Most diagrams, even if they can be interpreted as embeddings in infinite
objects, are nevertheless finite in extent if they are to be presented
conventionally. In many applications where diagrams are used to explain,
instruct, communicate, cogitate or conjecture, this finiteness implies
local extent. By this we mean that even if they are used to reason
about potentially unbounded domains or constructs, the fact that they can
be used at all suggests that only local properties are being examined.
We are interested in how the features of such diagrams can be described
in logic, and how diagram manipulations that represent actions can
be justified. In particular, we establish a correspondence between a
substructure construction which formalizes local extent and a class
of sentences preserved under extension from, and reduction to, this
substructure. The sentence class is called EnE because it has the form
of successively nested existential and negated existential subsentences,
and appear to cover most of the applications so far encountered. The hope
is that this understanding can be used to mark out the local regions of
diagrams that can be safely isolated for manipulations.