Filter pairs and natural extensions of logics

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