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

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 13 Feb 2009 00:28:54 -0600
Message-id: <FDD4C863-D0CB-400B-94DF-C55E929B2270@xxxxxxx>

On Feb 11, 2009, at 10:55 AM, Azamat wrote:

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).

None of the above makes sense. 

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 thegraph of L.]
So, one can say "an n-ary relation is an ordered class of n-tuples

Exactly. This is the standard mathematical "extensional" notion of relation, and is used in logical ("Tarskian") semantics. However, it is in many ways more natural to distinguish the relation itself from its extension (set of tuples), as apparently different relations can have the same extension 'by accident'. Of course, those who adopt an extensionalist discipline as a matter of principle would disagree with this. 

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;

Nonsense. Many relations relate things of different kinds. In fact, these are the most important relations in most ontologies. 

2. ordering of relations, their direction, a triadic 'giving', tetradic 'paying' or triadic 'betweenness';

What ordering are you referring to? The tuples are ordered by definition (that is what 'tuple' means). Other than that - essentially the ordering of the relational arguments - relations have no intrinsic order. 

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.

Wrong. This is not represented by the graph. In general, there is no way to tell, given the graph of relation (which just means, given the extension of the relation) what "kind" of relation it is. For example, any causal relation can also be interpreted as a (weaker) temporal relation, since there is a temporal relation with exactly the same graph (because causes never follow their consequences.)

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. 

Indeed, that is a rich collection of examples: but it is misleading to think that all relations are similar to social relationships between people. 

Any general ontology missing the class of relation as the fundamental category of reality is internally defective.

Well, the point can be argued. Certainly, one must use relations when describing reality. But a nominalist might balk at admitting that these relations exist in the same sense that physical things exist. 

Pat H

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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