A Proximity Approach to Some Region-Based Theories of Space

Journal of Applied Non-Classical Logics 12 (3-4):527-559 (2002)
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This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.



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References found in this work

Process and reality: an essay in cosmology.Alfred North Whitehead - 1929 - New York: Free Press. Edited by David Ray Griffin & Donald W. Sherburne.
The calculus of individuals and its uses.Henry S. Leonard & Nelson Goodman - 1940 - Journal of Symbolic Logic 5 (2):45-55.
A calculus of individuals based on "connection".Bowman L. Clarke - 1981 - Notre Dame Journal of Formal Logic 22 (3):204-218.
Region-based topology.Peter Roeper - 1997 - Journal of Philosophical Logic 26 (3):251-309.
Point, line, and surface, as sets of solids.Theodore de Laguna - 1922 - Journal of Philosophy 19 (17):449-461.

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