Point-free foundation of geometry and multivalued logic (Q989422)

\textit{A. N. Whitehead} introduced in [The concept of nature. Cambridge: Cambridge University Press (1920; JFM 47.0049.03)] and in [An enquiry concerning the principles of natural knowledge. Cambridge: Cambridge University Press (1919; JFM 47.0049.02)] a variant of mereology (the part-whole theory) based on \textit{events} and the \textit{extension} relation. Moving closer to a point-free geometry, he later replaced in [Process and reality. Cambridge: Cambridge University Press (1929; JFM 55.0035.03)] the notion of extension by that of \textit{contact between two regions}. In this paper, the authors reformulate Whitehead's philosophically worded theory into one in first-order logic, discarding superfluous axioms, and show that the move from an extension-based to a connection-based theory was necessary, given that ``while it is possible to define the inclusion from the connection relation the converse fails.'' They also show that by reinterpreting the inclusion-based theory as a theory in multi-valued (in fact, \([0,1]\)-valued) logic, it is possible to define the contact relation in terms of the extension relation in the resulting fuzzy structures called ``graded inclusion spaces of regions''.