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Re: [ontolog-forum] Relationship: n-ary vs binary

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Ravi Sharma" <ravisharma@xxxxxxxxxxx>
Date: Wed, 11 Feb 2009 20:56:44 -0500
Message-id: <C2BE8BD20D3042918B12CC6D929FCF21@DFPLWW81>


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?

Thanks for starting the Wiki page as well..


(Dr. Ravi Sharma)

313 204 1740 Mobile



From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Azamat
Sent: Wednesday, February 11, 2009 11:55 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Relationship: n-ary vs binary


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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