[om] Bug in definition of `strict_order'

Professor James Davenport jhd at cs.bath.ac.uk
Mon Jul 14 17:46:55 CEST 2003

On Wed, 9 Jul 2003, Scott L. Burson wrote:
> I just now learned about OpenMath as I was using Google to look for some
> definitions.
> I seem to have found a bug in the definition of `strict_order', or more
> precisely, in that of `antireflexive' on which it depends.  A strict order
> is irreflexive, meaning `forall(x) ~R(x, x)'.  The current definition of
> `antireflexive', however, says
I can't find antireflexive, but there is antisymmetric, which does indeed
have the definition you quote below, which I think is correct for 
>    forall(x, y) R(x, y) -> R(y, x) -> x = y
> But this definition does not work as part of the definition of
> `strict_order', because by irreflexivity and transitivity, there are no x, y
I don't quite follow this point. I do agree that the definition is over 
the top, as irreflexive and transitive imply anisymmetric, but I don't 
think it's wrong as such.
> such that `R(x, y) & R(y, x)'.  It could be written instead as something
> like
>    forall(x, y) (R(x, y) = R(y, x)) -> x = y
> and then it would work for both reflexive and irreflexive relations.
James Davenport
OpenMath CD Editor
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