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  1. Is Kagan's slipup one of formal or informal logic?

  2. Can you please unravel in high-school terms, but more broadly, how Alito's counterargument rebuffed Kagan's argument? I have music degrees, but nothing in math or philosophy.

  3. Can Kagan's and Alito's arguments be generalized into an Argument Form?

2. Justice Kagan concedes that Justice Alito devastated her hypothetical

Early in the challengers’ case, Justice Kagan tried to trap advocate Michael Carvin with a cleverly-designed (though not really on point, as we’ll see) hypothetical trying to make the government’s argument that HHS establishes an identical exchange to the one “established by the state” because of the statute’s use of the word “such”:

JUSTICE KAGAN: Can — can I offer you a sort of simple daily life kind of example which I think is linguistically equivalent to what the sections here say that Justice Breyer was talking about?

So I have three clerks, Mr. Carvin. Their names are Will and Elizabeth and Amanda. Okay? So my first clerk, I say, Will, I’d like you to write me a memo. And I say, Elizabeth, I want you to edit Will’s memo once he’s done. And then I say, Amanda, listen, if Will is too busy to write the memo, I want you to write such memo.

Now, my question is: If Will is too busy to write the memo and Amanda has to write such memo, should Elizabeth edit the memo?

(Laughter.) . . .

JUSTICE KAGAN: Because in my chambers, if Elizabeth did not edit the memo, Elizabeth would not be performing her function. In other words, there’s a — a substitute, and I’ve set up a substitute. And then I’ve given I’ve given instructions: Elizabeth, you write — you edit Will’s memo, but of course if Amanda writes the memo, the instructions carry over. Elizabeth knows what she’s supposed to do. She’s supposed to edit Amanda’s memo, too.

After Carvin answered, Justice Alito jumped in with his own hypothetical showing why Carvin’s answer was correct:

JUSTICE ALITO: Well, Mr. Carvin, if I had those clerks, I had the same clerks –

(Laughter.)

JUSTICE ALITO: — and Amanda wrote the memo, and I received it and I said, This is a great memo, who wrote it? Would the answer be it was written by Will, because Amanda stepped into Will’s shoes?

MR. CARVIN: That was my first answer.

(Laughter.)

JUSTICE KAGAN: He’s good, Justice Alito.

Even Justice Kagan had to admit that Justice Alito had bested her.

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I think it was Alito who is wrong here, by taking what Kagan is saying to an unreasonable extreme. I don't think Kagan slipped up, but she was appealing to some kind of (reasonable, I think) implicit principle that some properties (duties, obligations, but also things done too them) of a person get transferred over to a substitute who is tasked with performing the same job (even if these properties are not explicitly stated). Since Will is to have his memo reviewed by Elizabeth, and Amanda substitutes for Will, Amanda is to have her memo reviewed by Elizabeth. I think that Alito's objection wrongfully assumes that what Kagan is saying is all of the properties get transferred over two-ways, and that Will and Amanda are somehow now interchangeable. This is absurd, but from this interchangeability of Will and Amanda, he concludes that, because Amanda substitutes in for Will in writing the memo, and Amanda wrote the memo, that therefore the property of writing the memo gets transferred to Will as well.

There's not really much in terms of formal logic here, aside from noting that Alito's argument is a sort of reductio ad absurdum, where he tries to reduce (his understanding of) Kagan's suggestion to an absurd implication, and concluding that therefore the suggestion is wrong. He also employs some kind of indiscernibility of equivalently-tasked persons, which sort of reminds me of the indiscernibility of identicals (not to be confused with its converse, the identity of indiscernibles), a rule of formal logic.

(BTW I don't know anything about American politics or the context of the exchange; my analysis is entirely based off what is written in the question).

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