  We adjust the notion of finitary filter pair, which was coined for creating
and analyzing finitary logics, in such a way that we can treat logics of
cardinality $\kappa$, where $\kappa$ is a regular cardinal. The corresponding
new notion is called $\kappa$-filter pair. A filter pair can be seen as a
presentation of a logic, and we ask what different $\kappa$-filter pairs give
rise to a fixed logic of cardinality $\kappa$. To make the question
well-defined we restrict to a subcollection of filter pairs and establish a
bijection from that collection to the set of natural extensions of that logic
by a set of variables of cardinality $\kappa$.
  Along the way we use $\kappa$-filter pairs to construct natural extensions
for a given logic, work out the relationships between this construction and
several others proposed in the literature, and show that the collection of
natural extensions forms a complete lattice.
  In an optional section we introduce and motivate the concept of a general
filter pair.
