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[ontolog-forum] Relations for classes and predicates [Was: Inconsistent

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "doug foxvog" <doug@xxxxxxxxxx>
Date: Fri, 12 Feb 2010 00:54:14 -0000 (GMT)
Message-id: <4389.>
[I'm changing the subject line because in responding, i changed the main
topic of discussion]    (01)

Rob Freeman wrote:
> ...
> Maybe we need to back up a bit. We're always using the word "meaning".
> How much consensus do we really have? What is our fundamental model of
> "meaning"?    (02)

> To declare myself, I do think sets, loosely defined, are a good model
> for most of our intuitions about "meaning".    (03)

> Pat C says: "I can't imagine why you would want to identify concepts with
> sets".    (04)

> What about everyone else?    (05)

I consider the meaning of terms in an ontology to be a mapping to things
outside the ontology.  These mappings relate not just (or primarily) to
physical wavicles, but ever changing patterns and groups of them, categories
of such groups, and relations among such groups found useful for labeling by
human minds as well as other mentally considered categories and
relationships even though they might have no physical existance.    (06)

[Combinations of wavicles can form atomic nuclei (standard classes of which
have variations in isotope & excitement levels) which can combine with others
to form atoms (classes of which have variation in ionization level and
filling of
electron shells). These can be expanded to classes of molecules, crystaline
and polymolecular structures, larger structured and unstructured mixtures,
biological and geological structures, and physical artifacts.  Artifacts can
also be intangible, and such categories are useful if their existance and
properties are accepted as "real" by multiple people (agreements, nations,
conceptual works, games, monetary values, etc.).  Events in which such
things are involved, standard types of relations among such things, and
rules governing such things are all candidates for mapping to terms in an
ontology.  The definition of such classes, relations, and instances thereof
is conventional.  Defining exact boundaries for class or relation membership
can be almost impossible.]    (07)

Sets, being mathematical structures which are by definition fixed, do not
map very well with such categories or relations, whose instances change
with time and between contexts.  With a fixed time and context, a class
or relation has its membership fixed and so can be treated as a set.  But
i don't see the set-ness as crucial.  Sets can aid description and possibly
computation in describing rule macro predicates.    (08)

However, given this relationship between Set and both Class and Relation
Instance, we can define operators similar to the set operators for both
Class and Relations.    (09)

With Classes, key relations are:    (010)

membership: isClassMemberOf( Member1 Class1)    (011)

subclass:    isSubclassOf(Class1 Class2)
        isClassMemberOf(M C1) & isSubclassOf (C1 C2) =>
        isClassMemberOf (M C2)    (012)

disjointness: isDisjointWith(Class1 Class2)
        ~ ThereExists(x): (
                isClassMemberOf (x C1) &
                isClassMemberOf (x C2) &
                isDisjointWith (C1 C2))    (013)

mutual disjointness:    isClassMemberOf(S MutuallyDisjointSetOfClasses)
        (S == {C1, C2, ... Cn}) &
        isElementOf (Ci S) & ClassInstance (Ci) &
        isElementOf (Cj S) & ClassInstance (Cj) =>
        isDisjointWith(Ci Cj) OR
        (Ci == Cj)    (014)

class union: isClassUnionOf(C1 C2 C3)
        isClassMember(x C1) <=>
        (isClassMember(x C2) OR
         isClassMember(x C3))    (015)

multiple class union: isClassUnionOfSet(C SC)
        isClassUnionOfSet(C0 SC) =>
         ForAll (x): isClassMemberOf(x C0)
          ThereExists (Ci): isElementOf(Ci SC)
               (isClassMemberOf(x Ci))))    (016)

        isClassUnionOfSet(C0 SC) =>
         ForAll (Ci): isElementOf(Ci SC)
           ClassInstance(Ci) &
           ForAll (x): isClassMemberOf(x Ci)
               (isClassMemberOf(x Ci))) =>
        isClassMemberOf(x C0))    (017)

class partition: classPartitionedIntoSet(C SC)
        classPartitionedIntoSet( C0 SC) =>
        isClassMemberOf(S MutuallyDisjointSetOfClasses) &
        isClassUnionOfSet(C0 SC)    (018)

For binary relations, key relations are:
        subrelation (R1 R2) <=>
        (ForAll A: ForAll B: R1(A B) => R2(A B))    (019)

        inverseRelation (R1 R2) <=>
        (ForAll A: ForAll B: R1(A B) <=> R2(B A))    (020)

        inverseSubrelation (R1 R2) <=>
        (ForAll A: ForAll B: R1(A B) => R2(B A))    (021)

[inverseRelation (ForAll A: ForAll B: R1(A B) <=> R2(B A)) can be
 derived from inverseSubrelation
        inverseRelation (R1 R2) <=>
        (inverseSubrelation (R1 R2) & inverseSubrelation (R2 R1))
]    (022)

        negationRelation (R1 R2) <=>
        ~(ThereExists A: ThereExists B: R1(A B) & R2(B A))    (023)

        transitiveClosure (R1 R2) <=>
        TransitiveRelation(R2) &
        subrelation (R1 R2)
(see TransitiveRelation below)    (024)

Various relation types should be defined, each of which has implicit rules
Reflexive Relation
        ReflexiveRelation(Rr) <=>
        ((domain(Rr C1) <=> range(Rr C1)) &
         (ForAll x: isClassMemberOf(x C1)
          ( Rr(x x) )))    (025)

Symmetric Relation
        SymmetricRelation(Rs) <=>
        ForAll x: ForAll y:
        (Rs(x y) <=> Rs(y x))    (026)

Transitive Relation
        TransitiveRelation(Rt) <=>
        ForAll x: ForAll y: ForAll z:
        (Rt(x y) & Rt(y z) <=> Rt(x z))    (027)

Similarly, we should define irreflexive, asymmetric, and antisymmetric
relations.    (028)

-- doug    (029)

> We are always arguing about what things mean. But we are never going
> to agree how to resolves names to "meaning", if we don't agree what
> "meaning" is. What consensus do we have on a model for "meaning"
> itself, that we can argue constructively about a way of resolving
> names/labels to it?
> -Rob    (030)

doug foxvog    doug@xxxxxxxxxx    (031)

"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
    - Dr. Martin Luther King Jr.
=============================================================    (032)

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