The Logic of Kal va-ḥomer: A. Relevance (Column 735)
For my daughter Rivka, with whom this discussion took place this past Shabbat
Kal va-ḥomer
Kal va-ḥomer considerations in the Talmud rest on three halakhic premises that are known to us, from which we derive a halakhic conclusion regarding a case that is unknown (a lacuna in the law). For example (from Bava Kamma 25): if “tooth and foot,” which are exempt in the public domain, are liable in the damaged party’s courtyard, then “horn,” which is liable in the public domain—surely it is a fortiori liable in the damaged party’s courtyard?!
We can present all the assumptions involved in this reasoning in the following table (0 means exemption, and 1 means liability):
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 0 | 1 |
| Horn | 1 | ? |
Table 1: Standard kal va-ḥomer
The three data that appear in the table are laws we know, and from them we fill in the missing cell (the halakhic lacuna) and conclude that horn in the public domain is also liable in the injured party’s courtyard (i.e., the result is 1).
To complete the picture, I’ll add that even if the table were slightly different, we could still fill in the missing cell (for the sake of discussion I’ll now assume that tooth and foot are liable also in the public domain):
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 1 | 1 |
| Horn | 1 | ? |
Table 2: Standard binyan av
The difference here is the liability of tooth and foot in the public domain, which I am now assuming. In such a case we can again fill the missing cell with 1, only this time it is a binyan av rather than a kal va-ḥomer. There is no hierarchy here of stringency and leniency, but rather we analogize the domains to one another and/or the types of damages to one another.
Pilpul in the form of a kal va-ḥomer
Several times in the past I have brought the dialectical pilpul that obligates a lintel in tzitzit (see, for example, column 52 and elsewhere), which goes like this: If a four-cornered garment, which is exempt from mezuzah, is nevertheless obligated in tzitzit, then a lintel, which is obligated in mezuzah—surely it should be obligated in tzitzit?! This is an example of pilpul as defined in that column, since we have an argument that at first glance is persuasive, yet its conclusion appears absurd. I explained there that pilpul is a sort of riddle, like a paradox, for it challenges us to find the flaw in the argument. When one looks at this argument, it seems like any other kal va-ḥomer in the Talmud, and so it is hard to put one’s finger on the flaw, much like with a paradox.
In the past (see column 601 and elsewhere) I noted another possible way to treat paradoxes that we tend to ignore: to conclude that there is, in fact, no flaw in the reasoning, and that its conclusion—despite seeming preposterous—is actually correct. This too is an option that must be considered. But in our case, given that there is not a single decisor in existence who obligates a lintel in tzitzit, it would seem the halakhic conclusion is nevertheless wrong, leaving us the task of pointing to the defect in the argument that leads there.
The reverse kal va-ḥomer and its meaning
By similar reasoning, one could learn to obligate a four-cornered garment in mezuzah: If a lintel, which is exempt from tzitzit, is obligated in mezuzah, then a four-cornered garment, which is obligated in tzitzit—surely it should be obligated in mezuzah?!
At first glance this does not contradict the previous argument but rather adds to it. Yet on closer examination, a problem emerges. The first argument relied on the premise that a four-cornered garment is exempt from mezuzah, whereas from the present argument it emerges that it is obligated. And conversely, the present argument assumes that a lintel is exempt from tzitzit, whereas from the previous argument it emerges that it is obligated. Thus, ostensibly, these are two arguments that contradict one another.
But a second look shows that this is not so. Suppose indeed that a lintel is obligated in tzitzit. One could still learn to obligate a garment in mezuzah. Only now it would be a binyan av rather than a kal va-ḥomer (as in the second table above). That is, even if both arguments are made, it will not be a kal va-ḥomer, but the learned law will still be correct by force of binyan av. Thus, these two arguments do not contradict one another.
Why not learn “stringent-to-lenient”?
Seemingly, one could also learn here a “stringent-to-lenient” inference, but that leads straight to opposite conclusions. For example, to exempt a garment from tzitzit: If a lintel, which is obligated in mezuzah, is exempt from tzitzit, then a garment that is exempt from mezuzah—surely it should be exempt from tzitzit?! And likewise, to exempt a lintel from mezuzah: If a garment, which is obligated in tzitzit, is exempt from mezuzah, then a lintel that is exempt from tzitzit—surely it should be exempt from mezuzah?!
For these last two arguments it is easy to see where the flaw lies. The Torah itself says that a lintel is obligated in mezuzah and a garment is obligated in tzitzit, and such a kal va-ḥomer cannot overturn that. Kal va-ḥomer and binyan av are hermeneutic rules that can teach us a law not explicitly stated in the Torah, but they do not change laws that are stated explicitly. However, this is of course irrelevant to the first pair of arguments. There they truly fill a lacuna that exists in the Torah and do not change a law that is stated in it (the Torah does not state that a garment is exempt from mezuzah or that a lintel is exempt from tzitzit; it merely does not explicitly obligate them. That is a lacuna).
A hint of the problem
These notes already hint at some problem in the first pair of arguments. Just as the cell we fill must be empty (that is, it must reflect a lacuna in what is written in the Torah), so the three cells on which these arguments rely must be full; namely, they must contain a law that is explicitly stated in the Torah. Moreover, the hermeneutic rules tell us that we do not learn a kal va-ḥomer from a law that is halakha le-Moshe mi-Sinai (see Nazir 57a), that is, the three data cells must be laws explicitly stated in the Torah.[1] Thus, for example, in the example at the beginning of the column regarding horn damage in the injured party’s courtyard, the three filled cells are laws written in the Torah.
Is this the case in the first two arguments of the pilpul cited above? Definitely not. If we present the laws written in the Torah for this reasoning, the table looks like this:
| Object / Law | Tzitzit | Mezuzah |
| Lintel | ? | 1 |
| Four-cornered garment | 1 | ? |
Table 3: Pilpul kal va-ḥomer
Note that there are here two cells that represent laws written in the Torah (and not three, as in a standard kal va-ḥomer) and two lacuna cells (and not just one), and for the two lacunae the Torah contains no law—neither obligation nor exemption. This means that the arguments to obligate a lintel in tzitzit and a garment in mezuzah are based on two scriptural data points, not three. The third is obtained by our filling in a lacuna cell on our own. My claim is that when there are two lacunae, as in the pilpul’s Table 3, one cannot fill either of them, and the table must remain as it is.
Admittedly, on the face of it this is unclear. True, the Torah does not say that a lintel is exempt from tzitzit; it merely does not obligate it in tzitzit. But in practice the lintel is indeed exempt from tzitzit, so in practice one could fill that cell even if the law is not explicitly written in the Torah. If so, why not perform a kal va-ḥomer? After all, in practice the garment turns out to be more stringent than the lintel (from the right-hand column); why not therefore fill 1 in the left-hand column so as to preserve the hierarchy of stringency, as in any kal va-ḥomer? Admittedly, one can likewise go the other way and fill the lintel-tzitzit cell in the right-hand column with 1. That already changes the filling that we did based on the Torah’s lacuna (even though it does not obligate a lintel in tzitzit, we conclude that it is obligated in tzitzit). Alternatively, we could indeed ignore the lacuna and fill both lacuna cells by kal va-ḥomer or binyan av and get a table of all 1s. That would not be a kal va-ḥomer but a binyan av, yet binyan av is also a legitimate hermeneutic rule. But we do not do any of that in practice, for halakha rules that a lintel is exempt from tzitzit and a garment is exempt from mezuzah. Something here is still unclear.
The relevance assumption
As I understand it, the reason we do not make either of these arguments is that we assume that a lintel is not “exempt” from tzitzit; rather, tzitzit is wholly irrelevant to it. And the same holds for mezuzah with respect to a garment (it is not “exempt” from mezuzah; rather, mezuzah is irrelevant to garments). The implication is that each cell in such a data table has three possible values, not two: 0 – exemption, 1 – obligation, and X – not relevant. Accordingly, our table is really the following:
| Object / Law | Tzitzit | Mezuzah |
| Lintel | X | 1 |
| Four-cornered garment | 1 | X |
Table 4: A possible explanation for the kal va-ḥomer pilpul (irrelevance)
When we say that the obligation of tzitzit is irrelevant to a lintel, the claim is that there is neither exemption nor obligation. Consider a claim such as “virtue is triangular.” Is that statement true or false? Neither. It is meaningless, because virtue does not belong to the semantic field of geometric shapes. It cannot be described by them. The same holds for a lintel and tzitzit. A lintel does not lie within the semantic field of things obligated in tzitzit.
To understand this better, take the following example. The Talmud brings in several places a gezerah shavah that likens a slave to a woman. One consequence is that he is obligated in the commandments as a woman is. R. Akiva Eiger argues that the Bavli and the Yerushalmi disagree whether the comparison is toward stringency or toward leniency—that is, whether a slave is basically exempt from all commandments like a gentile, and the gezerah shavah teaches that he is nevertheless obligated in those commandments that a woman is obligated in (a slave is an “upgraded gentile”), or whether he is in principle obligated in all commandments like a Jewish man, and the gezerah shavah nevertheless exempts him from those commandments from which a woman is exempt (a slave is a “diminished Jew”). Note that these two claims are not equivalent. A practical difference arises regarding a prohibition such as shaving the sides of the head (hakafat ha-rosh, pe’ot). A woman is not “exempt” from this prohibition; rather, it simply does not apply to her (in practice she does not grow peyot). If a slave is obligated in whatever a woman is obligated in, he will not be obligated in the pe’ot prohibition, for a woman is not obligated in it. But if he is exempt from whatever a woman is exempt from, then a woman is not exempt from the pe’ot prohibition; it simply does not apply to her, whereas for a slave it does apply. On that view, a slave would be obligated in the pe’ot prohibition. The upshot is that there is a difference between saying that a woman is exempt from the pe’ot prohibition and saying that it does not pertain to her or is irrelevant to her. Exactly such a distinction I made with respect to tzitzit and the lintel: the question is whether the lintel is exempt from tzitzit (0) or whether tzitzit is altogether inapplicable to it (X).
If so, we have understood the meaning of X in the above table. It represents irrelevance. It is now clear that on this view one cannot derive from the right-hand column of Table 4 a hierarchy of stringency between garment and lintel, because the lintel is not exempt from tzitzit; it is irrelevant to it. Therefore, one cannot prove from here that the lintel is more lenient than a garment. Of course, the same can be said about the top row of Table 4: from it too one cannot derive a hierarchy of stringency between mezuzah and tzitzit. Likewise, one cannot derive from the bottom row an opposite hierarchy between tzitzit and mezuzah, nor from the left-hand column a hierarchy between garment and lintel. All such conclusions about hierarchies of stringency are incorrect if the entries in those two cells are X rather than 0.
What lies behind the relevance problem is this: when we look at the right-hand column, we wish to learn from it to the left-hand column. The assumption is that they share something in common, and that its degree is greater in the left-hand column, hence it is more stringent. But in light of the picture I propose here, there is nothing in common between the columns and/or the rows, and therefore one cannot infer from one to the other. And if a hierarchy exists in one of them, that does not mean it will appear in the other.
The question is: how do we know that such a lacuna expresses irrelevance (X) rather than exemption (0)? How did we decide that the lintel is not exempt from tzitzit, but that tzitzit is simply irrelevant to it? And from another angle: why should we not say the same of the cell for tooth and foot in the public domain (and interpret the exemption there as irrelevance rather than as exemption)? There is an intuition that in lintel and tzitzit it is indeed irrelevance, but this calls for grounding. Moreover, there is a kind of demand here for the reason of the text, which we are not supposed to make when interpreting verses. In short, we must understand why in the table of lintel and garment we fill X rather than 0.
To explain this, I will first give a somewhat formal account of the ordinary kal va-ḥomer inference.
Clarifying the kal va-ḥomer inference
One of my earliest articles dealt with kal va-ḥomer and examined its relation to the logical syllogism (a valid logical argument). I showed there that kal va-ḥomer is not a logically valid argument, and the indication is the possibility of pirchot (objections). A valid argument has no and cannot have pirchot. At most, one can point out an error in the argument. A pircha does not indicate an error in the argument but rather that the argument does not lead to the correct conclusion. Hence it is a non-necessary argument, i.e., not a logically valid one.
In the first book in the series on Talmudic logic (presented on the site as a pair of long articles from the journal BD”D: the first and the second), we delved deeper into the topic and developed a complete formal logic that deals with non-deductive arguments. The basis was kal va-ḥomer, binyanei av, hatzad ha-shaveh, and the pirchot on them, but it became clear to us that any non-deductive argument in any field can in principle be presented as a structure composed of these building blocks. I was surprised now to discover that I cannot find on the site a systematic presentation of the basis for this logical picture, and I will take the opportunity to do so here.
To that end, let us return to the kal va-ḥomer argument presented at the beginning of the column regarding horn in the injured party’s courtyard. We saw there that if the three laws in the table are given, one can infer the fourth (i.e., record 1: that horn is liable in the injured party’s courtyard). Why indeed is that so? What rules out the possibility that horn is exempt there? Before I explain that, I will ask another question.
I will first return to what I already hinted at above: every kal va-ḥomer argument can be presented in two forms—by columns and by rows:
- A column-based kal va-ḥomer looks at the right-hand column in the table, finds there a hierarchy whereby the second row (filled with 1) is more stringent than the first (filled with 0), and then applies this to the left-hand column. The assumption is that there too the second row (the empty cell with a question mark) is more stringent than the first (filled with 1), and therefore the appropriate fill is 1.
- A row-based kal va-ḥomer does exactly the same but rotated 90 degrees, so instead of the columns we look at the rows. First we look at the top row and find there a hierarchy between the columns (the second more stringent than the first), and now we apply this to the bottom row: therefore, if in the left-hand column there is a 1, then in the right-hand column there will certainly be a 1.
In my initial article I showed that each of these two arguments is not a deduction—that is, not a logically valid argument. For example, in the column argument, by looking at the right-hand column, we find a hierarchy (horn more stringent than tooth and foot in the public domain). Now comes a generalization to all domains: horn is more stringent than tooth and foot in every domain. Only then can we conclude that horn is more stringent than tooth and foot also in the injured party’s courtyard. This intermediate conclusion (non-necessary, since it is a generalization) on which the argument is based is that horn is more stringent than tooth and foot in all domains. By contrast, in the row argument the intermediate conclusion is different: that the injured party’s courtyard is more stringent than the public domain for all types of damages.
How do we know that these are not two formulations of the very same argument? First of all, because these two arguments rest on different assumptions. To see this more clearly, one can simply look at a pircha.
The meaning of a pircha
When a pircha is raised against a kal va-ḥomer, we are in effect adding two data to the three existing ones. For example:
| Damager / Domain | Public Domain | Courtyard of the Injured Party | The Moon |
| Tooth and Foot | 0 | 1 | 1 |
| Horn | 1 | ? | 0 |
Table 5: Standard kal va-ḥomer with a column pircha
Suppose there is a domain (the moon) in which tooth and foot are more stringent than horn. This is a column-type pircha on the kal va-ḥomer. Why? Because we see that the hierarchy assumed between horn and tooth/foot is not correct, or at least not general. The generalization whereby from the right-hand column one can infer that horn is always more stringent than tooth and foot, in any domain, is not correct. It holds in the public domain, but not on the moon. We can now wonder whether in the injured party’s courtyard that hierarchy holds (as in the public domain) or not (as on the moon). The pircha shows that we cannot infer a clear conclusion regarding horn’s liability in the injured party’s courtyard.
What about the row-based kal va-ḥomer? We saw that it assumes a generalization that leads to a completely different intermediate conclusion: that the injured party’s courtyard is more stringent than the public domain for all types of damages. Does the additional column in Table 5 refute this intermediate conclusion? Not at all. If so, ostensibly the row-based kal va-ḥomer remains intact even after the pircha. To refute it, we must find in the law a third row in which the hierarchy between the domains is reversed, for example:
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 0 | 1 |
| Horn | 1 | ? |
| Tail | 1 | 0 |
Table 6: Standard kal va-ḥomer with a row pircha
We have found a third damage type (“tail”) that is liable in the public domain and exempt in the injured party’s courtyard. This of course overturns the intermediate conclusion of the row argument that the injured party’s courtyard is always (for all damages) more stringent than the public domain.
Yet a survey of the Talmud yields a surprising conclusion: when a single pircha—row or column—is presented, the kal va-ḥomer is rejected, and that is that. No Talmudic sage anywhere entertains turning the kal va-ḥomer around and using the other formulation. If the column argument is refuted, they never deploy the row argument, and vice versa.[2] This is a puzzle that long bothered me, for it hints that, despite what we have seen, the two arguments are in fact merely two formulations of the same argument. Therefore, when one falls, the other falls as well. The question is: what is incorrect in what I have described thus far? As we saw, ostensibly these are two different arguments, since each assumes a premise that the other does not.
Non-deductive logic: kal va-ḥomer
Here we reach the logical analysis of the kal va-ḥomer argument that served as the basis for developing the entire non-deductive logic. Let us once again look at a standard kal va-ḥomer table like the one we saw:
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 0 | 1 |
| Horn | 1 | ? |
Table 1: Standard kal va-ḥomer
Let us begin with the column argument. It starts with the fact that horn is more stringent than tooth and foot. It is reasonable that this stringency is based on some property that we denote by α, of which horn has 2 units and tooth/foot has one unit. That is, horn has this property with strength 2α and tooth/foot with strength α. But this is not enough to conclude the result for the left-hand column. To do that, we must assume that the property α that governs the hierarchy in the right-hand column (liability in the public domain) is also the property relevant for the laws in the left-hand column. In other words, the greater the damager’s α-strength, the more it will bring about liability also in the injured party’s courtyard. This is essentially our generalization that this stringency exists in all domains. But note what this assumption says: if in the public domain a strength of 2α is needed to incur liability (hence only horn is liable, not tooth/foot), then in the injured party’s courtyard a strength of only α suffices (for there tooth/foot is also liable). From this we can infer that horn, which has strength 2α, will certainly be liable in the injured party’s courtyard.
Note what we have obtained: if we assume a hierarchy with respect to the parameter α between the rows in the right-hand column, we must tacitly assume that same hierarchy between the columns. If we assume that there is a parameter α that characterizes the damagers, that same parameter must necessarily also characterize the domains. To execute the column argument, it is not enough to assume a hierarchy between the rows (in the right-hand column and by generalization to all domains); implicitly we are also assuming a hierarchy between the columns (in the top row and by generalization to all damagers). The hierarchy between damagers is their severity: the greater their α-strength, the more stringent they are. Between the domains the situation is the reverse: the greater a domain’s α-threshold, the harder it is to incur liability there (a greater α-strength of the damager is required to incur liability there).
Thus, the row-based and column-based kal va-ḥomer arguments are in fact two formulations of the same argument, which assumes a hierarchy for the manifestation of the same property both between the rows and between the columns. Both assumptions are required to carry out the row argument and both are required to carry out the column argument. No wonder that a column pircha or a row pircha, each of which knocks out one of the hierarchies (of the columns or rows, respectively), topples both formulations. Therefore the Talmud is right never to “rotate” a kal va-ḥomer to save it from a pircha. The rotation will not help; it will not save the kal va-ḥomer.
In other words, when we examine a table like that of the damages, we ask ourselves what explains the three known data in Table 1. The proposed explanation is as follows:
| Severity of the “tooth and foot” damager – α
Severity of the “horn” damager – 2α Liability threshold in the public domain – 2α Liability threshold in the injured party’s courtyard – α |
Model 1: An explanation for a standard kal va-ḥomer
You can see why the injured party’s courtyard is more stringent than the public domain (because a lesser damager strength suffices to incur liability there), and why horn is more severe than tooth/foot (because its strength is 2α). But as noted, both assumptions are required to carry out the column argument and both are required to carry out the row argument; hence, in essence, they are two formulations of the same argument.
Non-deductive logic: a pircha on kal va-ḥomer
What happens when there is a pircha? It turns out that in such a case you will not be able to find a single-parameter strength model. Consider, for example, the column-pircha table (Table 5) above.
| Damager / Domain | Public Domain | Courtyard of the Injured Party | The Moon |
| Tooth and Foot | 0 | 1 | 1 |
| Horn | 1 | ? | 0 |
Table 5: Standard kal va-ḥomer with a column pircha
What model can explain this table? It is easy to see that there is no single-parameter model that can account for the five data points in the table. The reason is the reversed hierarchy in the left-hand column as compared with the right-hand one. This shows that there are at least two different parameters at play in the background. If you analyze the table and search for a model that explains it, you can arrive at several different models, but all will have at least two parameters. For example:
| Severity of the “tooth and foot” damager – α
Severity of the “horn” damager – β Liability threshold in the public domain – β Liability threshold in the injured party’s courtyard – α Liability threshold on the moon – α |
Model 2: An explanation for a standard kal va-ḥomer with a column pircha
Note that here you can no longer establish a hierarchy between horn and tooth/foot, nor between the injured party’s courtyard and the public domain.
Therefore, the filling of the missing cell (the lacuna) is not univocal. One can fill 0 or 1 there, and both will fit the model we proposed. It depends on the relation between α and β. Hence, in the presence of a pircha we have no way to infer from the data the law regarding horn in the injured party’s courtyard.
The question of model dimensionality: kal va-ḥomer
Admittedly, even in Table 1 of the standard kal va-ḥomer one can find another explanation for the three data. For example:
| Severity of the “tooth and foot” damager – α
Severity of the “horn” damager – β Liability threshold in the public domain – β Liability threshold in the injured party’s courtyard – α |
Model 3: An alternative explanation for a standard kal va-ḥomer
If you look at Table 1, you will see that this model explains its three data quite well. Only now the fill of the lacuna is 0 and not 1 (for according to this model horn has the property β, but to incur liability in the injured party’s courtyard the property α is required). In contrast, according to Model 1 for the kal va-ḥomer table, the appropriate fill is 1. So why is it that in a kal va-ḥomer we fill the lacuna specifically with 1?
The answer is that Model 1 (which is uni-parametric—it has only α) is simpler than Model 3 (which has two parameters, α and β). The assumption is that the model chosen to explain a given table is the simplest model that explains it (this is a version of Occam’s razor; see column 426).
An alternative formulation of the standard kal va-ḥomer
We can now formulate the logic of the kal va-ḥomer argument thus: Given Table 1 with three data, and we wish to know what to place in the lacuna cell (0 or 1). To check this, we fill 0 there and find the simplest model that explains the resulting table. Then we fill 1 there and find the simplest model that explains the resulting table. If one of the two models is simpler, our assumption is that it is the correct one, and therefore the fill it entails is the correct fill for the lacuna.
Let us now apply this to kal va-ḥomer. The data appear in Table 1. If we fill the lacuna with 1, we get:
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 0 | 1 |
| Horn | 1 | 1 |
Table 7: Standard kal va-ḥomer with fill 1
The simplest model that explains the four data in this table is, of course, Model 1:
| Severity of the “tooth and foot” damager – α
Severity of the “horn” damager – 2α Liability threshold in the public domain – 2α Liability threshold in the injured party’s courtyard – α |
Model 1: A model for a standard kal va-ḥomer with fill 1
If we fill the lacuna in Table 1 with 0, we obtain the following table:
| Damager / Domain | Public Domain | Courtyard of the Injured Party |
| Tooth and Foot | 0 | 1 |
| Horn | 1 | 0 |
Table 8: Standard kal va-ḥomer with fill 0
It is easy to see that we cannot explain these four data by means of a single-parameter model. The simplest model that arises here is, of course, Model 3:
| Severity of the “tooth and foot” damager – α
Severity of the “horn” damager – β Liability threshold in the public domain – β Liability threshold in the injured party’s courtyard – α |
Model 3: A model for a standard kal va-ḥomer with fill 0
Comparing these two models shows that Table 7 is simpler than Table 8, and therefore, by Occam’s razor, the fill of the lacuna in a standard kal va-ḥomer is 1.
A different angle: back to relevance
What underlies this analysis is that, to carry out a kal va-ḥomer, we must assume a connection between the data in the table—that they are all governed by the same parameters. One cannot build a hierarchy in one column based on parameter α and infer from it a conclusion for another column that is governed by another parameter, β.
The conclusion is that behind the kal va-ḥomer table—and indeed any data table from which we wish to infer conclusions—lies the assumption that all these data belong to the same semantic field; that is, that all the laws in the table are governed by the same parameter. Only for that reason can we infer that a hierarchy present in one row or column will also hold in another row or column.
Consider, for example, the following. If Reuven scored higher than Shimon in mathematics, is it correct to infer from this that he will also be better than him in literature? If we denote mathematical aptitude by α, then aptitude in literature is different and will be denoted by β. One gifted with more of α is not necessarily gifted with more of β. This is precisely the relevance I discussed above. To infer from certain data a conclusion for another datum, they must be relevant to that conclusion; that is, they must be determined by the same parameters that govern the conclusion as well. Perhaps from the hierarchy among students in mathematics one can infer something about their grades in physics, because aptitude in physics is akin to that in mathematics (not necessarily, but more plausibly), but not about literature (there it is a very different aptitude).
We can now return to the kal va-ḥomer pilpul (about lintel and tzitzit) and try to explain the flaw in the argument.
Back to the lintel
Consider now Table 3, which represents the kal va-ḥomer pilpul:
| Object / Law | Tzitzit | Mezuzah |
| Lintel | ? | 1 |
| Four-cornered garment | 1 | ? |
Table 3: Pilpul kal va-ḥomer
The question is what to place in the two lacuna cells. If we assume the upper-right cell is 0, we obtain by a kal va-ḥomer that the lower-left cell is 1. And conversely: if we assume the lower-left cell is 0, then by a similar kal va-ḥomer we obtain that the upper-right cell is 1. This is exactly what the pilpul does: it fills 0 in one of them each time and then concludes that the other is 1.
But if we look only at the data we have from the Torah, then, as we saw, in these two cells we should place 0. If so, we get something very similar to Table 8. What does that mean? That the model explaining the table is Model 3 (like a kal va-ḥomer with fill 0), namely, that each row is governed by a different parameter and likewise each column. This implies there is no relevance—that is, one must not infer conclusions from one column to the other or from one row to the other. And this in turn means that the correct fill for the two lacuna cells is not 0 but X (see Table 4). In other words, the very structure of the pilpul’s table—that is, the mere fact that it contains only two data and that its diagonal is empty (two lacunae)—teaches us that there is no relevance between the columns or between the rows, and therefore we must not infer conclusions from the data in one column/row to another column/row.
So why, in the standard kal va-ḥomer table (Table 1), do we assume there is relevance? Why do we fill the upper-right cell with 0 rather than X? Because that entry is a datum stated in the Torah. There are three data here, not two. The Torah itself says that this is an exemption and not irrelevance. But when there are two lacuna cells, we have no way to know which of them to fill with 0, and so the question of relevance remains open. It is possible that both are 0, and as we saw, when both are 0 this in effect means not 0 but X—that is, the lintel is not “exempt” from tzitzit; rather, tzitzit is irrelevant to it. And so too for mezuzah in a four-cornered garment. It is also possible that both are 1 (from the two opposing kal va-ḥomer arguments). In such a case, we decide that we are dealing with irrelevance (otherwise we fall into a paradox whereby filling both cells with 0 yields, by a two-way kal va-ḥomer, a contradictory result: that both should be 1).
The conclusion is that, in the logical model we posit, the lintel does not possess at all the property α that is responsible for the obligation in tzitzit; it only has the property β that is responsible for the obligation in mezuzah. And likewise for the four-cornered garment: it has the property α that is responsible for the obligation in tzitzit, but not the property β that is responsible for the obligation in mezuzah. There is no relevance here, and the fill of the two lacuna cells is X, not 0. Note: it is not that the other property appears in that object to a lower degree; rather, it does not have that property at all. It is irrelevant to it.
It is no coincidence that the rule-writers wrote that one can learn a kal va-ḥomer only if we have three scriptural data points. If we have only two, we should leave matters as they are (according to my explanation, the reason is that we are dealing with irrelevance rather than exemption). By the way, the lintel-in-tzitzit example is not mine; it is cited by several rule-writers precisely to illustrate this idea.
[1] Admittedly, there are also limits on “learning from something that was itself learned,” in the sugya in Zevaḥim 50 and its surroundings, but there it appears that in some cases one may learn a kal va-ḥomer from a law that itself was learned by exegesis rather than stated explicitly in the Torah.
[2] There are two exceptions in the Talmud (in Bava Kamma and in Niddah) where the Talmud does “rotate” the kal va-ḥomer, but in both the table is more complex than the one we saw here (cases of diyyoh—that is, where one of the cells that for us would be 1 is filled with, say, 0.5).
Discussion
If you fill in 0, you can indeed remain with alpha, but with two levels. That is still more complex than filling in 1, where the model posits alpha at the same level for all the slots.
I would just note that this is a weak refutation. In the articles we build a more complex model, because increasing the number of levels within the same parameter turns out not to be decisive. This was not the place to go into that. I only wanted to illustrate the principle.
A. I am taking the opportunity to ask again, because in the past in 537 it came up in passing and I asked, but the answer did not become clear to me. What does it mean, “This is a puzzle that bothered me for quite a long time”? The moon is a refutation on the columns and “proves” on the rows. Therefore rotating should make no difference. That is how I remember it appearing in Halikhot Olam, which I studied diligently at the beginning of my learning. The puzzle that bothered you is about the “yokhi’ach”—do you wonder about every “yokhi’ach” that appears in the Gemara (or do you accept “it proves” whether against a refutation or against the common denominator)? In the kal va-homer of the rows, we learn a hierarchy between the columns, that the damaged party’s courtyard is more stringent than the public domain, and therefore an ox goring should be liable in the damaged party’s courtyard; and against that comes the “yokhi’ach” from the moon and says that the damaged party’s courtyard may be like the moon and nothing more.
B. The relevance assumption is a major innovation: if it is technically possible, why shouldn’t there be relevance? What is the problem with putting a mezuzah on the corner of a garment and fringes on a doorpost? It also seems that you are claiming that one never derives from two data points but only from three, but it is clear that there are many such cases in the Gemara, and some have written that this is really a binyan av, as you discussed in Middah Tovah with the two-slide model.
Two questions and a little question:
1. If this assumption is indeed required, then presumably here and there a dispute should appear in the Talmud about whether there is relevance between two topics or not, and on that would depend whether to expound a given kal va-homer or not. Clearly, if we find such a dispute about relevance, that proves the relevance hypothesis (the hypothesis that the relevance assumption is relevant). But what if we do not find such a dispute? Would you agree that this would weaken the hypothesis? It is no worse than a dispute about whether something is a pesik reisha.
2. Why not make the effort and look for refutations? What is special about a garment, which is forbidden in sha’atnez? What is special about a house, which is obligated in a parapet? And if you come to derive that sha’atnez is also forbidden in a doorpost, the refutation would be: what is special about a house, which is obligated in tzitzit?
3. A little question about the example of the doorpost and mezuzah. Perhaps one can infer that there is relevance between mezuzah and tzitzit from the fact that in the recitation of the Shema there appear together the two mezuzah paragraphs and the paragraph of tzitzit. When there is a shared root with two legs (as you brought regarding building on Shabbat), there is already a basis for thinking that it is really one leg with two aspects—that is, that the mezuzah paragraphs and the tzitzit paragraph are connected to one another and go together.
A. If I understood you correctly, then a “yokhi’ach” is a challenge to the rows by way of a column-refutation (or vice versa). The claim is that if the kal va-homer assumption of the rows were correct, it should have implications for the new column. In my view this is a very problematic refutation. Therefore only the analysis I offered here explains why one does not rotate.
Why is this a problematic refutation? Because it is basically telling us that if we make a kal va-homer of rows between the domains and infer that the damaged party’s courtyard is more stringent than the public domain, then by the same token we could infer that the moon is more stringent than the public domain. But that is not a refutation, because you have shown that the kal va-homer regarding the moon is indeed problematic. So what? I am making a kal va-homer about the damaged party’s courtyard, and that is not threatened here in any way.
B. The question is not whether one can put a mezuzah on a garment, but whether the idea of mezuzah is relevant to a garment.
I will get to my article on Parashat Shemini about a kal va-homer from two data points in the next column.
1. I do not see any necessity that a dispute about relevance should appear in the Talmud, and I also do not know how to check whether such a thing appears. If I had to bet, I would bet that it does.
2. I did not understand the question.
3. First, appearing in the same passage does not necessarily indicate relevance. Second, if we accept your suggestion, then there is relevance and we are back to the difficulty. So why adopt it, on the model of “one can raise the difficulty only with some strain”?
A wonderful article, sharp, clear, and enlightening.
A kiss on the lips!!
A. This is the wording of Rabbi Yosef Karo in his rules of the Gemara, in Halikhot Olam, Gate 4, beginning of chapter 2:
https://hebrewbooks.org/pdfpager.aspx?req=15255&st=&pgnum=27
“But when the refutation is of the same kind as the kal va-homer, then it does not help to alter the parable within that kal va-homer. For with tooth and foot there would be another domain in which they are liable and the horn is exempt in it; when we object and say: what is special about tooth and foot, for they are liable in such-and-such a domain—namely a jointly owned domain—will you say the same of horn, etc.? Now even if he changes the kal va-homer from places, there is no gain in this, for when you say: if in the damaged party’s domain, where tooth and foot are liable, is it not logical that horn should be liable there? I will answer you: the jointly owned domain proves it, for tooth and foot are liable there and yet you exempt horn there.”
Perhaps the refutation is problematic, but clearly this is how the mechanism works, and therefore the Gemara does not rotate in such cases (there in the rules of the Gemara, and in Yavin Shemu’ah as well, they also discuss when rotation does help). The moon too is more stringent than the public domain, as we see from tooth and foot, and nevertheless it is exempt for horn; that means there is some unknown problem in drawing conclusions from a hierarchy of columns. True, one does not infer that there is some unknown problem in the very measure of kal va-homer in general, but one does infer that there is some unknown problem in inferring here from a hierarchy between columns. I completely accept your explanation of the matter, but it is not necessary for the mechanical operation of the derivations and the rotations.
B2. My question is that one can solve your question—why one does not make a kal va-homer to obligate a doorpost in tzitzit—by saying that there is a refutation (“what is special about a house, which is obligated in a parapet?”), without the innovative claim about the relevance assumption.
B3. Because the relevance answer is astonishingly novel and strained. And in the particular example you brought, of a doorpost and a garment, I see a great deal of relevance in itself: one puts a mezuzah on a doorpost so that we will always remember something whenever we enter and leave the house, and one puts tzitzit on a garment so that we will always remember something whenever we wear it. Their common denominator is that there is importance in remembering the matter constantly, and therefore they put both the mezuzah paragraphs and the tzitzit paragraph in the Shema, which is recited every day. And there is no problem or irrelevance in putting the mezuzah paragraphs on a garment and the tzitzit paragraph or the fringes of tzitzit on a doorpost. And as stated in B2, there is no difficulty from saying, if so, let us make a kal va-homer; for the craft of our fathers is in our hands, to find refutations: what is special about this, which has such-and-such, and what is special about that, which has such-and-such.
A. I did not know that they had already dealt with this question, but I did address their answer. Very weak. As I noted here, on that basis one could throw away the tool called kal va-homer altogether. That is exactly how I have answered those who suggested this explanation to me more than once.
B2. In my eyes relevance is not an innovative claim, so I disagree with your starting point. After all, it is obvious that if there is no relevance, there is no logic at all in making a kal va-homer. If you learn from one thing to another, you assume that they have something in common and that there are relations of leniency and stringency between them. That is not an innovation at all. Admittedly, the question of how one finds irrelevance is harder. Why assume that a doorpost and tzitzit are not relevant to one another? To that I answered that it follows from the very fact that there are only two laws in the table and not three. In the next column I will show exceptions to this and explain more. In short, when there are two laws in the table, the assumption is that there is no relevance unless there is a positive rationale that there is relevance. But if there is such a rationale, then we will accept relevance even in a table of two laws.
B3. The irrelevance is in the mode of remembrance, not in remembrance itself. Putting a parchment on a garment is probably not a sensible way to remember. Again, it is clear that the rationale here is ad hoc, but I think this is the prevailing intuition among those familiar with these materials. And as stated, the requirement of relevance is not forced.
A. The Gemara makes such a “yokhi’ach.” In Dicta I immediately found Menachot 5b: “And just as a blemished animal, which is permitted to an ordinary person, is forbidden to the Most High, so a tereifah, which is forbidden to an ordinary person, should certainly be forbidden to the Most High. Fat and blood prove otherwise, for they are forbidden to an ordinary person and permitted to the Most High.” Their very weak answer is the correct one, and it can then be explained further with your mechanism. One could have thrown away the tool called kal va-homer, but something more limited is being discarded. The attempt to infer from the three data points before us to the fourth datum fails in the case of the moon, so perhaps it also fails in the case of the damaged party’s courtyard.
B. I am indeed not familiar with these materials, though in the past I studied Halikhot Olam and some of the rules of the Gemara. And perhaps because of that, I know only of an irrelevant refutation, and not of an irrelevant kal va-homer or an irrelevant binyan av. In any case, from where I stand, the requirement of relevance is not accepted at all, and especially not here in the example you brought. If you come across a source that discusses this, I would be very happy to see it.
The question is what rationale underlies this “yokhi’ach.” I claim that the rationale is my analysis, and not the “yokhi’ach” in its literal sense, because that is weak. When they bring such a “yokhi’ach,” that is, a column-refutation against an argument of rows, my claim is that the rationale behind it is my analysis (I am not claiming that the Sages were aware of this, but this is what underlies their intuition).
By the way, I have no experience with Dicta. Is this not just a search for the word “yokhi’ach,” which can also be done in the Responsa Project? Is there a better tool there?
I understand that this is what you are claiming, and I also accept (enthusiastically) your explanation of the matter. I am only saying that the absence of rotation is not a surprising conclusion from searching the Gemara, but something to be expected. And still, the explanation is needed.
(For a simple search like this there really is no difference. It’s just that lately, and in the near future, I do not have access to Otzar HaHokhmah and the Responsa Project. Dicta is more flexible with distances and inflections and different forms of the word, which is both an advantage and a disadvantage, and Dicta is also accessible by phone.)
I got stuck at the beginning.
1.
In the section “The reversed kal va-homer and its meaning”:
“Binyan av”—who even mentioned that?!
Seemingly, what is required is to examine the measure of kal va-homer, not the halakhic results (which come from binyan av). If so, when we approach the “doorpost and garment” case with this measure, it cannot be applied forward and backward, nor can one arbitrarily take hold of one of the possibilities (forward/backward). That itself shows a deficient structure, a lacuna. That is, when one claims that “just as a garment, which is exempt from mezuzah, is obligated in tzitzit, so a doorpost, which is obligated in mezuzah, should also be obligated in tzitzit,” it is already implicit here that the datum that a garment is exempt from mezuzah is a datum clear to us. If so, already at this stage the reverse move has been blocked, since one cannot derive by kal va-homer that a garment should be obligated in mezuzah. And similarly the other way around.
Seemingly, a valid kal va-homer is only one that is relevant in one direction alone.
If so, for the dilemma one can remain with the starting points that on both sides there is a datum of “exempt,” and not a lacuna.
The discussion whether there really is such a datum or whether it is only “irrelevance” is not necessary for a dilemma that does not exist.
2.
You wrote that one can perform a binyan av after a kal va-homer. This is not clear to me:
Seemingly, in every hermeneutical measure there are the data on which it is based (hereafter: “the building blocks of the measure”) and there is the result of the derivation.
Now we have to clarify: A. What kind of building blocks are required for each of the measures in question (binyan av, kal va-homer) in order to carry out the derivation? Are Torah-level building blocks required (that is, data explicit in the Torah), or are rabbinic building blocks sufficient? B. What counts as the result of the derivation? Is the law in the result considered Torah law or rabbinic law?
There are several possibilities.
A. That the building blocks are always Torah-level (for both measures), and the result of the derivation is also Torah-level.
B. That the building blocks are always Torah-level, but the result is rabbinic.
C. That it is enough for the building blocks to be rabbinic (for both measures), and the result of the derivation is rabbinic.
D. That it is sufficient/necessary for the building blocks to be rabbinic, and the result is Torah-level.
E. The building blocks of kal va-homer are Torah-level, and the result is Torah-level; unlike binyan av, where both are rabbinic.
F. The reverse.
G. The building blocks of kal va-homer are Torah-level, and the result is Torah-level.
The building blocks of binyan av are Torah-level, and the result is rabbinic.
H. The reverse (of the measures).
I. The building blocks of kal va-homer are Torah-level, and the result is rabbinic.
The building blocks of binyan av are rabbinic, and the result is rabbinic.
J. The reverse (of the measures).
There are still other possibilities within D, but since it is implausible I will refrain from detailing them.
Now,
According to possibility A, one cannot perform a binyan av after a kal va-homer, because if so the building blocks of the kal va-homer change from Torah-level (because of the binyan av), and then there is no longer a kal va-homer. (And it needs investigation whether this itself shows that possibility A is impossible.)
According to possibility B, likewise, since binyan av also requires Torah-level building blocks, but they are not, since the result of the kal va-homer is rabbinic.
According to possibility C, likewise, since the building blocks of the kal va-homer change from rabbinic and then the kal va-homer is nullified (similar to A).
According to possibility D, it depends on whether it is necessary or sufficient. But this possibility is implausible, so I will not elaborate.
According to possibilities F, H, J, likewise. And that is obvious.
So we are left with possibilities E, G, and I, according to which alone it is possible to perform the binyan av after the kal va-homer.
But from your starting point it seems that you hold that if we have clear knowledge (that is, at least rabbinic force), that is enough for it to serve as a building block from which to teach a kal va-homer. In that case I do not see any suitable possibility (in order for it to be possible to perform a binyan av after a kal va-homer)—unless we adopt something like D, that it is necessary that the building blocks of kal va-homer be rabbinic (and only kal va-homer), and the result of binyan av be Torah-level. But that is not likely.
I would be glad for clarification on the matter.
As for me, I adopt possibility B (since these measures are not deductive in character). Therefore it indeed cannot be done.
1. Binyan av is a table in which there are three 1s and we complete the fourth as 1. Unlike kal va-homer, where there are two 1s and one 0. When there are two data points, one can complete either of the missing ones by kal va-homer, and then de facto a binyan av table results. One can view this as a binyan av on the two slots together. There is no impediment to that, and it is even the most reasonable thing were it not for the relevance question (because it is the simplest filling compared with all three other possibilities). Therefore it is not correct that a kal va-homer is valid only when it can be made in one direction alone (although some writers on the rules did write that). In the next column I bring counterexamples to this.
To remain with the assumption that on both sides there is an exemption and not a lacuna—this is the least plausible possibility (because the table obtained that way is explained only by a two-parameter model).
I did not go into the details of your section 2 because it is long, and I got the impression that in light of what I have now explained it is unnecessary.
The whole treatment of data tables of different kinds is laid out in the articles I linked to, and of course here I could not really get into all of that.
I do not understand why you are explaining the validity of performing a binyan av when I asked only about applying kal va-homer.
You merely noted that the assumption in my question is incorrect. In any case, I will wait for the next column on this matter.
As for section 2, it is another issue—regarding the circularity of a binyan av after a kal va-homer in the “garment and doorpost” case. I would be glad for a response.
I explained why in principle one can make a kal va-homer in two directions. In essence this is a simultaneous binyan av on the two slots. See in the next column the example where I also note that the wording of the Gemara implies that these are two kal va-homers and not a binyan av.
Regarding rounding the head for a woman:
1. The Gemara does indeed look for a source to exempt her.
2. When her hair does in fact grow and she removes it, she is indeed exempt.
That is, it seems that she too is exempt.
Seemingly, the fact that sidelocks do not grow for her is only generally true and not absolute, and therefore either this is relevant to women, though uncommon, and they are exempt; or at least because it is uncommon, the Torah exempted them.
Let me sharpen point 2.
If her hair grows and she removes it, the ruling is that she is exempt. Unlike the doorpost, where it is not ruled exempt from tzitzit.
That is, intuitively there is relevance to the prohibition of rounding the head in the case of a woman.
This should be discussed, because in Nazir 57 there is an amoraic dispute about the prohibition for a woman regarding rounding a minor’s head, and it appears that they disagreed over whether a woman has an exemption or not.
In any case, in Nazir 57b it is explained that a woman’s exemption from rounding is learned from her exemption from destroying the beard. And Rambam, Hilkhot Avodah Zarah ch. 12, halakhah 2, wrote that regarding destruction she is exempt because it is not applicable to her (as one of the explanations in the Gemara, Kiddushin 35). So what I wrote regarding rounding can be said regarding destruction. But the commentaries (Shakh and others) on Yoreh De’ah 181:7 did say this regarding rounding, which is learned from destruction. That is, they understand that a woman is not exempt, but rather it is simply not applicable to her, as in destroying the beard; and therefore they write that in the case of a slave there is no such exemption.
Something is escaping me here—
Why in the “garment and doorpost” sugya should we not fill all the slots with 1 and get a one-parameter table, rather than filling x in two slots (on the diagonal), which yields a two-parameter table?
That is exactly what I asked. And the answer is that the assumption is that there is no relevance between the columns.
?!
Why assume that if it is less simple than the alternative?
That is exactly my claim. Relevance is based on rationale, and therefore it precedes considerations of simplicity. Assuming there is relevance between the columns, we compare the levels of simplicity between the two filling options (0 in both slots or 1 in both). But if it is not relevant, then there are no filling considerations at all.
First of all, more power to you for the article; I read it and enjoyed the straightforward and clear analysis very much.
I just did not understand why, when we have 3 scriptural data points, this necessarily means that there is certainly relevance between the two columns in the table. Let us return for a moment to the example you brought regarding school subjects:
Reuven got 50 in mathematics, and Shimon got 100, whereas in literature Reuven got 100—we have here 3 data points, but would it be correct to infer from this that Shimon will certainly get 100 in literature?
Where did I say that it is certain? After all, in every case of a refutation we see that despite there being three data points there is no relevance. What I said is that given three data points, that is the starting point (that there is relevance) until proven otherwise. But when there is a clear rationale that these are independent parameters, as in the case of the exams, then it is indeed reasonable to infer that there is no relevance.
It is worth noting the sugya in Zevachim 16, which really touches on this matter.
And see Tosafot there, who wrote as follows: “Let each one stand in its own place.” This requires investigation… in the chapter Keitzad HaRegel (Bava Kamma 25b), where it says: “Then let tooth be liable in the public domain by kal va-homer,” and in the first chapter of Kiddushin (14a): “Then let a yevamah go out by a bill of divorce by kal va-homer,” and in Nazir, chapter Sheloshah Minin (44a), regarding one who is impure as one who renders impure, and at the end of Yesh Mutarot (Yevamot 87b), that the dead should be like the living by kal va-homer—one must examine all of them: why did it not say, ‘Let each one stand in its own place’?”
Indeed. And the conclusion of Tosafot’s words, “and this requires investigation,” is not a difficulty left unresolved, but a methodological instruction. We must examine in every such case why there is relevance, even though on the face of it there perhaps appears not to be.
I opened Zevachim there and did not see any connection to relevance. There there are four data points, two permissions and two prohibitions, none of which is explicit, but each is learned by the rationale that one should not differentiate where the text did not differentiate, as Rashi and Tosafot explain there. Just as in a kal va-homer from two data points, the third datum is not explicit but learned by the “rationale” that where no obligation is written, there is no obligation. [Impurity was permitted in a communal offering; one could minimize the novelty only to the High Priest, or one could say that we do not differentiate, and an ordinary priest too may offer a communal sacrifice in impurity. Impurity was forbidden in an individual offering; one could minimize only to an ordinary priest, or one could say that we do not differentiate, and the High Priest too is forbidden to offer an individual sacrifice in impurity. Mourning before burial was permitted for the High Priest; one could minimize only to a communal offering, or one could say that we do not differentiate, and also in an individual offering mourning before burial is permitted for the High Priest. Mourning before burial was forbidden for an ordinary priest; one could minimize only to an individual offering, or one could say that we do not differentiate, and also in a communal offering mourning before burial is forbidden for an ordinary priest.]
If one uses only three data points, one can learn the fourth law by kal va-homer in the opposite direction from the above rationale. If one uses two permissions and one prohibition, then one learns permission in the second prohibition; and if one uses two prohibitions and one permission, then one learns prohibition in the second permission. Altogether there are four possible ways to choose three data points in which to use the rationale that one should not differentiate where the text did not differentiate, and then infer in the fourth law the opposite of that rationale. Rava chooses one of the possibilities, and the questioner presents him with the three remaining possibilities. The questioner concludes that since we have no way to decide which three to use with the rationale, and which fourth result to infer by kal va-homer against that rationale, therefore we should use the rationale in all four laws and refrain from all the derivations.
If so, Uri’s point is as follows. In Zevachim, according to the questioner, one refrains from all the derivations. And with a kal va-homer from two data points, we saw in the columns that there are cases where one chooses one by rationale (and this is apparently what Rava does in Zevachim too), and there are cases where one makes both derivations simultaneously. So there are three questions here: A. why in Zevachim does one not choose one derivation by rationale? B. why in Zevachim does one not make all four derivations simultaneously? C. why in a kal va-homer from two data points do we not refrain from both derivations?
To the first question—why one does not choose one of the derivations—it seems that Rashi and Tosafot answer as above: that the questioner found no rationale to choose precisely one derivation, because they are all completely identical.
To the second question—why in Zevachim one does not make all the derivations simultaneously—it seems to me one should answer that then we would get that with impurity the distinction is not between communal and individual, but between High Priest and ordinary priest; and with mourning before burial the distinction is not between High Priest and ordinary priest, but between communal and individual. But this is an entirely different division from what one sees in the verses, because in the verses it seems that with impurity there is indeed a distinction between communal and individual, and with mourning before burial there is a distinction between High Priest and ordinary priest. Therefore it is implausible.
To the third question—why in the two kal va-homers from two data points we do not refrain from both of them—one could answer that this is always the last option, but it seems to me that the answer is simpler. In a kal va-homer from two data points, the lacuna is completed by a “rationale” from absence: if there is no source to obligate, then there is no obligation—for example, there is no source obligating grace after meals before eating, so there is no obligation. Such a rationale from absence is also certain, and therefore independent as well: certain, in the sense that it is obvious that as long as there is no derivation that fills the absence, there is no obligation; and independent in the sense that if elsewhere there is indeed a derivation that fills the absence, that does not affect the correctness of the rationale elsewhere. But in Zevachim the lacunae are completed by an interpretive rationale that instead of minimizing the novelty to the minimum novelty, we say that one should not differentiate where the text did not differentiate. That interpretive rationale is not certain but only a plausible consideration, and therefore it is also not independent: if in one place we see a derivation that reveals that this rationale is mistaken, then we already question it in the other places too.
Be that as it may, I did not see any connection here to relevance. That is, it seems quite clear that in all the derivations there is a great deal of relevance. True, I looked into it somewhat hastily, and if I erred, blame my mistake on me.
Where are the articles?
Do you mean the books on Talmudic logic?
Search here on the site for the articles on non-deductive inferences. It is the first book in the Talmudic Logic series.
You wrote that in practice girls do not grow sidelocks.
That is puzzling,
because the place of the halakhic sidelocks certainly exists in girls as well, namely between the eye and the ear (and according to the Ashkenazic invention, also upward a bit more).
What did you mean?
I probably confused it with destruction of the beard. See Rambam, ch. 12 of Hilkhot Avodah Zarah, halakhot 1–3.
Regarding the claim of irrelevance: in the sugya in Bava Kamma 88a (and in Tosafot s.v. “she-ken”), the Gemara uses as a refutation to a kal va-homer something that is not relevant: the Gemara learns by kal va-homer that a slave is more stringent than a woman, and refutes it from circumcision, where a slave is commanded in circumcision and a woman is not, even though a woman does not belong to the category of circumcision at all.
What do you think? Does it make sense that a refutation should work even in a case of irrelevance, whereas a kal va-homer would not?
No. They probably thought this was relevant. Either because absence too is a leniency, or because the Torah could have obligated women in something relevant to them, and if it did not do so, that is a leniency.
And what is the logical model in a binyan av? I can stay only with alpha and still not obligate. There is no hierarchy there. Is the hierarchy assumed a priori? Doesn’t that verge on ta’ama de-kra?