x1. In this paper we are concerned with some special theories of non-Fregean logic which determine correspondingly spe- cial elementary, i.e. axiomatic, strengthenings of that logic. The construc- tions presented here extend what has been done in [1] and [2] within the SCI-language. They also apply to the non-Fregean logic in comprehensive languages of kind W involving quantiers binding sentential variables and nominal variables, as well, . However, for the sake of sim- plicity, the underlying language considered here is the (...) SCI-language with quantiers 8 and 9. The reader may inspect the case when the numerical quantiers also occur in the language. (shrink)