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Here is a proof for double negation elimination. I wanna know if it's a proof of how law of excluded middle implies double negation elimination, since there's usage of rule of explosion (ex falso sequitur quodlibet) in this proof and I don't know whether it's an axiom or a derived rule.

Taroccoesbrocco
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Pooria
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    There are different versions of Natural Deduction, but usually, if we have $\bot$ as primitive (and thus $\lnot P$ is defined as: $P \to \bot$) we need the $\bot$E rule. – Mauro ALLEGRANZA Sep 25 '17 at 10:10
  • You can see this post for a review of the various rules regarding negation. – Mauro ALLEGRANZA Sep 25 '17 at 10:11
  • @MauroALLEGRANZA hmmm so you mean usually $\bot$ elim is an axiom and so that proof proves that implication? – Pooria Sep 25 '17 at 10:27
  • @MauroALLEGRANZA I guess doesn't have to be an axiom, just a law that isn't derived from axioms. – Pooria Sep 25 '17 at 10:40
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    In natural deduction, there are no axioms at all: only rules. – Mauro ALLEGRANZA Sep 25 '17 at 11:38
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    In "usual" axiomatic presentations of classical prop calculus, we can have many different versions: as you can see in List_of_logic_systems there are many possibility of equivalent versions of axiom(s) for negation. – Mauro ALLEGRANZA Sep 25 '17 at 11:41
  • @MauroALLEGRANZA So I guess it's a yes as a direct answer to the question, since you said when we have $\bot$ as a primitive which is the case in this proof, we have $\bot$ elim rule already defined(there's also no talk on proofwiki about deriving $\bot$ Elim rule (proofwiki.org/wiki/Rule_of_Explosion/Proof_Rule) which means it's already defined) and there's no problem using it. – Pooria Sep 26 '17 at 07:28
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    The page you have linked says: "by the tableau method of natural deduction". So, yes: the answer is YES. In Natural Deduction the Law of Explosion (or Ex Falso Quodlibet): $\bot \vdash \varphi$ is a primitive rule of the system. With it we can prove the equivalence between Excluded Middle and Double Negation. – Mauro ALLEGRANZA Sep 26 '17 at 15:48

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Apparently in the link you have provided they use classical propositional logic, this means that they are using a deductive system with the excluded middle and explosion rule.

In this context they are proving that in classical logic the double negation elimination is a derivable (hence provable) sequent.

Of course if you consider a system (i.e. logic) in which you remove the excluded middle (for instance in the intuitionistic logic) with basically the same proof you can derive the sequent $$p \lor \neg p, \neg \neg p \vdash p$$ from which it follows from implication-introduction rule that $$p \lor \neg p \vdash \neg \neg p \rightarrow p\ .$$

This would be a proof that in intuitionistic logic the law of middle excluded implies the double elimination principle.

Of course in intuitionistic logic the principle of explosion still holds. If you change the logic removing also the explosion principle you could lose this proof and so the implication could not hold.

I hope this helps.

Giorgio Mossa
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