Logical Loops and Self-Reference (Column 406)
With God’s help
Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.
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 “עובד” (we read “עובד”), I would say even if it happened on its own; and if it were written “עבד” (we read “עבד”), I would say only when he himself worked with it. Now that it is written “עבד” but we read “עובד,” 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.
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I have a hard time seeing the problem with Migo.
The argument is that when he has a claim that can be accepted in any case, other exemption claims will also be accepted and we will not address the weaknesses that they could have in another situation.
Therefore, there is no point in continuing the loop, right, the claim that he paid within the loan days is equivalent to the good claim that he paid after.
In fact, we agree that he returned the loan because in any case he could have claimed it.
Or in general, if it is agreed that I can claim exemption claim X, and I claim exemption claim Y (no matter if Y is weaker or more equal), I must be telling the truth because why would I lie without any benefit from it.
Me too. I referenced the sources where I discussed it. But that's not the issue here.
Is there a problem with Migo? After all, this is a formula for success in court. A fraudster who does not take advantage of this is a stupid fraudster.
And those who pursue justice should take into account that not all fraudsters are stupid, and therefore all these generalizations of Migo and the like for doing justice are contrary to the pursuit of justice and should be ignored.
And if you say that Migo applies only in the case where we know that the claimant is not a fraudster… the problem is that in such a case there is no need for Migo at all. Because if he is not a fraudster then we must believe everything he says.
The links in this column throw me back to two points that have remained silent for me. And one more from the past. And one more.
A. Tape theory as a solution to paradoxes.
Russell in the introduction there does not say this. He is looking for a set of axioms and rules of deduction from which we can formally deduce everything we want to deduce and not allow us to deduce incorrect things. If the method allows us to deduce all the mathematical claims that are already familiar to us and does not allow us to deduce any errors, then it is a good method that mathematicians can comfortably and confidently use. The entire book that is full of proofs of mathematical claims is intended only to convince us that the set is adequate and complete, where complete means everything that is ’interesting’. The set of axioms does not have to point to any truth but to be a good working tool. That is what Russell says there and not a general suggestion to ignore paradoxes. And to that end, all the counterarguments you wrote in the column are irrelevant. And the theory of taifs is an excellent mathematical tool. And here https://mikyab.net/posts/70517#comment-47457
B. Halachic pluralism.
I said all sorts of things about this here https://mikyab.net/posts/71518#comment-50537 and I will repeat some because I did not come to my satisfaction.
B1. The argument attacks pluralism by virtue of the fact that there is a deaf and wise sage who held to monism. Who is this sage who held to monism.
B2. Explain to me how you understand pluralism in Halachic law regarding any question you choose to illustrate with it, and then I will explain pluralism also regarding the question of this dispute itself. Even on a third reading of the article there, it was not clear to me what is special about this dispute more than any other dispute in which pluralism is required.
B3. There is no problem with excluding a particular position from the rule if there are good considerations in favor of this exception. If indeed your argument proves that such an exception is necessary to uphold the pluralist position, which is the correct position, then exceptions are fine. It is clear that sages can be mistaken about a minor matter and rule that the sun set and the sun rose at the end is not a rule but an error. But if there are no decisive reasons against the position, then halachic pluralism assumes that it is correct.
C. Blind before blind before blind.
C1. Here https://mikyab.net/posts/66820#comment-35769 I proposed a simple solution to the blind before loop. And it is that the one who fails causes an obstacle only to pass retroactively on a blind person, and a retroactive offense according to a suggestion that you raised elsewhere is not forbidden (like Tori Eben who thought that a question about a sacrifice should be permitted even if it is made retroactively with help. And this is except for vows and oaths in which the ate is for the sake of suffering and the ate is for the sake of its conditions is a defect).
C2. And it must be said very simply that if in every blind person there is an infinite pre-blind person, then this is the non-self that contains all infinity and this is one non-self because this is defined as such that in such an act (which has an infinity ‘pre-blind person of the first level’) one transgresses the non-self of a blind person. And there is no reason to think that one really transgresses an infinite number of non-selfs.
C3. And it is also not clear what is the result if one transgresses the pre-blind person once or an infinite number of times. One is not subject to this non-self because it is the non-self of generalities. Is there a situation in which one has to choose between an act with five negatives and an act with four negatives and choose the act with four negatives? I am not aware (and this does not mean anything) that there is a difference between two negatives and one negative. Not in the case of two acts of one negative versus one act of one negative (like slaughtering instead of feeding carrion), but rather these and these are one act, in this case ten negatives and in this case eight negatives. And not in the case of a negative and an act that is more serious than the full one.
So in human law there is no issue. In terms of making repentance, one makes the decision and not its results. In terms of the higher worlds, for those who are interested in their construction and destruction, it is clear that there is a converging geometric column of destruction here (sort of like the nice joke that appears in the response I linked above) and they planned the spiritual destruction so that all the infinity before the blind will create a finite amount of destruction in total.
D. The Law of the Tapestry in Halacha (Red Cow and Pesik Risha).
I remember, as I think I saw once in the book Teharat HaBayit by the Rabbi, which gives another example. In a bath, the majority of those who abstain and observe it observe it from Torah, the majority and do not observe it, or the minority and observe it observes it from the Rabbis. The question is, in a majority and do not observe it, since they abstain from the Rabbis, then now they observe it so that they can bathe, and so there will also be a separation from Torah. And Rabbi Ovadia brings up the Red Cow (that if one observes it because of the laws of the Red Cow and their implications, then it is not a separation that invalidates) and the Aruch (that if one does not observe it because of the laws of Shabbat and their implications, then it is not a separation that invalidates) in order to suggest (perhaps in the name of the latter) that even in a separation, if one observes it because of the laws of immersion and their implications, then it is not considered a separation.
But in the case of a buffer, there is only the hierarchical component and no paradox is created when it is examined.
And perhaps I have even scratched my memory and this example of a buffer was brought up in the Enlightenment in Two Carts and not in the Purity of the House [and here it was not brought up with its companions precisely because it does not create a paradox]
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, I think it is not correct because it does not allow for the formulation of legitimate claims, as I wrote. In this context, the example of Goedel's theorem is interesting, as is any self-reference in mathematics.
B1. Me, for example. And if I am small in your eyes (: All the best to you for all the sources I mentioned there. All halakhic discourse is monistic, at least in the Messiah according to his perfection (talk about error in ruling and fear of teaching, etc.).
B2. Pluralism is the perception that there is no halakhic error. Every position of a halakhic scholar is acceptable/valid.
B3. Your sources are evidence to contradict. This exactly shows the monism that underlies halakhic discourse. I explained in the column that exceptions should be justified in themselves, and it is not enough that they prevent the problem.
C1. I don't know what to solve here. There is a loop that results in countless offenses for everyone. Even if there is a situation in which this does not happen, so what? Why is it important?
C2. I did not claim this for every ‘blind man’. Although it is perhaps possible to expand there as well.
C3. The נפמ”m are The problems that Rabbi Kahneman came to solve. There are several of them.
D. Not tapped (unless I tapped). I am not familiar with this example. Indeed a beautiful example of the application of self-reference that stops and does not continue forever.
A. Do you mean that in your opinion this 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 the thinker is to hand over a deposit if he commits rape, Beit Shammai is liable and Beit Hillel is exempt. Can you formulate for me the halakhic pluralism on this dispute in such a way that it is impossible to formulate the dispute in the methods of jurisprudence.
B3. If we assume that pluralism is correct, then monism is an error, as was taught in 2:11 that the sun set and finally rose, and therefore it is excluded. 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 God decided that there should be no monism and that is His right, we are only discovering what He decided.
C1. Maybe it is important. If it is better to slaughter once than to eat two rotten olives, then perhaps it is also better to commit an act in which no one commits than an act in which an infinite number of non-non-nons commit. And there is room to say that even two acts in which no one commits are better than one act in which there are infinite non-nons. This opens up problems. If a retroactive offense is really nothing, then there is only one non-non and everything is convenient.
C2. I really didn't understand why you didn't claim this for every blind person. And I assumed an extension. In the response I quoted, you did present it for every blind person.
D. This is indeed found in the Purity of the House and is presented 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 the Purity of the House, Part 3, Page 8.
In mathematics, everything is a matter of definition. If you define and remain consistent, it is legitimate. The question is, does this solve any real problem, philosophical or mathematical? I don't think so.
B2. I don't understand the question. Pluralism says that both are right. You can rule as you wish. It may be possible to rule in practice, but it is arbitrary (not related to the truth).
B3. If it is a mistake, then whoever said it was wrong. If so, there is no basis for assuming that the scholars of halachah are not wrong.
C2. In the case of an ordinary blind person, it is not clear to what extent it can be said that my offense is in the scope of the other's failure. I cause his action to become an offense but do not fail him. Beyond that, in the case of an ordinary blind person, there is your opinion that the Torah defined the infinite number of offenses together as the offense of a blind person. But this is not true regarding the other's failure in the case of a blind person on me.
B2. “I can rule as I please”. So why in the monism-pluralism controversy can't the pluralist say that I can rule as I please. Exemption is opposed to obligation just as monism is opposed to pluralism. My argument is that the meta-controversy is not special and does not excite the pluralist.
As for the rest of the matter of pluralism, I am not convinced yet, but I will leave it to 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 truth and its opposite cannot go along with it. This has halakhic implications (in the fallacy of a person who thinks differently) but these are derivatives of the meta-halakhic question.
As in a dispute over opinions, there too in most cases it is impossible to accept all positions together (because these are factual claims). So one can say that there is no truth (which is nonsense, but there are those who say so), but one cannot say that there is and that there is no truth at the same time.
And in the question of whether the act is exempt, isn't it a derivative of the opinion regarding the true halakha? I didn't understand why it is possible to allow all the implications without committing to any monistic truths.
True halakhah is not a fact. One can relate to this in a pluralistic manner. I am a monist and think that there is halakhic truth, and there is still an obligation to act autonomously. Such conduct cannot be applied to thinking. What I think, I think.
Does not fold
Because as mentioned, if pluralism deals with thinking about what is truth, then even about the guilty-exempt, it cannot be defined and each side denies the other. And if pluralism deals with practical instructions, then even in the consequences of monism-pluralism, pluralism can be applied and all practical possibilities are legitimate.
Therefore, if there is a contradictory problem in pluralism, it does not concern self-reference.
I repeat myself, but necessity will not be spared. I simply do not understand how the answer relates to the question.
The pluralist claims that there is no halakhic truth and therefore there is no reason to say that both the opinion that X is permitted and the opinion that =X is prohibited are equally correct (or not). Both are considered ‘halakhic truth’. With regard to facts, there is no such possibility because here it is clear that there is truth, and therefore a contradiction is not possible.
C2. Even in the case of the other's failure in front of a blind person (a monk who offered a cup to the offerer), the infinitive is only “I cause his act to become a transgression” as in all cases of blind people. When the offerer offers back, he fails the monk in desecrating monasticism and turns the monk's reaching out into a transgression in front of a blind person and therefore fails the monk in front of a blind person as well. Therefore, the monk failed the offerer in two transgressions in front of a blind person and he himself transgresses twice in front of a blind person. Therefore, the offerer failed in three transgressions and transgresses three times in front of a blind person. And so on.
Likewise, in all cases of blind people, the act of transgression of the blind person turns the act of stumbling into a transgression in front of a blind person and therefore transgresses because of the offerer. Thus the blind man failed in the offense itself and in the offense before the blind man, therefore the hinderer failed in two offenses and passed twice and therefore the blind man failed in the offense and twice before the blind man. Therefore the hinderer failed in three offenses and passed three times. And so on. What is the difference?
But on second thought I don't understand at all why there is infinity. A person passes before the blind man even if the blind man did not fail. Therefore the blind man's offense does not change anything. When the monk reached out, he passed before the blind man a point because he allowed the one reaching out to commit an offense (an offense before the blind man). When the one reaching out reaches out, he passed before the blind man because he allowed the monk to commit an offense (drinking wine). It was not the monk's drinking that turned the act of the one reaching out into an offense before the blind man. Where is the infinity?
Apparently you are right, but my feeling is that there is a difference between reversing the act of failing that somehow was within the scope of before the blind to something that still has a prohibition and turning an act into a prohibition. But I am not sure.
There is a dispute as to whether there is a prohibition before the blind even when the blind does not fail. Apparently the discussion is only on the side that there is a prohibition only if the blind fails. But again I am not sure there is no room for discussion on the other side as well. The possibility of passing also depends on how many offenses there are.
In the liar's paradox, the relationship is direct.
In the examples given, the relationship is not direct but includes another intermediary factor along the way, and the examples are also very similar to a catch-22.
Perhaps the title should have been Catch-22 in Halacha.
Does the self-reference that you proposed against the Halachic pluralist to arrive at the conclusion that he is wrong create a paradox for his method? Because you proposed it under the heading of self-references that do not lead to paradox, but it seems that it does lead to paradox for the pluralist method and therefore his method can be rejected.
And as a rule: Does a proof by a negative method that uses self-reference = using self-reference that leads to paradox?
Sorry, after reading the column a second time, I noticed that you explicitly replied.
Is Russell's prohibition of making a claim that refers to a claim higher than it the same as making a grammatical error, for example, "I will see" instead of "I will see"?
Because already, we have gained that there are no more questions (like the ones you asked above) about his teachings, because if making self-reference is a grammatical prohibition, then it is a grammatical prohibition, period.
And perhaps we can reduce his teachings (because of one of your questions above) and say: self-reference is permitted, but self-reference that leads to a paradox is prohibited, like saying "a big house" instead of "a big house", meaning you can say "a big house" until tomorrow, but what will I do and it is not grammatically correct, that's how language is structured, so the same thing with paradox.
And if no one from the 20th century can create a grammatical prohibition of the spoken language, I'll tell you okay, but if they told you that when the language was formed, linguists said "Okay, so "I" goes with "see" in the future, house is masculine and not feminine, paradoxes are forbidden, ice cream is feminine and not masculine, etc., etc., etc."
I hope I asked a good question?
I didn't understand. It's clearly a grammatical prohibition, and that's precisely why it doesn't solve any problem. It created a language in which there is a grammatical prohibition to express the problem. It's not a solution. These are exactly the difficulties I raised against it.
And if this language he created was the language spoken by all of us since the beginning of the world, there would be no problem with this language and the difficulties you raised again were not difficulties, right?!
So the whole problem is just that it was a man from the 20th century who created it? That's not a fundamental problem.
Imagine there was a paradox X if I said "big house" instead of "big house", what would you answer me, there is simply no paradox because they say "big house" and not "big house". So the same thing with the theory of tips.
What does it matter who created language and when? Language is not the problem. 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 that it does not exist.
Hello Rabbi,
Why is the example you gave of a monk giving a glass of wine to a person who offered him a glass of wine in return? Apparently, the very fact that he received the glass of wine itself causes the "before me", even if he was not the one who gave the glass in the first place, right?
I'm reading now and I really don't understand what I wrote. The Rabbi from Ponivez is of course talking about your case. I don't know why I described successive physical transfers there. Strange.
Above in the comments there is a reference to your words in Mecca that indeed, in every ordinary pre-urbanization there is the pre-delpre to infinity.