[4] placed an approximation space (U,≡ ) in a type-lowering retraction with its power set 2 U such that the ≡ -exact subsets of U comprise the kernel of the retraction, where ≡ is the equivalence relation of set-theoretic indiscernibility within the resulting universe of exact sets. Since a concept thus forms a set just in case it is ≡ -exact, set-theoretic comprehension in (U,≡ ) is governed by the method of upper and lower approximations of Rough Set Theory. Some central features of this universe were informally axiomatized in [3] in terms of the notion of a Proximal Frege Structure and its associated modal Boolean algebra of exact sets. The present essay generalizes the axiomatic notion of a PFS to tolerance (reflexive, symmetric) relations, where the universe of exact sets forms a modal ortho-lattice. An example of this general notion is provided by the tolerance relation of “matching” over U.


Modal Logic Approximation Space Tolerance Relation Complete Boolean Algebra Proximity Space 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Peter John Apostoli
    • 1
  • Akira Kanda
    • 1
  1. 1.Department of PhilosophyThe University of PretoriaPretoriaSouth Africa

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