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Re: [ontolog-forum] Binary versus N-ary relations

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "doug foxvog" <doug@xxxxxxxxxx>
Date: Mon, 10 Sep 2012 15:58:04 -0400
Message-id: <7c9dd9f931cf7d00b89a467e08e51e5e.squirrel@xxxxxxxxxxxxxxxxx>
On Fri, September 7, 2012 14:52, Chris Menzel wrote:
> On Fri, Sep 7, 2012 at 11:00 AM, doug foxvog <doug@xxxxxxxxxx> wrote:
>> On Thu, September 6, 2012 15:44, Andries van Renssen wrote:
>> > Doug,
>> > Your statement that occurrences cannot be higher arity relations,
>> > because
>> > they are first class objects implies that you see an opposition
>> > between 'first class' objects and occurrences.
>> > I don't know what 'first class objects' are, and I don't see why
>> > higher arity relations cannot be first class objects.    (01)

>> I guess we have different understanding of what is meant by "relation".
>> I was taking you to be using two definitions (as many in this forum do):
>> * A predicate or function, which when applied to a set of values returns
>>   a truth value.
>> * A logical statement made using such a predicate/truth function.
>> To make the distinction, i'll call the second meaning a relation
>> instance.    (02)

> I find that confusing. Instances of relations are usually understood to be
> things in the world, actual real world connections like an instance of
> marriage or paternity between two specific individuals, not linguistic
> entities.    (03)

Thanks for this new definition.  It really helps me understand why we
have been miscommunicating.  This meaning of "relation" means either
"a situation <i>of</i> some relationship holding among some things" or "a
situation <i>during which</i> some relationship hold among some things".    (04)

I don't know what restrictions might hold on the relationship or the things,
yet.  For example, is 12 being greater than 11 such a relation?    (05)

What really drove me away from the possibility that this was referred
to was the reference to an "arity" of such a "relation".  To quote from
  "In logic, mathematics, and computer science, the arity of a function
   or operation is the number of arguments or operands that the function
   takes. The arity of a relation is the dimension of the domain in the
   corresponding Cartesian product."
Since we are discussing a combination of logic and computer science,
i took "arity" as having such a definition.  Not only does this suggest
that the thing that has "arity" is a function (a superclass of predicate) or
operation, but that the thing that has a fixed set of arguments (ignoring
the meaning of "variable arity" for the moment.    (06)

The meaning of "relation" that i've been using also is defined in Wikipedia:
  "In set theory and logic, a relation is a property that assigns
   truth values to k-tuples".    (07)

I will call a situation <i>during which</i> some relationship holds between
non-datatype individuals a "relationship situation".    (08)

> Why not just stick with something like "relational statement" or
> "atomic statement" or some other piece of more or less conventional
> terminology?    (09)

I was just trying to figure out what is being referred to.  I guessed wrong.
If that is not what you were referring to, we don't have to include the
word "relation" in naming it.  "Logical statement using a predicate"
is fine with me.    (010)

>> I agree that an ontology can treat anything it desires as a "first class
>> object" -- something that it can make statements about by including
>> it as an argument to a predicate or function.    (011)

> Those two aren't the same, if by "first-class object" you mean (as I think
> it usually means) something that is in the range of the individual
> variables of the language.    (012)

What's the distinction here?  You refer to something in the range of
individual variables, and i refer to something that a system can make
statements about.  Certainly a system should be able to make statements
about anything it represents as a variable.    (013)

Is the distinction on the modifier "individual"?    Although a system with
restricted logic can not use a variable to refer to a class, a predicate,
or a statement, a less limited system can.  Such less limited systems
can thus treat thus treat classes as first class objects.    (014)

So, by saying "an ontology can treat anything it desires as a 'first class
object'", i am not claiming that this is possible in any ontology language.
It is not.  I was referring to properties of ontologies, not of every
that can represent restricted classes of ontologies.    (015)

> You can make statements about relations in
> higher-order logics, but the universe    (016)

By "universe", i take you to mean something about the "real world" as
opposed to a feature of an encoding.    (017)

> is still falls into into a hierarchy
> of individuals ("first-class objects"),    (018)

I don't accept this equivalence.  "Individual" is a property of the "real
world", while "first-class object" is a property of an encoding.  Many OWL
ontologies have classes as "first classes objects".  For example,
the Wine Ontology has wine varieties by vineyard as instances of classes
of wine varieties.  It does not consider the meta-classes as meta,
because it does not define the basic classes to be classes or to have
their own instances.    (019)

> properties of/relations on individuals,    (020)

The Wine Ontology (and others) includes a set of properties of and
relations on wine varieties by vineyard.  These have real-world classes
as arguments (although the ontology does not define them as such    (021)

> properties of/relations on lower typed entities, etc.    (022)

I'm not sure what your distinction between "individuals" and "lower
typed entities" is.    (023)

I would say that the universe (something outside the system of
encoding) includes various things that can be modeled:    (024)

* individuals -- things that can not have instances
  + aspatial atemporal individuals
     - predicates / properties
     - numbers
     - statements ...
  + spatial or temporal individuals
     - physical objects
     - situations (including events)
     - non-physical created things ...
       :organizations ...
* types of things
  + types of individual
     - types of <class of individual>
  + metatypes    (025)

>> My objection to equating occurrences to relation instances is that
>> i take the meaning of a relation instance to be fully implied by the
>> relation itself plus its arguments.  I take any statement about an
>> actual occurrence to, in the vast majority of cases, only partially
>> describe the occurrence.  Many additional statements may be made
>> about the same occurrence -- statements that are not implied by
>> the first statement.    (026)

> Very much unlike most of the things you say, I can't make any sense of
> this at all.    (027)

This is because of the miscommunication.  By "relation instance" i
was referring to a statement encoded using a predicate (relation).
You seem to consider me to have meant a situation in the real
world.  Yes, it makes no sense using that interpretation.    (028)

-- doug foxvog    (029)

> -chris    (030)

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