Question

Hello,
I apologize in advance for the length; I would be very grateful if you still find the time to read it.
I recently read the Rabbi’s two articles on the logical interpretive rules. I understood that they also came out in book form in the Talmudic Logic series, but I don’t have access to it, so I apologize if the answer to my question is there. One of the central claims that emerges from the model presented in the articles is that reversing a kal va-chomer does not really have any significance. You showed how this conclusion fits the Talmudic passages about dayyo and the Talmud’s symmetrical treatment of refutations by row and by column.
But even though we do not find such examples in the Talmud, as is well known there are several places where medieval authorities (Rishonim) — mainly Tosafot — reverse a kal va-chomer in order to escape a refutation. For example, in the passage in Chullin (25a):

Rav Adda bar Ahavah said to Rava: Then let an earthenware vessel become impure from its outside by a kal va-chomer: just as all vessels, which do not become impure through their airspace, do become impure from their outside, so an earthenware vessel, which does become impure through its airspace, should all the more so become impure from its outside!

Tosafot there asks: why not refute it by saying, “What is true of all vessels is that they can become a primary source of impurity”? And it answers: “One can say that the kal va-chomer is not being made from the force of ‘vessel’ but from the force of ‘outside’ and ‘airspace.’” Later authorities there explain that this means a reversal of the kal va-chomer. Seemingly, according to the way you described reversal in the model in the article, the reversal should not help at all.
I think I may have a solution to this question, and I wanted to present it here and see how you think it fits with your model.
In my opinion, there is an inherent asymmetry between the rows and the columns of a kal va-chomer (for convenience, I’ll define the columns as the ones on which the ordering relation is built). When I define an ordering relation between all vessels and earthenware vessels, that relation is supposed to be preserved for every row I add to the table. When I add a row to the table, I am free to choose any relevant property of vessels. The property can be “become impure through their airspace” or “become impure from their outside,” but it can also be “become a primary source of impurity,” because that too is a property of vessels that we regard as relevant to impurity.

By contrast, when adding a column to a kal va-chomer, the restriction is much tighter. In the reversed kal va-chomer I define an ordering relation between locations of impurity: airspace and outside. The properties I choose for the rows are properties of locations of impurity: “imparts impurity in all vessels” or “imparts impurity in earthenware vessels.” If we want to reverse Tosafot’s original refutation, we would get: “A primary source of impurity proves it, for it imparts impurity in all vessels and does not impart impurity in earthenware vessels.” But I cannot add that column to the table, because the properties “imparts impurity in all vessels” and “imparts impurity in earthenware vessels” are not true with respect to the subject ‘primary source of impurity.’ When I added a row, I was free to attach whatever verb I wanted to ‘primary source of impurity,’ and I chose the appropriate verb, ‘become.’ But once I reverse the kal va-chomer, I am bound to the verbs of the existing rows: ‘imparts impurity.’ When one says “a primary source of impurity imparts impurity in all vessels,” the word “imparts impurity” does not belong there — or at the very least, it does not belong in the same sense in which it belongs with airspace and outside.
For that matter, this can be seen as a dimensional problem. The “units” in which one measures the impurity of airspace in all vessels, or of outside in all vessels, are not the same units in which one measures the impurity of a primary source of impurity in all vessels. I cannot compare the values of the ‘outside’ column to the values of the ‘primary source of impurity’ column, and therefore I cannot claim an ordering relation between them: “outside > primary source of impurity” / “outside = primary source of impurity.” Placing the ‘primary source of impurity’ column in the table is mistaken. To allow comparison, the units within each row need to be uniform. That is unlike the units within each column, where there is no problem comparing subjects on the basis of several properties, even if each is measured in a different unit.
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I do not think this solution contradicts the model or the claims you presented in the article. I think the problem with the “reversal” you carried out in the article in the chapter on dayyo is that it is not really a “genuine” reversal. When you “reversed” the kal va-chomer, all you changed was whether one looks at the ordering relation between the columns or between the rows. But the headings of the columns and rows remained the same headings.
If we reverse the kal va-chomer in Chullin in the same way you reversed it in the article, we get something like this:

Instead of discussing the ordering relation between the vessels, we discuss the relation between the locations of impurity as actions (which actions operate in more vessels). In this form the refutation still stands, because the action “becoming a primary source of impurity” really does prove that one cannot infer that actions operate in earthenware vessels from the fact that they operate in all vessels. Or, in terms of your model, one can build an ordering relation between “becoming a primary source of impurity” and “imparting impurity from the outside” because the units in which each of their rows is measured are uniform: “success of an action.”
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Maybe this solution looks like a linguistic-semantic trick, but I think there is a substantive difference here between the arguments. When I claim that earthenware vessels are more stringent than all vessels, I expose myself to refutations from all the existing properties of vessels — not only properties of “whether contact in a certain place renders them impure,” but also the property of “whether they can become a primary source of impurity.” That is a property of vessels.
When I reverse the kal va-chomer, and I claim that the location ‘outside’ is more stringent than the location ‘airspace,’ I avoid refutations that are properties of vessels (such as “become a primary source of impurity”), but at the same time I expose myself to other refutations that are properties of the locations ‘airspace’ and ‘outside.’ If there were some law regarding those locations that was unrelated to types of vessels, I could refute it from that. Suppose, for example, that the outside did not become impure from a creeping thing, while the airspace did. I would refute it by saying: “What is true of the outside is that it does not become impure by a creeping thing.” That would be a refutation of the reversed kal va-chomer that could not be used to refute the original kal va-chomer.
I’ll conclude by saying that this explanation also clarifies when one cannot escape a refutation by reversing. If the refutation has the same ‘units’ as the previous rows, and the units of the whole table are uniform, then we can also place it as a column in the reversed kal va-chomer. For example: