Revisiting analogical proportions and analogical inference

In this paper, we consider analogical proportions which are statements of the form "(a) over right arrow is to (b) over right arrow as (c) over right arrow is to (d) over right arrow", understood as a comparative formulation between vectors of items described on the same set of attributes. Analogical proportions and analogical inference have been extensively studied in the last decade, in particular by the authors of this paper. Some important remarks have been made regarding these proportions on i) the role of ordered pairs in them; ii) the large number of them associated with taxonomic trees; and more recently iii) their close relationship with multi-valued dependencies. We offer a renewed presentation of these facts together with some new insights on analogical proportions, emphasizing the role of equivalence classes of ordered pairs. Moreover, not all consequences had been drawn for a better understanding of analogical inference. This is the main purpose of this paper. In particular, it is advocated that analogical proportions whose four members are equal on some attributes are better predictors in general for classification purposes than analogical proportions for which there does not exist such attribute. This is confirmed by experimental results also reported in the paper. Thus this paper can be read both as an introductory survey on recent advances on analogical proportions and as a study on the impact of particular patterns on analogical inference, a topic never considered before.