A Modal View on Abstract Learning and Reasoning

We present here a view on abstraction originating from the relation between formulas in a partially ordered language L and their extension on a set of instances W . In Formal Concept Analysis, this relation is materialized as a lattice G . Particular self-maps on either L or the powerset P ( W) are known to ensure structure-preserving reductions of the lattice G and have been shown to be in one to one correspondence with abstractions , defined subsets of either L or P ( W) closed under union. We investigate specifically extensional abstractions (subsets of P ( W) . Such an abstraction comes down to a change in granularity: extensions are now considered as union of abstract instances , that is, union of predefined subsets of instances. The main contribution of the paper is the investigation of the class of (non normal) monotonic modal logics whose semantics relies on such abstractions, and that we call abstract modal logics .