the use of logical diagrams has shown that Warntz was ahead of his time in thinking that the spatial and geometrical component of logical diagrams would be useful analogs for spatial maps. 
As stated above, Warntz' paper calls on the work of Charles Sanders Peirce (above) and his existential logic diagrams, which he sees as mappings from non-spatial sets to geographical maps. Looking at the complexity of Peirce's systems, there is both an alpha and beta form depending on the required complexity, one wonders how deeply Warntz explored the subject of existential graphs. An important aspect of these graphs that Warntz thought useful for regional geographic analysis was the fact that a logic diagram can be drawn as a two-dimensional figure with spatial relations that are isomorphic with the structure of some logical statement. This is very important if one is going to try to apply set theory of the type Warntz is envisioning here, simply because these spatial relations are usually of a topographic nature.Logic diagrams, especially the type developed by Peirce (simple examples shown above with a page frm Peirce's notebook below), stand in the same relation to the various logical algebras as maps of areas stand in relation to their particular algebraic functions; they are simply other ways of symbolizing the same basic structure. 
In much of what Warntz has to say here we are reminded of the long way we have come when talking about set theory, topology and the formal properties of spatial structures and their relationship to cartography. One only has to look at books like Varzi and Casati's, Parts and Places: the Structures of Spatial Representations (MIT, 1999)  to get a feel for how our language and conceptual grasp of these topics has improved since Warntz and others involved in the early development of computer cartography were experimenting with what at the time were radically new ideas.
The current project envisioned here, which grew out of my research for the 20th century volume of the History of Cartography, will provide a mathematical and philosophical analysis of both of the Harvard papers mentioned above, along with others from this formative period that apply set theory and logical analysis, in an effort to show not only how constructivist methods migrated from mathematics to geography, but also to show how these new levels of abstraction changed the foundational ontology of geographic and cartographic objects. Using the philosophical debates that took place over things like existence theorems in the mathematical literature as a basis, this study will show that a foundational shift in the ontology of geographical objects opened the door to new conceptualizations of geographic space and formed the theoretical basis for the development of spatial logics and the current use of topological and abstract algebraic methods in geographical analysis.
 It is interesting to note that many early mathematical geographers had an interest in Wittgenstein. Waldo Tobler, in a private communication, told me recently that he was persuaded by Peter Gould (1932-2000) to take up the reading of Wittgenstein.
 The two papers are; Geography and an Existence Theorem: A Cartographic computer solution to the localization on a sphere of sets of equal-valued antipodal points for two-continuous distributions with practical applications to the real earth (1968) and The Sandwich Theorem: A basic one for geography (1971).
 A Note on Surfaces and Paths and Applications, William Warntz, Discussion Paper Number 6, Michigan Inter-University Community of Mathematical Geographers, 1965.
 The Philosophy of Maps, edited by John Nystuen, Michigan Inter-University Community of Mathematical Geographers Discussion Paper 12, 1968.
 For recent research on the logical status of Venn diagrams and the nature of spatial logic see, Eric Hammer (1995), Logic and Visual Information, Stanford CA: Center for the Study of Logic and Information; Nathaniel Miller (2007), Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, Studies in the The