Re: 1/[0,2]=NaI
Ralph Baker Kearfott wrote:
> Nate Hayes wrote:
>> Arnold Neumaier wrote:
>>> Nate Hayes wrote
>>> (in: Re-submission of motion 5: multiple-format arithmetic):
>>>
>>>> Arnold Neumaier wrote:
>>>>> Nate Hayes schrieb:
>>>>>> Arnold Neumaier wrote:
>>>>>>> Nate Hayes schrieb:
> .
> .
> .
>
>> My position on this subject therefore allows you to compute
>> 1/[0,2]=[1/2,Inf) in your range enclosure example, i.e., it allows
>> to avoid NaI in this case because it implies
>>
>> 1/[0,2] = 1/[0,0] \union 1/(0,2]
>> = {empty} \union [1/2,Inf)
>> = [1/2,Inf)
>>
>
> According to my understanding, THE ABOVE (taking the union of limiting
> of point values) is the basic idea underlying csets, for what it's
> worth.
To the best I understand, this is what John advocates in his proposal...
that zero is dropped from the domain:
1/[0,2] = 1/([0,2] \intersect (0,Inf)) = 1/(0,Inf) = [1/2,Inf).
This essentially causes a limiting condition near zero.
>
>> In the context of predicate logic, though, such an interpretation is
>> not correct. For example, if I seek values of y such that
>>
>> (for all x \in [0,2]) y = 1/x
>>
>> is true, there is no value of y when x=0 to make the conditional
>> equation true.
>
> But doesn't classical interval arithmetic (which I thought is the
> basic thing we are standardizing) seek
>
> (the set of all y such that there exists an x \in [0,2] with
> y=1/x) ??
> That is different from what you have above.
It is.
>
> So the predicate is undefined, i.e., NaI (this is why in Moore's
>> classical arithmetic division by zero is undefined).
>
> Huh?
For example, the logical formula
(for all x in [-4,-1]) [1,3] \union sqrt(x)
is undefined (NaI) because it is not true for all x.
This is different than the set-theoretic point of view which says
[1,3] \union sqrt([-4,-1]) = [1,3] \union {empty} = [1,3].
Nate