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2.2 Identity in HI

Minimal
Identity
 		 P1-P6 found an identity relation which is weakly reflexive, symmetric, and transitive;
T1* for allx(for somey(x = y) if-then (x = x))10 
(Identity is weakly reflexive.)
T2* for allxy(x = y if-then y = x) 
(Identity is symmetric.)
T3* for allxyz((x = y  &  y = z) if-then x = z) 
(Identity is transitive.)
identicals that are indiscernible, and indiscernibles that are identical if self-identical.
T4* for allxyz(x = y if-then (x ex Z equivalent y ex Z))
(Identicals are indiscernible.)
T5* for allxy((x = x  &  y = y) if-then (for allz(x ex Z equivalent y ex Z) if-then x = y))
(Self-identical indiscernibles are identical.)
P1-P6 thus found a minimal identity relation.11

From P2, moreover, it follows that an individual is self-identical just in case there is some haecceity it exemplifies; and from P5, that a haecceity is self-identical just in case some individual exemplifies it.

T6* for allx(x = x equivalent for somey(x ex Y))
Hence identity is strongly reflexive for individuals and haecceities just in case every individual exemplifies and every haecceity is exemplified.
T7* for allx(x = x) equivalent for allxfor somey(x ex Y))
From T5*-T7* it thus follows that indiscernibles are identical just in case every individual exemplifies and every haecceity is exemplified.
T8* for allxy(for allz(x ex Z equivalenty ex Z) if-then x = y) equivalent for allxfor somey(x ex Y))
Hence indiscernibles are identical just in case identity is strongly reflexive.

Thus P1-P6 invite elaboration. Is every haecceity exemplified? Does every individual exemplify? In view of T9*, we have not two questions here but one.

T9* for allxfor somey(y ex X) if-then for allxfor somey(x ex Y)12

cont.

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