The Logic of Kal va-ḥomer: A. Relevance (Column 735)

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.

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):

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):

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:

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:

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:

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:

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:

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:

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.

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:

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:

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:

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:

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:

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:

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:

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).