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SUO: Re: Intension Versus Extension Again




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AP:  Here's a practical question on the merged ontology.
     We currently have the axiom

     (<=>
       (disjoint ?C1 ?C2)
       (not
         (thereExists (?I)
           (and
             (instance-of ?I ?C1)
             (instance-of ?I ?C2)))))
     
     two classes are disjoint if no individual exists
     which is a instance of both classes and vice versa

     (see

     http://ontology.teknowledge.com:8080/rsigma/SKB.jsp?req=SC&name=disjoint&caseSensitive=on&skb=Merge>)

AP:  This looks problematic to me and Ian and I have discussed changing it from
     bi-implication to simple implication.  One could make a case for the above
     version if classes are extension.  The counter argument (which could be
     labelled an argument for intentional classes) would say that Equine and
     SinglyHornedAnimal are not disjoint just because no one has reified Unicorn
     in the knowledge base.

AP:  What do folks think?

JA:  Could not both C1 and C2 be empty?

IN:  That's just the case the raises the problem.
     If C1 and C2 have contradictory meanings associated with them and
     both of them have no extension (i.e. both of them are associated
     with the empty set), then there is no actual object which is an
     instance of C1 and is not an instance of C2.  Hence, the two
     classes would not be disjoint, according to the standard
     extensional criterion of disjointness.  However, since
     it is impossible for any instance to be a member of
     both classes, the classes are disjoint according
     to an intensional criterion.  The question, then,
     is whether we want to go with an extensional or
     an intensional criterion of disjointness.

Ian,

I think that we are not talking the same set of dialects yet.
I will explain the three most common ways that I would read
an expression like the one that you folks were asking about,
taking them in order of increasing frequency in my own usage
but also in order of increasing conceptual overhead in the
interpretive environment that is required to support them:

1.  The minimum overhead choice is the "plural reference" option,
    one that I would normally entertain only under the most severe
    constraints of nominal thinking.  In this case, the variable names
    "C1" and "C2" would be taken as signs in a "sign relation complex",
    each sign of which could have no objects, one object, or many objects
    as their denotations.  As a relational table, the set-up might look so:

    ---------------------------
    | Object  |  Sign   | Interp  |
    ---------------------------
    |   e11   |  "C1"   |   ...   |
    |   ...   |  "C1"   |   ...   |
    |   e1j   |  "C1"   |   ...   |
    ---------------------------
    |   e21   |  "C2"   |   ...   |
    |   ...   |  "C2"   |   ...   |
    |   e2k   |  "C2"   |   ...   |
    ---------------------------

    Whether one takes these signs as terms in a public language medium
    or as concepts, that is, mental symbols in a private frame of mind,
    one says that the objects e11, ..., e1j fall under the sign "C1",
    and says that the objects e21, ..., e2k fall under the sign "C2".

2.  The next choice is the "intensional" option.  This time, the objects
    being denoted are "intensions", in other words, "properties", say,
    with "C1" denoting property f1 and with "C2" denoting property f2. 

    ---------------------------
    | Object  |  Sign   | Interp  |
    ---------------------------
    |   f1    |  "C1"   |   ...   |
    ---------------------------
    |   f2    |  "C2"   |   ...   |
    ---------------------------

    Typically, this order of sign-relational development arises
    from the previous type of plural reference situation, where
    f1 is a property that entities e11, ..., e1j have in common,
    f2 is a property that entities e21, ..., e2k have in common,
    and so on.

3.  Last but not least, one has the "extensional" option.  Here,
    the objects being denoted are constituted as sets of elements,
    where "C1" denotes the set g1 and where "C2" denotes the set g2.

    ---------------------------
    | Object  |  Sign   | Interp  |
    ---------------------------
    |   g1    |  "C1"   |   ...   |
    ---------------------------
    |   g2    |  "C2"   |   ...   |
    ---------------------------

    This is an alternative species of sign-relational development
    that frequently arises from either of the previous situations:
    g1 is a set or class that contains the elements e11, ..., e1j,
    g2 is a set or class that contains the elements e21, ..., e2k,
    and so on.

This is how I understand the various pictures.
The question, I think, is not so much how to
choose among them as how to integrate their
sundry different advantages and utilities.

Right at the moment, though, I am mostly working on
sign relations as cast in the extensional framework,
and so let me say how I read your example from that
point of view.

The first thing that I notice is the variables "C1" and "C2",
which are read in extension to range over a set of sets, TBA,
but there is no condition that makes them refer to different
sets, and thus it is perfectly possible that C1 = C2.  If so,
then it is not the usual thing to say that a set is disjoint
from itself, even if that set is empty, so there seems to be
a difficulty with the formulation here, at this boundary case,
one that I imagine requires some sort of explicit stipulation.

Jon Awbrey

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