Regarding the claim that the generalization underlying the comparison is “not hidden at all” — I think that in concrete comparisons it definitely is hidden, meaning it isn’t really fully defined.
When I say this frog knows how to swim and therefore that frog also knows how to swim, it does indeed pass through some broad generalization, because there is nothing special about that particular frog as such, but I don’t know how to define exactly what a frog is. It’s possible that tomorrow I’ll see some kind of animal that I’ll think is a “frog” but it won’t know how to swim. The generalization about swimming depends, for example, on body structure and habitat. And the actual set of criteria that led me to compare frog A to frog B is something a person cannot always formulate. Even when he says “these and those are frogs,” that does not always capture exactly the criterion he used for the comparison. Since, after all, we are all casuists and not positivists, as was explained there in Two Wagons.

Right?

Absolutely right, in my opinion.

I still have trouble separating this argument from the first argument. What you are basically saying is that even before I thought about Plato at all, from the very fact that I decided to look for a reason (or correlation or explanation) for Socrates’ being mortal—namely, his being human—I already generalized that all those in whom that reason exists (simply by virtue of its being a reason or a correlation, etc.) will also be mortal. But that is exactly the same generalization as in the first argument. There is only one generalization here.

To Naakesh,

I disagree with you. Usually when a person makes an analogy it is because he already suspected that there is a connection between the frog and its swimming, and he has already generalized it in his mind. He still does not say it explicitly, out of caution. He wants to gather evidence for the generalization before he states a general claim and acts on it, and before he is sure of it. So it is only a matter of self-awareness of our cognitive process. But once that is clear to a person, then the moment at which the generalization was made (in that first case) will also be clear to him.

(Who said there are two generalizations at the foundation of every analogy? If you quote the passage from Two Wagons where that is written, it might be possible to try to address it.)
First, the “hiddenness” of the induction when making an analogy is simply because in the wording of the analogy the induction is not presented. So even if, when one thinks about it, one sees as clearly as the sun halfway across the sky that there is an induction here, it is still called “hidden.”
Second, the induction as a sweeping rule is also hidden in an essential way, and here I disagree with you in return, although perhaps there is really no disagreement. “Self-awareness of our cognitive process” is exactly the point. Usually there is no possibility of formulating our cognitive process precisely (that is, to formulate a positivistic rule and from now on let machines derive deductions from it). Someone making an analogy cannot always (and usually cannot) formulate explicitly and precisely the induction in the background, even though he uses it itself (because the boundaries are gray and winding).
Third, you wrote that it is not clear why they considered analogy from the outset as an additional mode of inference, etc. For some reason, and please take this in good spirit, this reminds me of the story about a certain fellow who stood in the rain, at a loss as to how he would get home, and cried out to the Holy One, blessed be He, to save him, and he would take upon himself this and that. Barely a moment after he finished praying, a splendid Mercedes stopped next to him and offered him a ride. The man hurriedly said to himself, “Never mind, Blessed God, I’ve already managed,” and got into the car.

Indeed נכון. These are two sides of the same coin.

I’m sorry. I really still don’t understand. How are they two sides of the same coin (the generalization)? After all, the first side is simply the fact that there is a generalization at the basis of the analogy, and that is understandable. The second side refers to the decision how to generalize (and not to the generalization itself. The Rabbi there spoke about the induction in the fact that I choose דווקא one parameter and not another that is irrelevant—the letter their names begin with). How exactly is that a second side? It seems as if the Rabbi was saying there that the very fact that we decided that something is relevant and something else is not—that is the induction (“something like an induction”). And I didn’t understand how.

I’m pressing the point precisely because I feel that you are raising something here that is a real point. This has really bothered me for years, and only now did I remember that there is a site where I can ask about it. I’d be glad if the Rabbi could explain it in greater detail. I mean, how the Rabbi would rewrite that passage there.

I’m not going to rewrite the passage here, and I’m not sure anything needs to be changed. My claim is simple: when you choose a relevant parameter of similarity, you are in effect making an implicit generalization. That reflects, from another angle, the fact that there is a generalization at the basis of the analogy. What isn’t clear?

What isn’t clear is why the relevance of the parameter matters. Even if I choose an irrelevant parameter, then too I am making an implicit generalization (admittedly not a good generalization, but still).