# Reduction ad absurdum in Chiswell&Hodges

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?

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: • I suspect that in the rule you're quoting, "there are $a$ and $\neg a$" means these two formulas are conclusions, not assumptions. In any case, your statement of this rule isn't clear enough for useful discussion. – Andreas Blass Jun 22 '14 at 21:05

## 1 Answer

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.

• Please, check the edit to my question. I gave the simplest derivation that I could think of. – Graduate Jun 23 '14 at 11:45
• Your derivation works. But in fact C&H (if you look very carefully) allow you to discharge "assumptions" that actually appear nowhere, so the bottom three lines of your proof already suffice! – Peter Smith Jun 23 '14 at 12:38