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Nick:
> And:
> > > There are three reasons you might count something as tu'o
> > >
> > > First, there's only 0 or 1 of them. Dumb reason. Something like this
> > > may have been attempted with ledu'u
> > I'm not sure what you have in mind here, but if the reference to
> > ledu'u is a clue then the argument was that when in the mass of
> > all worlds there is exactly one of something, it is undesirable
> > (for reasons that I can spell out yet again, if necessary) to
> > *have* to quantifier over all broda in order to refer to the one
> > broda. So this would really by like your third case
>
> You may not have to, but there is one there: you can put a quantifier
> in the prenex, su'osu'epa da. In the third case, you can't: the prenex
> is simply sidetracked. Nothing alike. But let's not dwell on that
I'm not convinced that su'osu'epa is the right number. Maybe
{mo'e tu'o lo namcu} is the right number.
At any rate, I accept the important distinction between
{vei mo'e tu'o lo namcu ve'o lo broda} and {tu'o lo broda}. So
we are making progress. However, the way I think this should
be done is:
{vei mo'e lo-kind namcu ve'o broda} (Substance)
{lo-kind broda} (Kind)
or, if you insist:
{vei mo'e lo-kind namcu ve'o lo-kind broda} (Kind)
but this gives rise to infinite recursion unless there is a
PA that means "mo'e lo-kind namcu").
> > > Second. the cardinality of the set is trans-infinite. This is what
> > > holds for substances
> > What does "trans-infinite" mean?
>
> Damn. I meant transfinite. I won't mangle algebra here, but I remind
> you that the infinite cardinality of real numbers is greater than that
> of natural numbers, because real numbers are uncountable
>
> > > In my ontologies, I have been quantifying with prenexes over
> > substances
> > > and bits of substances. I can say that if x is water, all conceivable
> > > bits of x are water --- so I am saying all. Similarly, I can speak
> > of x
> > > + y being a real number, for all real numbers x and y
> > Quantifying over a substance is not the same as quantifying over
> > bits of substance. The latter makes sense, and to me at least, the
> > former doesn't
>
> I now see why. But our inner quantifier for collectives quantifies over
> bits of collective (lei re prenu), not over the collective.
I understood it to be {ro fi'u PA} where the numerator is the number
of members in the collective, PA is the number of things with the
te memzilfendi, and ro = denominator.
> So I
> believe our inner quantifier for substances quantifies over bits of
> substance. Of which there are uncountably many, by definition. So
> aleph-1 is an appropriate and distinctive inner quantifier. This is not
> just tu'o
As I have said in other messages today, if the predicate means
"is a bit of broda-stuff" then the inner quantifier is {ro fi'u
ci'i no}. If the predicate means "is broda-stuff" then the
inner Q is {ro fi'u vei mo'e lo-kind namcu ve'o} (ideally
shortened by a PA meaning "vei mo'e lo-kind namcu ve'o").
> > > The set of natural numbers has cardinality aleph-0
> > > The set of real numbers has a cardinality, and it is aleph-1
> > > That means that there are proper subsets of real numbers that are
> > > countable: N is a subset of R. It also means it is feasible to speak
> > of
> > > 'all' over a transfinite set. It's just that the set is not countable
> > Bearing in mind that I know next to no maths, so am probably talking
> > out of my netherparts, I am guessing that 'all' means 'every member
> > of' or 'every subset of', and not 'everything that is a set of real
> > numbers'
>
> Correct. I can say "all real numbers'. I can say "all possible bits of
> water" -- whether they are physically separate or not. They are {ro}.
> They are {ro su'e ci'ipa}. By analogy with collectives, that, not tu'o,
> is the inner quantifier of substances as lojbanmasses
Okay, if you are saying that loi coerces Substance into Bit of Substance.
{PA-kind (lo PA-kind) broda} should do for Substance.
Excellent progress.
> > With caveats about my possibly missing your point, I don't think
> > you correctly characterize our idea about tu'o and ro. We can
> > count namcu, I take it: {ci namcu} is meaningful, isn't it. So
> > {ro namcu} means (or is equivalent to) "every namcu", because
> > ro gives you the cardinality of the set of all namcu. So for
> > {ro -real-number}, ro = aleph-1, whatever the thuc that is
> >
> > But {tu'o broda} was to be used where the contrast pa/re/ci/../ro
> > made no sense -- how do you count something that has no boundaries
> > or fixed size? You can't. Can mathematicians?
>
> You can't count real numbers either. That's the point of aleph-1 being
> greater than aleph-0. Whenever you start counting (say, 1,2,3,4...), I
> can always devise a number you will necessarily miss. It's the same for
> bits of substance
You CAN count real numbers. Give me a list of n real numbers and
I will tell you what n is. But you can't give me a list of bits
of substance, because they aren't individuable (if you tried
to give me a list, you'd be giving me a list of Individual of
Bit of Substance). Numbers are (if I remember your scheme correctly)
atoms, while Bits of Substance are Substance. (If I haven't got that
quite right, I think you'll at least know what I mean.) What you
can't do is count the real numbers in a number-span -- that is, if
you count them, you always get to ci'ipa.
> > If 'broda' were 'bit of substance' rather than just 'substance',
> > then we could quantify. The cardinality would always be infinite,
> > but the quantification can be done as a proportion of all the
> > bits that exist
>
> OK, yes; but my point is, the inner quantifier of a fractionally
> quantified lojbanmass is not the cardinality of substance. It is the
> cardinality of bits of substance. In {pi ro lei vo prenrbitlzi}, 4
> isn't counting The Beatles. It is counting members of The Beatles ---
> atomic bits of Beatledom,
Okay. It hadn't fully dawned on me that you were trying to do
lojbanmasses. I'd thought you were trying to do Substance. You're
right about lojbanmasses, and lojbanmasses don't do Substance,
afaik.
> > but looking at the last
> > para, we weren't restricting ro to countably many things. We were
> > restricting ro to the cardinality of sets of countable things
> > {(LE) tu'o broda} was understood to mean that it was meaningless to
> > try to distinguish between pa/re/ci broda
>
> That's not what we've been doing with collectives in SL, though: they
> take cardinality of atomic bits, not cardinality of the collective
> itself, as the inner quantifier
Okay. SL lojbanmass doesn't do Substance. As for cardinality of the
collective, {loi re lo ci broda} in SL gives you a pair, even if there
are three things that are broda.
> > > The cardinality of Q, the rational numbers, is also aleph-0. And I
> > see
> > > why And wants Q to fraction-quantify collectives, and R to
> > > fractional-quantify substances. It may be too late for Standard
> > Lojban
> > > to demand this though
> >
> > I thought I was proposing Q for both collectives/sets and substances?
> > Q set/collective = Q members of. Q substance = Q bits of
>
> Yes, but the point is, the fractional quantifier of substances can also
> be a real number, whereas the fractional quantifier of collectives
> (retaining atomicity) can only be a rational number. I can have 1/pi of
> the water in the glass. I cannot have 1/pi of humans
I see. This was just my impoverished maths.
> > Sure. When we were proposing to use tu'o as an outer quantifier, this
> > is what it meant (though see comments to first reason above). As
> > an inner quantifier that states a cardinality, tu'o would mean
> > that there is no cardinality (not because the set is infinite, but
> > because there is no criterion for defining what counts as a single
> > member)
>
> By the time we get to PA lo tu'o lo ... , you've got tu'o ambiguous
> between Kind and Substance. You don't want that
No. I want PA-kind and lo-kind. Then all is hunkydory.
> > I have glossed over this, because I can't keep up and because it
> > doesn't cover +specific
>
> Right now, I think +specific is fine print
Not really, because in my own workings towards XSs I have found that
one of the pitfalls is an excessive asymmetry between o-gadri
and e-gadri, whereas -- elliptizabilites apart -- they ought to
be symmetrical.
> > This is all great stuff, but it'll be a long haul
>
> But you see what I'm doing, right?
>
> * Keep loi the lojbanmass
> * Keep the lo = individual and the lo broda = su'o lo broda
> * Allow consistent disambiguation of lojbanmass into collective and
> substance, whensoever needed
> * Allow Kinds
>
> .... basically, do what you want done with gadri (or at least, that
> which we both want done), but violating as little of CLL as possible
>
> Damn it, this is fun..
I do know what you're doing. I knew it was what you were going to
do. And I am confident that you can succeed.
The only real difference between KS and XS is that the former
favours SL-compatibility over elegance and the latter favours
elegance over SL-compatibility.
--And.