazirite’s drinking that turned the other person’s act into a transgression of “before the blind.” Where is the infinity?
Because as stated, if pluralism deals with thinking about what is true, then it also cannot be defined regarding liable/exempt, and each side negates the other. And if pluralism deals with practical rulings, then one can also apply pluralism to the implications of monism-pluralism, and all practical possibilities are legitimate.
Therefore, if there is a contradictory problem in pluralism, it does not concern self-reference.
I am repeating myself, but necessity is no disgrace. I simply do not understand how the answer addresses the question.
Prima facie you are right, but my feeling is that there is a difference between turning an act of causing stumbling—which in any case was already in the category of “before the blind”—into something that has another prohibition in it, and turning an act into a prohibition. But I am not sure.
There is a dispute whether there is a prohibition of “before the blind” even when the blind person does not stumble. Prima facie the discussion is only according to the side that there is a prohibition only if the blind person stumbles. But again, I am not sure there is no room for discussion even according to the other side. Even the enabling of a transgression depends on how many transgressions there are.
The pluralist claims that there is no halakhic truth, and therefore there is no problem saying that both the view that X is permitted and the view that non-X is forbidden are both equally correct (or not). Both are in the category of “halakhic truth.” With regard to facts, no such possibility exists, because here it is clear that there is truth, and therefore contradiction is impossible.
Does the self-reference you proposed against the halakhic pluralist, to arrive at the conclusion that he is mistaken, create a paradox according to his own view? Because you proposed it under the heading of self-references that do not lead to paradox, but it seems that in fact it does lead to a paradox according to the pluralist’s own view, and that therefore one can in fact reject his view.
And in general: does a proof by negation that uses self-reference = use self-reference that leads to a paradox?
Sorry, after the second reading of the column I noticed that you answered this explicitly.
Is Russell’s prohibition against stating a proposition that refers to a proposition higher than it like making a grammatical mistake, for example saying “I see” instead of “I will see”?
Because if so, we have gained that there are no longer any questions (such as the ones you asked above) about his theory, because if making a self-reference is a grammatical prohibition, then it is a grammatical prohibition, period.
And perhaps one can narrow his theory (because of one of your questions above) and say: self-reference is permitted, but self-reference that leads to paradox is forbidden—like saying “big house” with the wrong gender agreement; that is, you can say it until tomorrow, but what can I do, it is grammatically incorrect; that is how the language is built. So the same with paradox.
And if you say that no person from the 20th century can create a grammatical prohibition in the spoken language, I will say to you: fine, but suppose they told you that when the language was taking shape, the linguists said, “Okay, so ‘I’ goes with such-and-such future form, ‘house’ is masculine and not feminine, paradoxes are forbidden, ‘ice cream’ is feminine and not masculine,” etc., etc.
I hope I asked a good question?
I did not understand. Clearly it is a grammatical prohibition, and that is precisely why it solves no problem. He created a language in which there is a grammatical prohibition against expressing the problem. That is not a solution. Those are exactly the difficulties I raised against him.
And if this language he created had been the spoken language for all of us since the beginning of the world, there would have been no problem with that language, and the difficulties you raised would again not have been difficulties, right?!
So the whole problem is only that this is a man from the 20th century who created it? That is not an essential problem.
Imagine that there were paradox X if I said “big house” with the wrong gender agreement instead of the correct form—what would you answer me? Simply: there is no paradox, because one says the correct form and not the incorrect one. So the same with type theory.
What difference does it make who created the language and when? The language is not the problem. The language represents the problem. If it had been created with the creation of the world, then we could not express the problem, but that would not mean it does not exist.
Hello,
Why is the example you gave one where a nazirite gave a cup of wine to someone to hand back to him? Seemingly, the very fact that he received the cup of wine itself also causes a “before the blind of before the blind,” even if he was not the one who gave the cup in the first place, no?
I’m reading it now and really do not understand what I wrote. The rabbi of Ponevezh is of course speaking about your case. I do not know why I described there successive physical transfers. Strange.
Above in the comments there is a reference to your words elsewhere, that indeed in every ordinary case of “before the blind” there is a “before the blind” of “before the blind” ad infinitum.
And in Tosafot, s.v. “af oved,” there, they ask: “And if you would say: if a male mounted it, why is it invalid? Surely it is certainly not desirable to him to invalidate a heifer whose value is high for the sake of such a small matter.” And they answer: “If it were valid, it would be desirable to him, and therefore it cannot be validated.”
The Ran too, in his novellae there, was troubled by this, and brought two answers: “If a male mounted it, it is invalid. And if you would say: why? Since by invalidating it, it is undesirable to him, for because of a small benefit he would not incur a great loss, as stated in the chapter ‘Ein Ma’amidin.’ One can answer that since the act מצד itself is desirable to him, we do not validate it on account of its invalidation [for then its invalidation would be the cause of its validation], and furthermore, if you validate it, it will certainly be desirable to him.”
And then you wrote: “The first answer seems similar to that of Tosafot.”
And I ask: isn’t the Ran’s second answer actually exactly like Tosafot? What is the difference?
So in your eyes the Ran’s two answers are identical? I explained in the column what the difference is.
You explained in the column what the difference is between the Ran’s 2 answers, and you explained what the difference is between the Ran’s first answer and Tosafot.
But I am asking that the second answer is like Tosafot, and you did not address that in the column.
I no longer remember the details. It was a long time ago. But in the passage you cited, I explained that the Ran’s first answer is like Tosafot. And the difference between the second and the first in the Ran was explained. So what is the problem?
Briefly, there is a difference between saying that lack of desirability does not include lack of desirability stemming from the thing itself, and saying that a circular loop is created here. These are different claims.
This is Tosafot: “And one can answer that if it were valid, it would be desirable to him, and therefore it cannot be validated.”
This is the Ran’s second answer: “And furthermore, if you validate it, it will certainly be desirable to him.”
Do you see a difference? What is it?
A wonderful example of the sensitivity needed for the melody of the text, which is expressed in a letter or a word. Think, and you will see.
But surely you agree that you did not address this in the column, so address it now
because I tried to understand what practical difference there is between them, and even artificial intelligence did not manage to bring me a practical difference, apart from a philosophical word-laundering
Everything was explained in the column. Did you try to think as I suggested to you?
According to your approach, the Ran’s two answers are the very same answer (namely, the answer of Tosafot). So first of all, it is clear that you are not right. Now I will show you this from the Ran’s own wording.
In his first answer he writes, like Tosafot, that if it had remained valid it would have been desirable to him, and that is not considered lack of desirability. That is, this is a novelty in the parameters of “desirability”: it is determined without taking the law itself into account. This novelty is not compelled on its own terms; it is said ad hoc only to resolve the difficulty.
And so is the language of the Tosafot you cited: “And one can answer that if it were valid, it would be desirable to him, and therefore it cannot be validated.” That is, when the invalidation itself is what causes the lack of desirability, it cannot be validated.
But the Ran’s wording in his second answer is: “And furthermore, if you validate it, it will certainly be desirable to him.” Here the melody is that this follows logically and necessarily from itself: if you validate it, then certainly (!!!) it will be desirable to him. That is, you want to validate it because it is undesirable to him since it was invalidated, but if so, continue the process and you will see that you return to invalidating it. That is, he takes the questioner and tells him that on his own assumptions this cannot be. This is not an ad hoc novelty said to resolve the difficulty, but a logically necessary conclusion from the questioner’s own premises.
(This is the melody of the text I was talking about. Notice the word “certainly” in the Ran’s formulation. It says everything.)
Forgive me, forgive me, forgive me—
It seems to me that for some reason you understood Tosafot’s words not according to their plain meaning: “And one can answer that if it were valid, it would be desirable to him, and therefore it cannot be validated.”
Let me translate that, and see whether you agree with my translation.—
“And one can answer that if the heifer were valid [because of that reasoning in the question], this would again be a case of desirability to him [and so we would again return to invalidating it], and therefore it cannot be validated” [that is, therefore such reasoning as in the question does not work, because nothing comes of it—it is a loop].
Exactly like the Ran’s words in the second answer: “And furthermore, if you validate it, it will certainly be desirable to him,” meaning “it will certainly be desirable to him,” i.e. then we will again invalidate it, and the wheel turns in the world.
Whereas the Ran’s first answer—
“And one can answer that since the act מצד itself is desirable to him, we do not validate it on account of its invalidation [for then its invalidation would be the cause of its validation]”—
means: it simply does not sound logical to him that the cause of its invalidation should be its validation; it just does not sit well with him. Like an intuition.
Or, if you prefer, it does not sound legally coherent.
Doesn’t that sound much better???
I’ve exhausted this. Everything was explained.
With migo I have trouble seeing the problem.
The claim is that when he has an argument that could in any case be accepted, then other exempting claims will also be accepted, and we do not take into account the weaknesses they might have had in a different setting.
Therefore there is no reason to continue the loop. Right: the claim that he paid within the loan period is equivalent to the stronger claim that he repaid afterward.
In essence, we agree that he returned the loan, since in any case he could have claimed that.
Or more generally: if it is agreed that I can make exempting claim X, and I make exempting claim Y instead (whether Y is weaker or equivalent), then I am surely telling the truth, because why would I lie with no benefit whatsoever?