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Re: [ontolog-forum] Visual Complexity

To: "Cassidy, Patrick J." <pcassidy@xxxxxxxxx>
Cc: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Thu, 8 Feb 2007 12:09:30 -0600
Message-id: <p0623091dc1f1179b9326@[]>
>    I thought that in Mereology the singleton mereological grouping
>(Bill) was identical to Bill.    (01)

Yes.    (02)

>    If so, then we cannot have Bill and the mereological (Bill) as
>distinct entities.    (03)

Correct.    (04)

>  That is
>How mereology differs from sets, no?    (05)

Indeed.    (06)

>  Have I misunderstood?    (07)

No.    (08)

Pat(-2)    (09)

>-----Original Message-----
>From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
>[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of
>Christopher Menzel
>Sent: Thursday, February 08, 2007 12:11 AM
>To: [ontolog-forum]
>Subject: Re: [ontolog-forum] Visual Complexity
>>  That is the distinction between set theory and mereology:
>>>  If sets are there in the world, then no - there's nothing
>>>  abstract going on.  I'm real.  Chuck is real.  And the set
>>>  of me and Chuck is real.
>>  In mereology, Bill and Chuck are each parts of the collection
>>  that consists of Bill and Chuck.  You can call that pair C,
>>  but C is not a new entity.  It is just Bill and Chuck.
>John, that is not correct.  The mereological sum of Bill and Chuck -- 
>call it Bill+Chuck -- is typically defined in mereology as the 
>smallest thing that has Bill and Chuck as parts.  (Equivalently, it 
>is the unique thing X such that anything that overlaps X either 
>overlaps Bill or overlaps Chuck.)  It is not "just Bill and Chuck".  
>It is a third thing distinct from the two of them.
>>  But in set theory, if Bill and Chuck are members of the set S,
>>  then S is a third entity that is different from each of them.
>>  You have three things:  Bill, Chuck, and {Bill,Chuck}.
>And in mereology you have Bill, Chuck, and Bill+Chuck.
>>  You
>>  can also construct a fourth thing, which is the set whose only
>>  member is the set whose members are Bill and Chuck: {{Bill,Chuck}}.
>Now THERE is where mereology and set theory part company.  Mereology 
>does not support an unbridled hierarchy of sums parallel to the 
>hierarchy of sets in set theory.  As you note:
>>  You can form an infinite number of distinct sets whose starting
>>  elements are just Bill and Chuck:
>>      Bill, Chuck, {Bill}, {Chuck}, {Bill,Chuck}, {{Bill},Chuck},
>>      {Bill,{Chuck}}, {{Bill},{Chuck}}, {{Bill}}, {{Chuck}}....
>>  You don't even need to start with anything.  You can build up
>>  all of mathematics out of just the empty set:  {}, {{}}, {{},{}},
>>  {{{}}}, {{},{},{}}, {{{}},{{}},{{}}}...
>By contrast, in mereology, the sum of Bill and Bill+Chuck is just Bill
>+Chuck; likewise, the sum of Bill's left arm and Bill is just Bill. 
>In set theory, as you note, you get the distinct entities {Bill, 
>{Bill, Chuck}} and {BillsLeftArm, Bill}.  But in mereology and set 
>theory alike, the sum/set of Bill and Chuck is a third thing distinct 
>from Bill and Chuck.
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>    (010)

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