Theoretical view of order

Most of this is fine, but I can't figure if this is meant to apply to 
relations like '<' or '<='.

E.g.
If R is '<' then totality doesn't apply for x R x.
If R is '<=' then anti-symmetry doesn't apply for x R x.

Or maybe you meant to say that the x R y relations apply only for x != y?

#g

At 05:45 PM 5/26/01 +0200, Ziv Hellman wrote:

> > Can anyone define (designate) order itself without using
> > order?   Has anyone
> > studied this?  Are there any URLs to their thoughts ?
>
>"Order" is a concept very well-studied in logic and set theory. In fact, 
>it is sometimes surprising just how little one needs to define/designate 
>order -- basically just a syntactic binary symbol and a couple of simple 
>axioms. This is actually a good little example of the interplay between 
>syntax, constraints, models and our subjective interpretations, that has 
>caused much locking of horns on this list, so it is worthwhile going 
>through in some detail the standard model-theoretic point of view on the 
>matter.
>
>To avoid needless complications, for the rest of this posting I will 
>equate the word "order" with "total anti-symmetric order" that in plain 
>language means that I am assuming everything can be placed somewhere in 
>this one order that is being defined -- so if a and b are objects either a 
>is greater than b or b is greater than a, i.e. avoiding the possibility 
>that they are "apples and oranges" and cannot even be compared  -- and 
>that no two or more objects are "equally placed" in the order, i.e. 
>discounting the possibility of objects being in "a tie for second place" 
>and so forth.
>
>Begin with any binary relation symbol you like (hard-core RDFers believe 
>everything in life can be reduced to binary relations so there should be 
>no shortage of these), call it R for short and write a R b for a graphic 
>representation of the relation. Add to the mix the following axioms:
>
>1) Totality: For each x and each y, either x R y or y R x .
>2) Anti-symmetry: For each x and each y, x R y implies not (y R x).
>( One might object at this point and ask if this is not begging the 
>question because distinguishing between x R y and y R x seems to require a 
>prior notion of "ordering". But this is really a surface artefact of the 
>linear form of graphically writing out the relation and does not touch 
>upon the deeper matter of how objects are related, that has nothing to do 
>with any particular way of expressing that relation. To give a concrete 
>example, we recognise that saying "Fred is John's father" is the same as 
>"John's father is Fred" despite the different ordering of the names in the 
>sentences, but that both are different from "John is Fred's father", 
>because Fred and John are involved in a relationship that at some abstract 
>level is set down forever no matter how one expresses or writes down a 
>representation of that relationship )
>
>3) Transitivity: For each x and each y and each z, if x R y and y R z, 
>then x R z.
>
>And that's it. Any model including the symbol R and these three axioms is 
>now going to have an "order" imposed on it. What is remarkable here is 
>that there is no need to attach any "meaning" to the syntactical symbols 
>and axioms: they exist independently of any subjective meaning we wish to 
>attach to them. We could interpret R as meaning "greater than", "less 
>than", "before", "after", "older than", "having better eye-sight", "owning 
>larger tracts of land", and it really will not matter -- in terms of 
>formal structure the models will all be the same. Conversely, the formal 
>syntax and axioms _themselves_ cannot supply meaning, that has to be found 
>in the semantics of what happens in particular models we work with. It is 
>quite conceivable that during the modelling of some complicated system one 
>will inadvertently define or derive equivalent axioms and suddenly notice 
>that one has an order that was not expected aforethought, with all the 
>attendant implications.
>
>The only "weakness" in this approach is that whilst it certainly includes 
>the standard notion we have in our head of 0 < 1 < 2 < 3 < .... -- the 
>"quintessential" order as it were -- as a model, it also includes lots and 
>lots of wild and strange models of order, from the relatively "tame" 
>negative and positive integers together, . . .  < -2 < -1 < 0 < 1 < 2 < . 
>. . which is infinitely ordered in both directions, or anything like  . . 
>. a < b < c . . . q < r . . . where the elliptic dots ... stand for all 
>sorts of infinities and ordinals galore. One can of course add axioms to 
>eliminate certain irregularities, like decreeing that there is one and 
>only one element that is first in the order, and so forth, but it seems 
>that no matter what one adds there are always non-standard models that pop 
>up to bugger one's thinking on the matter.
>
>Cheers,
>
>Ziv

------------
Graham Klyne
GK@NineByNine.org

Received on Sunday, 27 May 2001 04:46:41 UTC