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By limiting the range of the predicate variables in a second-order languageone may obtain restricted versions of second-order logic such as weaksecond-order logic or definable subset logic. In this note we provide aninfinitary strongly complete axiomatization for several systems of this kindhaving the range of the predicate variables as a parameter. The completenessargument uses simple techniques from the theory of Boolean algebras.

A parametrised axiomatization for a large number of restricted second-order logics