Reduction ad absurdum in Chiswell&Hodges

Question

About that natural deduction system that is used in the beginning of Chiswell&Hodges. Can I introduce the sequent $\{a,\neg a\} \vdash b$ where $b$ is an arbitrary proposition? In other words this rule says that from absurdity anything follows.

What confuses me is that the rule in the book says that if there are $a$ and $\neg a$, and you have some assumption, for example $\neg c$, you can remove the negation sign from the assumption and introduce $c$. But the proposition $b$ is not an assumption in my system, it is arbitrary. I think I can put $\neg b$ as an vacuous assumption in the derivation, but is it legit to let it appear out of the blue?

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EDIT: If I want to prove an arbitrary statement $\psi$ from absurdity, I don't think this way will work (what rule should I apply?):

I have come up with this. Here I have found a way to put $\psi$ as an assumption:

Answer 1

It might be clearer if you look at the rules for negation in their sequent forms on pp. 26/27 of their Mathematical Logic. Two of them, simplified, are

($\neg$E) If $\Gamma\vdash\phi$ and $\Gamma\vdash\neg\phi$ are both correct, then $\Gamma\vdash\bot$ is correct.

(RAA) If $\Gamma\cup\{(\neg\phi)\} \vdash\bot$ is correct, then so is $\Gamma \vdash\phi$

So we have the following argument

$A, \neg A, \neg B \vdash A$ and also $A, \neg A, \neg B \vdash \neg A$ [two applications of the basic property of $\vdash$, the Axiom Rule of p. 8]

$A, \neg A, \neg B \vdash \bot$ [by $\neg$E]

$A, \neg A, \vdash B$ [by RAA]

So that's how the RAA gets used [in C&H's system in its sequent version] to derive any conclusion you like from a contradictory pair.

Now think how this is reflected in the natural deduction version of their system.