Some propositional attitude verbs, such as certain, have local implicatures, sometimes in addition to global ones. The local/global distinction is relevant only in complex sentences; in simplex sentences the local and global implicatures are one and the same. According to some authors, local implicatures are derived strictly in the pragmatic component, that is to say, after the meaning of the entire sentence has been computed. Other authors argue that local implicatures require access to the meaning of the embedded clause before the computation of the meaning of the entire clause is completed, and therefore its derivation is done alongside the compositional computation of the meaning of the entire sentence. The primary goal of this paper is to argue for the latter approach. For ease of reference, the authors call the former approach to local implicatures 'non-grammatical' and the latter approach to local implicatures 'grammatical.' The paper also addresses some challenges for the grammatical approach, which predicts a correlation between NPI-licensing and cancellation of local implicatures. The most interesting of these challenges is perhaps the one posed by the fact that interrogative clauses produce local implicatures and, at the same time, license NPIs.