[topicmapmail] Re: Logicians do not rule the world (fortunately)

[Eirik Jensen Opland]

Hi Murray,

Sorry to extend this beyond infinity (c.f. your ad infinitum), but I think  
there's an important point missed in this discussion; the distinction  
between symmetry and bidirectionality.

On Wed, 26 Apr 2006 09:35:51 +0200, Murray Altheim <murray06 at altheim.com>  
wrote:

> Before this continues ad infinitum,
>
> I think what Steve Pepper's lighthearted subject line expresses
> quite well is that people generally don't like logic, or perhaps
> even logicians, who are thought of as cold, calculating, possibly
> not even entirely human beings.

You may be right that some topic mappers don't like logic in it's purest  
sense, or don't care so long as their technology does what they want. With  
all due respect for people who have only a moderate interest in logic, I  
must say I do like logic and I think it's important that people like you  
feel you can safely do inferencing on them. I would argue that TM  
associations are inherently bidirectional, but only become symmetric (in  
the logical sense) if the role types are the same.

You were discussing the expression of symmetry, so I will start with the  
definition. I will use the definition from the wikipedia definition of  
symmetry, i.e. "for all x, y: xRy -> yRx". Please correct me if it's not  
sufficiently accurate.

The above relation has implicit roles, i.e. the subject and the object. x  
is the subject because it's written before the R. y is the object because  
it's written after R. What it means to be subject/object depends on the  
definition of R. The key point here is that the statement has the  
implicitly defined roles subject and object, which are not the same. One  
could lay it out as in TM structure:
   x <-- subject <-- R --> object --> y
Where 'x' is a role player, 'subject' is a role type, 'R' is an  
association type, 'object' is the other role type and 'y' is the other  
role player. That's an association. It is bidirectional, in the sense that  
it doesn't matter at which end you start inspecting (in this case reading)  
it. It gives the same meaning.

However, it is not symmetric, since 'subject' and 'object' are different  
roles.

One way to say that R is symmetric is to assert that the subject and the  
object roles are interchangable. In other words, one could assert that:
   x <-- subject <-- R --> object --> y
implies
   x <-- object <-- R --> subject --> y

This could be written more concisely as:
   x <-- role1 <-- R --> role1 --> y
where role1 is a kind of placehoder for both subject and object.

This is effectively what topic maps do. Rather than having relations with  
direction, they make role types explicit. If both role types are the same,  
then the association is symmetric. Otherwise it's not.

Having explicit role types also adds expressibility, as one can make more  
specific role types than just 'subject' and 'object'. This is particularly  
useful for associations with more than two roles, as they necessitate  
additional role types.

The role type type also improves the expressiveness further for relations  
with more than two roles. You can express partial symmetry, that allows  
your inferencing to differentiate associations like:

Parenthood(father, mother, child)       (not symmetric)
Parenthood(parent, parent, child)       (partially symmetric)
FrendshipClique(friend, friend, friend) (very symmetric)

Surely these kinds of distinctions must useful for inferencing, aren't  
they.

The way I see it, TM associations have three main constraints regarding  
symmetry and direction:

1. They are inherently bidirectional.
2. You can't have a symmetric association with different role types.
3. You can't have an asymmetric association with the same role types.

These are constraints on the modeller, as he has to take into account  
assumtions in the model, such as repeated role types implying symmetry. I  
don't see a big problem with these constraints, but if you see any, I'd be  
eager to hear about them.

> Logic is a hard tool, a very
...
> been on about here, really.
>
> Symmetry is a property of relations in certain kinds of logic.
> It's also a word in English. So long as we're using it in the
> latter sense, and we're clear about that, then everything's fine.
> If we want to begin to consider things more formally, to be able
> to use tools that rely on formal relations, then we need to
> begin to define things a great deal more than they're currently
> defined. It's all possible in Topic Maps (e.g., I've long
> conjectured that XTM would be a reasonable syntax for interchange
> of Cyc expressions, and have played around enough with that to
> see both its possibilities and its limitations), but it's not
> possible without definitions. One must have a model in order to
> be precise in making a statement or specification about something,
> and logic provides a model, a framework. Absent that, we're just
> talking in English, which is fine, but it's not logical.

I agree with the importance of these distinctions. I usually try to stick  
as closely as possible to the logical meaning of things when I'm aware of  
it.

>
> Murray
>
> ...........................................................................
> Murray Altheim <murray06 at altheim.com>                              ===   
> = =
> http://www.altheim.com/murray/                                     = =   
> ===
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-- 
Eirik Jensen Opland, Ontopian         <URL: http://www.ontopia.net >