I would like to know what the definition of a short proof is.


In Lance Fortnow’s article “[The Status of the P Versus NP Problem](http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext)”, Communications of the ACM, Vol. 52 No. 9, he says,



> 
> If a formula θ is not a tautology, we can give an easy proof of that fact by exhibiting an assignment of the variables that makes θ false. But if θ were indeed a tautology, we don’t expect short proofs. If one could prove there are no short proofs of tautology that would imply P ≠ NP.
> 
> 
> 


I have tried to find a definition of a “short proof”, but have not been able to.



What defines a "short proof"?
The statement you quoted is somewhat sloppy, since there is no precise notion of a short proof for a single formula. There is, however, a notion of short proofs for a class $C$ of formulas, when the class contains formulas of arbitrarily high length. One says that $C$ admits short proofs if there is a polynomial $p(x)$ such that, for every natural number $n$, all formulas in $C$ of length $n$ have proofs of length at most $p(n)$. 


