Abstract We define LE- $$\mathcal {ALC}$$ , a generalization of the description logic $$\mathcal {ALC}$$ based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE- $$\mathcal {ALC}$$ allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE- $$\mathcal {ALC}$$ , provide a tableaux algorithm for checking the consistency of LE- $$\mathcal {ALC}$$ knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE- $$\mathcal {ALC}$$ with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical $$\mathcal {ALC}$$ with acyclic TBoxes is PSPACE-complete.

Abstract In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$ . The structure in our axiomatization is a triple $(O,L,s)$ , where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to O$ . In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.