if F , G are two formulas , h[f] is the height of the formula f ,then h[ G a F ] is less or equal to sup( h[F] , h[G] ) + 1 if F , G are two propostional formulas , h[f] is the height of the formula f ,
then h[ G a F ] is less or equal to sup( h[F] , h[G] ) + 1 , a is one of the connectives , my question is , what is sup ??? and how to compute sup ?!! 
 A: The term $\sup$ stands for supremum. For finite sets, it coincides with $\max$, the maximum. So for example $\sup(3,7)=7$, and $\sup(4,4)=4$.  It is surprising that $\sup$ was used instead of the more common $\max$.  
A: $\sup$ is most likely the supremum:

In mathematics, the supremum (sup) of a subset S of a totally or
  partially ordered set T is the least element of T that is greater than
  or equal to all elements of S. Consequently, the supremum is also
  referred to as the least upper bound (lub or LUB). If the supremum
  exists, it is unique meaning that there will be only one supremum. If
  S contains a greatest element, then that element is the supremum;
  otherwise, the supremum does not belong to S (or does not exist). For
  instance, the negative real numbers do not have a greatest element,
  and their supremum is 0 (which is not a negative real number).

As André Nicolas already mentioned, there is no difference between the maximum and the supremum given that the formulas whose height you need to compute are of finite length. Infinitary logic, however, admits formulas of infinite height.
