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Nick:
> This is unimaginably behind the times (i.e. outflow from yesterday),
> but then, that's the nature of this. When I think how many
> definitions and redefs have flowed in the past 5 days..
>
> I will be leaving town Monday for a month; I will be taking printouts
> of everything pertinent and coming up with an ontology and a
> kludgesome on the plane. And I will keep trying to keep up
Please don't try too zealously. I need a break too.
> OK. So
>
> (1) A number can be singled out out of the mass of all numbers. It is a spisa
> Real numbers are not atomic: you can always divide them into x < n/2
> and x >= n/2. They are therefore stuff
I'm afraid I don't understand that. Probably it's because I don't
think of one number as a bit of another. And (perhaps naively!) I
don't think that is mere mathematical naivety on my part; I think
it is a valid way to conceptualize numbers.
> A bit of stuff can only be singled out of the mass of all the bits of
> stuff if it is physically separate. So the middle 1/9 of a glass of
> water is not a lo djacu}, because it is not physically distinct from
> its neighbours.
True for all but the most contrived pragmatic contexts. The real
criterion is not physical separateness but simply fixed size or
fixed boundaries in conceptual space.
> However, a glassful of water is distinct from the
> Pacific Ocean, because it is a spisa
>
> For 3-D stuff, spisa is defined as I gave it in Ontology #3: a
> sectioning of 3D space, such that the lines of sectioning pass
> through non-stuff. The lines delimiting the glass pass through glass,
> not water. So the contents of the glass, the water, being encased by
> non-water, are a spisa of water. So it can be refered to as {lo
> djacu}. The middle 1/9 of a glass of water is not so encased. So it
> is not {lo djacu}
OK as a rule of thumb. Not as an outright definition.
> For numbers, a spisa of numberhood is defined by the property = x,
> for some x. 5 is surrounded by numbers that are not 5
>
> Numbers have the peculiarity that every possible bit of a number is a spisa
> 3D objects have the peculiarity that every possible bit of a spisa
> are non-spisa
How about a chessboard and "lo -square-shaped-thing"? Some bits of
a chessboard are lo square.
[...]
> "But I wanted the bits counted by pimu to be equal", you might say.
> Sure, and here, you have a 3-D specific notion: volume. Bits are
> equal, *not* because they have the same cardinality (which is still
> aleph-1 on both sides), but because they describe an equal volume.
> That's different from what you're doing with finite sets. And volume
> is inapplicable as a concept to real numbers
>
> So: for finite sets and collectives --- things of finite cardinality
> n --- pimu loi means a thing of cardinality n/2
>
> For stuff of transfinite cardinality, we do *not* mean pimu loi is a
> thing of half that cardinality, because half of aleph-1 is aleph-1.
> Rather, we mean, for 3D objects, any portion occupying 1/2 the space
> of the piromei
I don't remember if I had said "equal size", but certainly that's
what I intended, not "equal cardinality". We already knew that all
bits of substance have the same cardinality regardless of size.
IOW, "x in every y arbitrarily delimited but equally sized bits".
As far as I can tell, pimuloi can be analysed in either of the
following ways, which are different, but seem to achieve the
same result:
A.
(pa loi) pimu loi = a collective/substance glomming together 1
in every 2 arbitrarily delimited but equally sized bits of
a collectie/substance
B.
re loi = two collectives
pa loi = a single collective
pimu loi = a half of a single collective
Or rather, with your KS use of lo/loi:
A'.
(su'o lo) pimu loi = a collective/substance glomming together 1
in every 2 arbitrarily delimited but equally sized bits of
a collectie/substance
B'.
re lo = two collectives
pa lo = a single collective
pimu loi = a half of a single collective
In B/B', pimu is simply a multiplier like other cardinals. In
A/A', pimu is a proportion of all bits. I didn't spell out details,
but the intent of XS4.1 was that it would be possible to express
each of these two meanings distinctly.
As for KS (or SL), in A', Q lo would be used for cardinal
quantification and Q loi for fractional quantification. In B',
lo would be used for cardinaiities of su'o and loi for cardinalities
of me'ipa.
If one does have to mimic SL's use of lo/loi, A' seems slightly
less arbitrary to me. Furthermore, A' explains why piro takes
loi, whereas if we took it as a cardinality it would be equal
to pa and should therefore take lo.
So I think the analysis that is most consistent with both SL and
'logic' (in the broad sense of internally consistent, compositional
etc.) is A'.
> Where space is irrelevant, as in real numbers, then *all* fractional
> quantifications are the same, and the only distinction is between
> piro and na'ebo piro = pida'a
Okay.
> So And, your 1 out of 2 is inapplicable to |R, and I'm right: pimu
> describes only volume for a 3D onject, not really quantifuing over
> bits
I don't remember whether at some point I said "1 in every 2 real
numbers" made sense. If I had, it would just have been because
you'd brought them into the discussion. And I accept that that
makes no sense.
But perhaps obtusely, I still think that fractions working as
fractional quantifiers as described above are viable, though
not the only way of making sense.
However, I have helpfully offered a solution, A', that looks
compatible with SL.
--And.