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There are two types of errors and Java's mistake was making them all the same.

That’s not true. Java has 2 types of exceptions checked and unchecked. Checked exceptions are what this thread has been calling errors, and unchecked exceptions are what this thread has been calling exceptions. Maybe it was a mistake to call them both exceptions, but Java also has 2 types of errors.

I'd say Java named them appropriately. While you are right that they almost cover the same intent, error state is not dependent, whereas checked exceptions force a dependency on the caller[1]. They are not quite the same thing.

[1] Ish. If we are to be pedantic, technically checked exceptions are checked by the exception handlers, not the exceptions themselves. If you return a 'checked' exception rather than throw it, Java won't notice. However, I expect for the purposes of discussion we are including exception handlers under the exception umbrella.

Checked exceptions are NOT like errors-as-values. It's only resemblance is that checked strictly forces the exception to be handled, similar to errors-as-values. But the handling itself is still the same as regular exceptions: out-of-band and not composable with anything else.

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