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3.311 Reflexivity and Complexes

Adding P10 to the postulates of HI sufficed to make identity strongly reflexive for individuals and haecceities. However, adding P10 to the postulates of C does not suffice to make identity strongly reflexive for complexes. For, from P7 it follows that a complex is self-identical just in case the individual and haecceity which constitute it are such that the individual exemplifies the haecceity*:
T18 for allxy(x.Y = x.Yequivalentx ex Y)
Hence every complex is self-identical just in case every individual exemplifies every haecceity:
T19 for allxy(x.Y = x.Y)equivalentfor allxy(x ex Y)
Moreover, it follows--from P2, P6 and T4*--that every individual exemplifies every haecceity just in case some haecceity is such that every individual exemplifies it*:
T20 for allxy(x ex Y)equivalentfor someyfor allx(x ex Y)
Therefore, for identity to be strongly reflexive for complexes, not only must every haecceity be exemplified, but some haecceity must be universally exemplified:
T21 for allxy(x.Y = x.Y)equivalentfor someyfor allx(x ex Y)
Identity in C is thus not strongly reflexive for complexes unless no two complexes are distinct.

Reflexivity and C-Complexes
 

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