Analogy between concepts

Abstract

Reasoning by analogy plays an important role in human thinking, in exploring parallels between situations. It enables us to explain by comparing, to draw plausible conclusions, or to create new devices or concepts by transposing old ones in new contexts. A basic form of analogy, called Analogical Proportion (AP), describes a particular relation between four objects of the same kind, e.g. “A calf is to a bull as a foal is to a stallion”. It is only recently that researchers have started to study APs in a formal way and to use their properties in different tasks of artificial intelligence (AI). This paper follows this line of research. Specifically, we are interested in giving the definition and some properties of an AP in lattices, a widely used structure in AI. We give general results before focusing on Concept Lattices, with the goal to investigate how analogical reasoning could be introduced in the framework of Formal Concept Analysis (FCA). This leads us to define an AP between formal concepts and to give algorithms to compute them, but also to point to special subcontexts, called analogical complexes. They are themselves organized as a lattice, and we show that they are closely related to APs between concepts, while not needing the complete construction of the lattice. To finish, we relate them to another form of analogy, called Relational Proportion, which involves two universes of discourse, e.g. “Carlsen is to chess as Mozart is to music”, which leads to the more compact way of saying “Carlsen is the Mozart of chess”, which is not anymore a relation between four objects of the same kind, but can be interpreted as well in FCAs framework.