Re: Motions 20 and 21 on comparisons
I'm sorry, John,
but I think you made a mistake in your critique of
our paper
Am 17.09.2010 22:27, schrieb John Pryce:
> P1788
>
> Juergen and Arnold have recently spoken in favour of simplicity for
P1788 comparisons. I agree.
>
> 2. Lack of canonical-ness.
> Consider "starts" which in M21 Table 1 is
> (\exists_a \forall_b a<= b) \land (\exists_b \forall_a b<= a)
\land (\exists_b \forall_a a< b) (2)
>
> Using compactness one can see that, for nonempty compact intervals,
this is equivalent to the result of swapping round each (\exists \forall
...), namely
> (\forall_b \exists_a a<= b) \land (\forall_a \exists_b b<= a)
\land (\forall_a \exists_b a< b) (3)
>
> But these give different results when either A or B is empty. Namely
(2) gives false, while (3) gives true.
why true ?
the first term is already false for A=empty B != empty
> (\forall_b \exists_a a<= b)
there is no a ! So in my understanding of logic the result is false
Our paper is consistent with emptysets
In remark 3 and table 2 we define the extension of the overlaps relation
to empty intervals. That does not mean that the other conditions all
deliver false.
We deal with that cases in remark 7. That admittedly leads to a problem
when the standard comparisons are introduced. We resolved that by an
additional rule in Remark 10.
It turns out that the rule is identical to that in motion 13.04
Another case is the comparison of unbounded intervals. Here our position
is to compare the bounds with the rule infty == infty. (see remark 4)
BTW we obtain Entire == Entire
All other interpretations of the sets cannot be handled without
decorations. But the paper and the motions 21 and 13.4 are about bare
intervals.
Juergen
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