In this work, the third derivative of the energy with respect to the number of electrons, the so-called hyper-hardness, is investigated to assess whether this quantity has a chemical meaning. To achieve this goal a new working expression for hyper-hardness is developed and analyzed. It transpired from this analysis that hyper-hardness, just like hardness, can measure the reactivity or the stability of electron systems. Interestingly, positive values of hyper-hardness point to quite stable species such as noble gases and molecules. On the other hand, radicals almost always display large negative values of hyper-hardness.