Interesting question!

It seems to also show up in the first episode of American Horror Story: Freak Show [2].

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After thinking a bit on it and reading about it, I would go with neither would go to jail.

It is better for a criminal to get away with murder than to have an innocent person go to jail for that crime.

It is better that ten guilty persons escape than that one innocent suffer.

AKA Blackstone’s ratio

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There are some other possible solutions to the problem, however. You might throw both twins in prison but treat only the guilty one as if he were a convict. When the siblings were released, for example, the good twin would have all the rights of a normal citizen, while the evil twin would have lost the right to vote, be registered as a sex offender, etc. You might even compensate the good twin according to the relevant payout rules for wrongful convictions. (Those who were wrongly incarcerated for a federal crime can get up to $50,000 per year, or $100,000 if they were on death row.)

All of the above assumes that one twin is unambiguously guilty, and the other is unambiguously innocent. In real life, it’s hard to imagine such a clear-cut case. For example, a jury might be inclined to believe that the “good” twin acted as an accomplice, or perhaps an accessory, to the crime after the fact. This charge would apply if one sibling knew that the other had committed a crime—which seems likely under any circumstances—and that he intentionally provided assistance or comfort to his sibling rather than calling the police at the first opportunity. If the good twin were convicted of an accessory crime in federal court, he’d be subject to at most half the prison term appropriate to his evil brother. In some states, however, it’s legal to harbor a fugitive if that person happens to be your sibling.

One more way that a “good” twin might be convicted, even if he took no part in the actual committing of the crime: In some states, he might be found guilty of not stopping his brother. Although as a general rule, common-law tradition dictates that you can’t be held accountable for something you didn’t do, 10 states have so-called “duty to rescue” statutes. These require innocent bystanders to call the police or reasonably attempt to aid a victim in distress. (In four of these states, siblings of the offender are exempt from the law.) If one twin tried to stab someone, the other might be expected to grab his arm or drag both of them to the ground. The penalty for failing to rescue is usually a fine, though some jurisdictions allow for up to a year in prison.

[1] Source: https://slate.com/news-and-politics/2010/01/a-siamese-twin-commits-murder-the-explainer-s-2009-question-of-the-year.html

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I believe that within our Constitutional framework, the only thing to be done in this situation is to release the conjoined twins. Even if the jury sentenced the conjoined twins to death, the court would have to commute the sentence and release the twins. The guarantees of due process under the Fifth and Fourteenth Amendments prohibit punishing an innocent actor.

He asks the important question: “Which do we hold higher, the punishment of an innocent life or freeing a guilty one?”

[2] Source: https://www.bustle.com/articles/44313-how-are-conjoined-twins-punished-if-one-commits-murder-asking-for-bette-dot-on-ahs