Analogy plays a very important role in human reasoning. In this paper, we study a restricted form of it based on analogical proportions, i.e. statements of the form
a is to b as c is to d
. We first investigate the constitutive notions of analogy, and beside the analogical proportion highlights the existence of two noticeable companion relations: one that is just reversing the change from
w. r. t. the one from
, while the last one called
proportion expresses that
what a and b have in common, c and d have it also
. Characteristic postulates are identified for the three types of relations allowing to provide set and Boolean logic interpretations in a natural way. Finally, the solving of proportion equations as a basis for inference is discussed, again emphasizing the differences between analogy, reverse analogy, and paralogy, in particular in a three-valued setting, which is also briefly presented.