A major problem that this analysis faces is the possibillity of narrow-scope readings in the case of adjuncts. For example, the following Dutch subordinate sentences are all systematically ambiguous between a wide-scope reading (adjunct modifies the event introduced by the auxiliary) or a narrow-scope reading (adjunct modifes the event introduced by the main verb).
Firstly note that the treatment of adjuncts as presented in [7], cannot be maintained as it simply fails to derive any of these sentences because the introduction of adjuncts is only possible as sisters of saturated elements. The fact that arguments and adjuncts can come interspersed (at least in languages such as Dutch and German) is not accounted for.
A straightforward solution to this problem is presented in [3]. Here adjuncts and arguments are all sisters to a head. The arguments should satisfy the subcat requirements of this head - the adjuncts modify the semantics of the head (via a recursively defined adjuncts principle).
The main problem for this treatment of adjuncts is that it cannot explain the narrow-scope readings observed above. If adjuncts modify the head of the phrase they are part of then we will only obtain the wide-scope readings.
If we assume, on the other hand, that adjuncts are on the subcat list, then we will obtain both readings straightforwardly. In the narrow-scope case the adjunct is on the subcat list of the embedded verb, and then inherited by the matrix verb. In the wide-scope case the adjunct simply is on the subcat list of the matrix verb. In the next section we present a treatment of adjuncts in which each adjunct is subcategorized for. By means of lexical rules we are able to obtain the effect that there can be any number of adjuncts. We also sketch how the semantics of modification might be defined.