  Category theory gives a mathematical characterization of naturality but not
of canonicity. The purpose of this paper is to develop the logical theory of
canonical maps based on the broader demonstration that the dual notions of
elements & distinctions are the basic analytical concepts needed to unpack and
analyze morphisms, duality, canonicity, and universal constructions in Set, the
category of sets and functions. The analysis extends directly to other
Set-based concrete categories (groups, rings, vector spaces, etc.). Elements
and distinctions are the building blocks of the two dual logics, the Boolean
logic of subsets and the logic of partitions. The partial orders (inclusion and
refinement) in the lattices for the dual logics define the canonical morphisms
(where `canonical' is always relative to the given data, not an absolute
property of a morphism). The thesis is that the maps that are canonical in Set
are the ones that are defined (given the data of the situation) by these two
logical partial orders and by the compositions of those maps.
