Azamat
I remember reading with great interest that part of your book
which really opened up my mind to understanding relationships
what was it? chapter X?
I hope it resonates sooner rather than later
PDM
On Thu, Feb 12, 2009 at 8:54 PM, Azamat <abdoul@xxxxxxxxxxxxxx> wrote:
On Thursday, February 12, 2009 3:56 AM, Ravi
wrote:
But for practical
applications we need to know how to test whether a given relation (say a few
triples described by same relation type) are conforming to Cartesian product of
a set?
As a product of three sets, X, Y, and Z, a general
Cartesian product is reduced to an ordered triple of class variables: X x Y x Z
. The _expression_ on a Cartesian grid is defined as the construct
consisting of all ordered triples <x, y, z>, where the members are the
elements, individual variables, of the respective sets. Your experimental
results or observations of some tripley related things could be depicted on 3D
graphs or charts, on which, for instance, weeks (Time, X) will be connected
with Dow-Jones (Index Number, Y) and interest rates (Economic fundamentals,
Z); or commodity production figures with prices and calendar years,
etc.
Since the relations can be of many sorts, causal,
temporal, spatial, quantitative, qualitative, formal, logical, to establish a
specific type one looks at the nature of relatives, or components, origin and
direction of dependency, correlation (statistics), as well as their formal
properties.
RS: Are sets the same as subject and
object pair, tied by the relation?
If i got the question right, yes and no. As such,
everything is connected with anything. For the sake of analysis, it is commonly
identified two types of relationships: simple, pure or
homogeneous, and complex, heterogeneous.
The first type is composed of the same kinds
of things as the relatives:
1. substances related with substances,
individuals with individuals, objects with objects, as space
relations;
2. states with states; qualities (quantities) with
qualities (quantities);
3. changes with
changes, processes with processes, actions with actions, events with events, as
causality and time relations;
4. relationships with relationships, as
analogy and proportion.
The second type deals with different levels of
relatives:
1. whole/part, with many different
sorts;
2. universal/particular, as generalization or
instantiation;
3. class/member, as membership or
subsumption
Kant viewed relation as one of the canonic classes
of things, ranking it as: i. the substance-property relation
(subject-predicate) logic; ii. causality-dependence; iii. reciprocal relation
(community, reciprocality, a relation of mutual dependence, action, or
influence).
Summing up:
Relationship is an ultimate criterion of any
reference ontology candidate; for RELATIONSHIP IS A MUST IN STANDARD ONTOLOGY.
Many thanks for the discerning
reading.
Azamat Abdoullaev
----- Original Message -----
Sent: Thursday, February 12, 2009 3:56
AM
Subject: Re: [ontolog-forum]
Relationship: n-ary vs binary
Azamat
Great clarity and
definition of relation – I guess
what you describe ought to also factor in predicate analysis.
But for practical
applications we need to know how to test whether a given relation (say a few
triples described by same relation type) are conforming to Cartesian product
of a set? Are sets the same as subject and object pair, tied by the
relation?
Relation is a canonic class of any ontology. It is
characterized by substantial properties and formal attributes. Of the material
properties, there are their reality, nature and type and direction of
dependency. Of the second, there are transitivity, symmetry, reflexivity, and
n-ary, or cardinality, terms, or tuples, of domains, elements,
components, or arguments).
The typical mathematical reading of relation is
an extensive set of ordered elements (as ordered pairs, Kuratowski, Wiener,
Skolem; well-ordering axiom).
There are two formal definitions of relationship
deserving attention, http://en.wikipedia.org/wiki/Relation_(mathematics):
i. [A relation R over the sets X1, …, Xk is a subset of their
Cartesian product, written R ⊆ X1 × … × Xk.].
ii. [A relation R over the sets X1, …, Xk is a (k+1)-tuple R = (X1, …, Xk, G(L)), where G(L) is a subset of the Cartesian product
X1 × … × Xk. G(L) is called the graph of L.]
So, one can say "an n-ary relation is an ordered class
of n-tuples, or it is an ordered class of (n+1) tuple". Three things are of
importance here:
1. the components of relations are of the same kind
and sorts, objects, persons, qualities, quantities,
times;
2. ordering of relations, their direction, a
triadic 'giving', tetradic 'paying' or triadic
'betweenness';
3. the key sense of relationship is represented by the
graph, indicating its nature and kind: if it's causal relation, temporal
relation, spatial relation, semantic relation, logical relation,
etc.
Think of the complex case of social networks, where
social relationships described in terms of nodes (agents) and ties
(relationships), of different sorts and kinds, like as emotional,
friendly, economical, political, or commercial links and
connections.
Any general ontology missing the class of relation as
the fundamental category of reality is internally defective. For example,
there is a comprehensive scheme of categories of reality proposed by Chisholm,
who divided Entia into Contingent things (States (Events) and Individuals
(Boundaries and Substances) and Necessary things (States and Nonstates
(Attributes and Substances). Since the class of relationship is
deprecated, the scheme is missing the adhesive of all things,
Relationship, as well as Time and
Space.
Azamat Abdoullaev
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