Analogical Proportions and the Factorization of Information in Distributive Lattices

Analogical proportions are statements involving four entities, of the form ‘A is to B as C is to D’. They play an important role in analogical reasoning. Their formalization has received much attention from different researchers in the last decade, in particular in a propositional logic setting. Analogical proportions have also been algebraically defined in terms of factorization, as a generalization of geometric numerical proportions (that equate ratios). In this paper, we define and study analogical proportions in the general setting of lattices, and more particularly of distributive lattices. The decomposition of analogical proportions in canonical proportions is discussed in details, as well as the resolution of analogical proportion equations, which plays a crucial role in reasoning. The case of Boolean lattices, which reflects the logical modeling, and the case corresponding to entities described in terms of gradual properties, are especially considered for illustration purposes.