How does the introduction rule and the elimination rule of falsehood ⊥ look like in a calculus of natural deduction?
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"Introduction rule" or "induction rule"? – David G. Stork Jul 10 '15 at 16:09
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lol. I mean "introduction rule". What would be an induction rule of ⊥? – asdfusername Jul 10 '15 at 16:13
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You can see also this post. – Mauro ALLEGRANZA Jul 10 '15 at 18:33
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You can avoid having any introduction rule for $\bot$ provided you define $\neg A$ to mean $A \to \bot$; then the '$\bot$-introduction rule' is really just $\to$-elimination, namely $$\cfrac{A \quad \neg A}{\bot}$$
As for $\bot$-elimination, it's quite simply $$\cfrac{\bot}{A}$$ That is, from a contradiction, you can deduce anything.
Clive Newstead
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The question is a little unclear, but it looks like you might want $\bot\rightarrow\phi$ for any (formula) $\phi$ and $\phi\wedge\neg\phi\rightarrow\bot$.
UserB1234
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