On epistemological criteria of identity

1
On epistemological criteria of identity

• What is identity?
• Definitions of identity
• Criteria of identity
• ontological criteria of identity
• epistemological criteria of identity

2

• (Q1) What is an ontological category of identity?
• (Q2) What is it for two objects to be identical?
• (Q3) What is it for you to know that two objects
  are identical?

3

• (D1) Domain(Zn)x1 ?x2, , xn ltx1, ,
  xngt??Zn??xn ?x1, , xn-1 ltx1, , xngt??Zn.
• (D2) A relation R is a weak congruence in Zn iff
• (i) Domain(R)?Domain(Zn),
• (ii) ?S?Zn ?x1, , xn, y1, , yn Rx1y1??Rxnyn
  ? (ltx1, , xngt?S ? lt y1, , yngt?S).
• (D3) A relation R is a weak congruence on Zn iff
• (i) it is a weak congruence in Zn,
• (ii) Domain(Zn)?Domain(R).
• (D4) A relation R is a strong congruence on Zn
  iff
• (i) Domain(R)?Domain(Zn),
• (ii) ?x1, , xn, y1, , yn Rx1y1??Rxnyn ?
  ?S?Zn (ltx1, , xngt?S ? lt y1, , yngt?S).

4

• (D5) Id(X) WC(?(Xn)).
• (F1) WC(Zn) is a singleton.
• (D6) Id(Domain(Zn)) ?WC(Zn).
• (F2) ?WC(Zn)?WC(Zn).
• (F3) SC(Zn)?WC(Zn).
• (D7) Id(X) SC(?(Xn)).
• (F4) SC(Zn) is an equivalence relation on
  Domain(Zn).
• (D8) Id(X) ltx, ygt ?n?N ltx, ygt?SC(Zn).

5

• (D9) An ontological criterion of identity is a
  relation which
• (i) is necessary and sufficient for identity,
• (ii) is not circular with respect to identity.
• (D10) An epistemological criterion of identity
  for a person P is an equivalence relation R
  such that if P believes xy, this belief is
  justified for her by her believing Rxy.

6

• (C1) For every person, no belief justifies itself
  (for her).
• (F5) An epistemological criterion of identity for
  non-minimally rational person is necessary for
  identity.
• (F6) For every non-empty set AEQ(Z2) of
  equivalence relations there exists in AEQ(?(X2))
  its greatest lower bound ?AEQ(Z2).
• (C2) ?Si?S ?Rj?AEQ(R2) ?x, y (Rjxy ? Six ? Siy).
• (C3) ?SX.
• (C4) ?R?AEQ(R2) X?Domain(R).
• (C5) Id(X)?AEQ(R2).

7

• (D11) A relation R is at least as close to a
  relation S as a relation T is to a relation U
  iff
• ?x, y (Txy ? Uxy) ? (Rxy ? Sxy).
• (D12) A relation R is closer to a relation S than
  a relation T iff
• (i) R is at least as close to S as T is to S,
• (ii) T is not at least as close to S as R is to
  S.

8

• (F7) ?AEQ(R2) is closer to Id(X) than any other
  relation in AEQ(R2).
• (F8) The criterion is neither impredicative nor
  circular (for P).
• (F9) ?AEQ(R2)?WC(S).
• (F10) ?AEQ(R2)?WC(AEQ(R2)).
• (C6) ?Rj?AEQ(R2) ?Si?S ?x, y (Six?Siy) ? Rjxy,
• (F11) ?AEQ(R2)SC(S).