# Natural deduction: understanding bottom elimination (¬e)

I am new to natural deduction and upon reading about various methods online, I came across the rule of bottom-elimination in the following example.

I do not understand the step in line 10.

Upon inspection, my initial thought would be that the assumption of ¬p and p both being true is absurd, hence anything can be inferred ( in this case 'p'). However, if this were the case where would you stop (this seems to be an overly powerful tool)? So I assume that this idea is wrong.

Could someone help me understand the rule?

NOTE: I came to StackExchange due to the lack of resources and specific information online.

• This strikes me as more appropriate for Mathematics. Perhaps it should be migrated there? May 10, 2020 at 6:34

Usually in practice we weld the two steps together and just say that from $$p$$ and $$\lnot p$$ anything follows, but in formal logic this is a combination of two rules of inference:
1. $$p$$ and $$\lnot p$$ both together entail falsehood $$\bot$$,
2. from $$\bot$$ anything follows.
We often take $$\lnot p$$ to be an abbreviation for $$p \Rightarrow \bot$$, in which case the rule "from $$p$$ and $$\lnot p$$ follows $$\bot$$" is just a special case of modus ponens "from $$p$$ and $$p \Rightarrow r$$ follows $$r$$".
• Anything can be made true if you already proved or assumed $\bot$. And you cannot "repeatedly use $p$ and $\lnot p$" unless you already know that they are both true. May 9, 2020 at 16:49
• In fact that is the whole point of rule 2: if you know something that is false (namely $\bot$) then you can conclude anything you like. The rule does not say "everything is true"! May 9, 2020 at 16:51