In my course notes it is stated that we can infer $\exists x\neg F$ from $\neg\forall xF$ (where $F$ is a formula) in classical logic, but not in intuitionistic logic. However, I came up with this natural deduction $$ \cfrac{\cfrac{\cfrac{\quad}{F}\scriptstyle1}{\forall xF} \quad \begin{array}{c}\\\neg\forall xF\end{array}}{ \cfrac{\cfrac{\bot}{\neg F} \scriptstyle1}{\exists x \neg F} } $$ and it seems fine and intuitionistic to me... What am I missing?
Possible mistake in natural deduction
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logic
proof-verification
natural-deduction
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1You cannot apply $\forall$-intro that way; if $Px \vdash \forall x Px$, then by $\to$-intro we get : $\vdash Px \to \forall x Px$, but this is not valid (cosider e.g. $(x=0) \to \forall x (x=0)$). – 2017-02-06
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0The restriction to $\forall$-intro is : the quantified variable $x$ must not be *free* in any assumption: in this case $x$ is free in the assumption $F$. – 2017-02-06
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0Oh, I see. That's what I was missing! Thank you :) – 2017-02-06