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Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems (view source)
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, 21:20, 21 January 2008→3.2. Differential Extensions of Propositional Calculi: markup
Using the isomorphism between function spaces:
Using the isomorphism between function spaces:
: ('''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B''') ≈ ('''B'''<sup>''n''</sup> → ('''D'''<sup>''n''</sup> → '''B''')),
: ('''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B''') <math>\cong</math> ('''B'''<sup>''n''</sup> → ('''D'''<sup>''n''</sup> → '''B''')),
each ''p'' : ''U''&prime' → '''B''' has a unique decomposition into a ''p''′ : '''B'''<sup>''n''</sup> → ('''D'''<sup>''n''</sup> → '''B''') and a set of ''p''″ : '''D'''<sup>''n''</sup> → '''B''' such that:
each ''p'' : ''U''′ → '''B''' has a unique decomposition into a ''p''′ : '''B'''<sup>''n''</sup> → ('''D'''<sup>''n''</sup> → '''B''') and a set of ''p''″ : '''D'''<sup>''n''</sup> → '''B''' such that:
: ''p'' : '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B''' ≈ ''p''′ : '''B'''<sup>''n''</sup> → ''p''″ : ('''D'''<sup>''n''</sup> → '''B''').
: ''p'' : '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B''' <u>≈</u> ''p''′ : '''B'''<sup>''n''</sup> → ''p''″ : ('''D'''<sup>''n''</sup> → '''B''').
For the sake of the visual intuition we may imagine that each cell ''x'' in the diagram of ''U'' has springing from it the diagram of the proposition ''p''′(''x'') = ''p''″ in d''U''.
For the sake of the visual intuition we may imagine that each cell ''x'' in the diagram of ''U'' has springing from it the diagram of the proposition ''p''′(''x'') = ''p''″ in d''U''.