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Re: [ontolog-forum] Foundation ontology, CYC, and Mapping

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Thu, 04 Feb 2010 08:17:24 -0500
Message-id: <4B6AC8E4.2030807@xxxxxxxxxxx>
Dear Matthew and Duane,    (01)

MW> It is the most basic thing about sets that they are defined
 > by their membership, which does not change.    (02)

Yes.  If you write something like the following,    (03)

    S1 = S union {x}    (04)

mathematicians don't say that S has changed, but that S1 is
a different set.    (05)

In programming languages, it is common to write    (06)

    x = x + 1;    (07)

but when that operation is carefully analyzed and described,
it's described as "the value stored at the location designated
by x has been replaced or updated with a new value."    (08)

There are also some programming languages, such as ML or Haskell,
called *functional* or *single-assignment* languages in which
no variable can be reassigned a different value -- i.e., no
statement of the form x=x+1 is permitted.    (09)

MW> So if you have something that has members, but the membership
 > can change, then what you know for certain is that it is not a set.
 > Some people use the word type for such things. A type will have,
 > at a point in time, a set which is its membership.    (010)

The usual distinction:    (011)

   The identity conditions for a set are *extensional*:  if S1 and S2
   have the same members, they are identical; otherwise they are two
   different sets.    (012)

   The identity conditions for a type are *intensional*; two types
   T1 and T2 are identical if their definitions are equivalent;
   their set of instances in a particular world w is called their
   *denotation* in w.  Those worlds may be, for example, the physical
   world or a particular computer system at different times.    (013)

John    (014)

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