[FOM] Question about Congruence

[hdeutsch@ilstu.edu]

Is the following fact well known?  It is of some philosophical  
interest, as explained below.

(*) Let R be an equivalence relation defined on a set A and let B be a  
set. Then B is closed under R iff each element E of the partition P of  
A induced by R is such that if some element of E is in B, then they  
all are.

I'm sure there is a significant generalization of this.  What is it?

One area of philosophical interest where this might be helpful  
concerns the puzzles of coincidence, e.g. the problem of identity  
through change. A popular solution is the 4D one, according to which  
objects have temporal as well spatial parts.  But 4D theorists (e.g.  
T. Sider, Four Dimensionalism, Oxford, 2001) have been unable to give  
an uncontroversial or decisive answer to the question: "What  
conditions must two temporal parts of things meet in order to be  
temporal parts of the same object.  The fact above gives at least a  
logical condition they must meet--not that it answers the more  
pressing question, assuming that the relation of being two temporal  
parts of the same temporally extended object is an equivalence  
relation. But in general, the fact mentioned above gives a general  
answer to what has become known as "Wiggins' Challenge"--the challenge  
to give some sort of systematic restriction on Leibniz' Law, if one's  
theory demands it--as in the cases of Gibbard's theory of contingent  
identity or Parsons' theory of indeterminate identity. (See K.  
Koslicki, "Almost Indiscernible Objects and the Suspect Strategy," JP,  
Feb., 2005 for discussion and references.) Parsons' indeterminate  
identity is not transitive, but there may something in the vicinity of  
(*) that would be helpful.  Thanks, hd



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