 The difference to the original truth definition of second-order logic is thus that the interpretations of the second-order variables cannot be quite arbitrary, they have to be found in \(\G\). The smaller the \(\G\) is, the weaker the logic. On the other hand, if \(\G^\star\) is the set of all relations and functions on M, then the truth definition for the general model \((\mm,\G^\star)\) is the same as the original truth definition. We call such general models full and indeed identify \((\mm,\G^\star)\) with \(\mm\). The other extreme is that \(\G=\emptyset\). Then we are back in first order logic. To distinguish second-order logic with general models from the original second-order logic, we call the latter full.