[FOM] Question about Congruence

[hdeutsch@ilstu.edu]

At the urging of Vaughan Pratt, and in deference to mathematical  
custom, please replace "Let B be a set" by "Let B be a subset of A in  
(*) in my earlier posting. hd



Quoting hdeutsch at ilstu.edu:




  Let me clarify the point of my posting "A Question About Congruence."

  In Koslicki, JP, Feb. 2005, Gibbard, Parsons, and me myself, are
  discussed as three differing efforts to solve problems of coincidence
  by placing restrictions on Leibniz' Law.  Gibbard and Parsons are
  criticized for not sufficiently motivating their restrictions.  My own
  restriction is based on the simple technical fact I asked about in my
  earlier posting. The relevant paper is my "Identity and General
  Similarity" in Phil Perspectives, 12, 1998. Koslicki sees that my
  restriction is motivated and is systematic, but claims that it is too
  "course grained."  She does not realize that the condition on the
  right (see my earlier posting) is equivalent to the condition on the
  left--which is itself equivalent to a restricted form of LL.  It's
  just the condition for R to be a congruence w.r.t. B (or a property
  having B as its extension).  As I point out in my posting--and this
  point I hope is up for discussion--congruence relations should be
  playing more of a role in discussions of metaphysical puzzles such as
  identity through change.  But, e.g. discussions of 4Dism, such as
  Sider's, do not apparently see the point.

  I was asking two questions in my earlier posting.  First, is the
  simple fact about closer under equivalence well-known?  Secondly, does
  anyone have anything to say about the philosophical import of the
  principle?


>
> Quoting Vaughan Pratt <pratt at cs.stanford.edu>:
>
>> hdeutsch at ilstu.edu wrote:
>>>> I don't see anywhere in the proof where the assumption that B is a
>>>>  subset of A is needed.  I originally formulated the question as
>>>> you suggest.  But then "realized" that I didn't need that
>>>> assumption.  Am I wrong? h.
>>
>> It depends on your type discipline.  If you regard the entire
>> mathematical universe as one huge class of individuals, and view R as
>> merely a subset of A^2, then you're right.  But many mathematicians
>> don't take set theoretic foundations quite so literally, and when they
>> see a set B unrelated to A and likely to be disjoint from it (if
>> disjointness is even meaningful to them when there is no relationship
>> at all), instead of regarding the restriction of R (a relation on A) to
>> B as the empty relation they view R as simply undefined on B, and get
>> quite confused until the other interpretation is explained to them.
>>
>> Disallowing nonmembers of A in B avoids the question of what R means on
>> nonmembers of A, given that it was originally only defined on A.
>>
>> Vaughan Pratt
>
>
>
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