Why is intuitionistic negation nonconstructive? Can someone simply describe why intuitionistic negation is

not constructive and why intuitionistic proof is constructive?


  
    in intuitionistic logic the notion of falsity has a 'subordinate' status,
    
    i.e. intuitionistic logic essentially rests upon a certain disparity between verification
    
    and falsification in favor of verification. Unlike intuitionistic truth, intuitionistic
    
    falsity is a non-constructive notion representing simply a non-truth of a sentence"
  


That is a part of paper entitle "Dual Intuitionistic Logic and a Variety of Negations:

The Logic of Scientific Research" by Yaroslav Shramko.
Intuitionistic negation is perfectly constructive, because $\lnot A$ is simply an abbreviation for $A \to \bot$ (i.e., "A implies false"). You may be thinking of the principle of double-negation elimination (i.e, $\lnot\lnot A \to A$). This principle is indeed nonconstructive.

The reason we call intuitionistic logic constructive is that existentials and disjunctions are "constructive". Concretely, this means that:


if you have a proof of $\exists x.\;A[x]$, then you can find a specific value $v$ such that $A[v]$ is provable. 
if you have a proof of $A \vee B$, then you can either find a proof of $A$, or a proof of $B$


The axiom of the excluded middle is not constructive, because it asserts $A \vee \lnot A$ for any $A$, without giving a procedure for deciding which of $A$ or $\lnot A$ is provable. 

Double-negation elimination lets you derive the excluded middle. This is because $\lnot\lnot(P \vee \lnot P)$ is derivable in intuitionistic logic. Here's a proof, given as a lambda-term:

f : ¬¬(P ∨ ¬P)
f : (((P ∨ (P → ⊥)) → ⊥) → ⊥ ) 
f = λk : (P ∨ (P → ⊥)) → ⊥. 
     let g : P → ⊥ = λp:P. k (inl p) in
     let h : (P → ⊥) → ⊥ = λn : P → ⊥. k (inr n) in 
     h g 


We can combine this with double-negation elimination to conclude that $P ∨ ¬P$ holds generally. Hence it follows that double-negation elimination is not a constructive principle in general.