A Modal View on Abstract Learning and Reasoning

We present here a view on abstraction based on the relation between sentences in a partially ordered language L and truth values of these sentences on a set of instances W . In Formal Concept Analysis, this relation is materialized as a lattice denoted as G that relates L and the powerset P(W )). We show here that projections on a lattice (here either L or the powerset P(W )) that are known to ensure structure-preserving reductions of G, are equivalent to abstractions, defined here as sets of subsets closed under union (regarding P(W )) or under minimal specialization (regarding L) and order them in a lattice of abstractions. We then discuss specifically abstractions A of P(W ) and discuss the properties, of abstract implications. We then exhibit the class of (non normal) monotonic modal logics the semantic basis of which relies on such abstractions, and discuss how reasoning may be performed at variable abstraction levels.