Suggested answer for Exercise 4
The Case Filter states that all DPs must have case. Sentence (1a) is ungrammatical because the DP John, which is the thematic subject of the embedded sentence, does not have case. As the inflection of the embedded sentence is non-finite (infinitival), it cannot assign case to the DP John. In sentence (1b) the embedded sentence is finite, the finite inflection can assign nominative case, hence the subject DP ‘John’ gets nominative case from the finite inflection and the sentence is grammatical.
Sentence (2b) is ungrammatical as the non-finite I cannot assign nominative case to the subject DP of the subordinate sentence. The sentence can be improved when the subject has accusative case, as the ECM verb believe can assign accusative case to the DP in the specifier position of its complement.
In (3a) the DP John does not have case, as the non-finite I of the subject subordinate clause cannot assign case. In (3b) the preposition for saves the sentence as it can assign accusative case to the specifier of its complement.