Local Extent in Diagrams

Norman Foo

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.

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