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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.
> 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. 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.
> 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.
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.
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,
> The amount of possible portions of substance (not physically separated) > as physical objects in 3-D space is at least as great as the number of> possible 3-D spaces, since such a space can uniquely contain a given > portion of substance >> A 3-D space can be delimited by at the very least four real numbers: an> origin (3 numbers as coordinate) and a radius, forming a sphere. The > number of possible spheres in a 3-D space is aleph-1 ** 4. The number> of possible parallelepipeds is constrained by 3 points, and is aleph-1> * 9. The number of possible spaces of arbitrary shape will be> determined by at least alpeh-0 points, and is thus aleph-1 * aleph-0 (I> forget if that's aleph-1 or aleph-2) > > We can choose to restrict ro to countably many things, but I doubt we > should. So we're still stuck, if so. In the following, I'll use not > tu'o as an inner quantifier, but ci'ino and ci'ipa for aleph-0 and > aleph-1. I retain tu'o for its true meaning (see below.) I have no idea what the penult para means,
No maths, And. Tut-tut. :-) My point is that the number of possible 3-D spaces also has transfinite cardinality. The number of bits of 3-D space is uncountable. And that number delimits the possible number of bits of substance in 3-D space.
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.
> 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.
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.
> So.. > > pa lo ci'ino Atom > tu'o lo ci'ino Kind of Atom So how do we say the equivalent of SL "lo ci broda"? I guess the sensible answer is that you can replace ci'ino but not ci'ipa by the actual cardinality.
Consistently, I meant to say su'e ci'i no.
> pisu'o loi ci'ino = > su'o fi'u ro loi ci'ino Collective of Individual > tu'o loi ci'ino = Kind of Collective of Individual If "pi mu loi ci'i no" gives you the collective of one in every 2 people, how do you get the distriutive? How do you get "a certain Q of". This question applies to all your examples.
Blind spot I have to fix: a full equivalent of se su'omei (any one individual in the collective). This will likely be PA lu'i.
> pisu'o loi ci'ipa != > su'o fi'u ro loi ci'ipa Substance > tu'o loi ci'ipa = Kind of Substance Why "!="?
Substances can be fractionally quantified by any real number between 0 and 1, not just any rational number. Misstated above.
> pisu'o lo ci'ino = > pisu'o loi ci'ipa nysi'e be pa lo ci'ino Individual-Goo (Substance of > Individual) > > pa lo ci'ipa = > pa lo ci'ino selci be piro loi ci'ipa Individual of Substance > > pisu'o loi ci'ino lo su'o lo ci'ipa = Collective of Substance > loi su'o lo ci'ipa Where "su'o" is the total number of individuals of substance?
Yes. Here I was starting to get sloppy, though. See KS1.
> The majority of properties are inherently atomic, group, or substance.> So the innermost quantifier, aleph-0 or aleph-1, is usually left out > with impunity. So how do we state the actual cardinality? Surely the cardinality of the set of my parents is not ci'ino.
Like I said, read {su'e ci'ino}. It can be {re}. The point is, it can never be any finite number for a *substance*. By definition of substance. (That's why I was doing the ontology. Bugger intuitive understanding: give me a formula, and I'll know what's going on.) You cannot meaningfully say {piro loi re djacu} under any world, whereas you can meaningfully say {piro loi re remna} if there are just two people. f there are re djacu, you can always split them apart into {vo djacu}. I am ignoring the atomic model (which makes water a collective, not a substance --- and space is certainly not atomic).
The Founder understanding is that the inner quantifier counts spisa (physically separate bits of the substance); but this is crap, as Jorge and his Jorge-solids can tell you.
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.
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... -- Life Dr Nick Nicholas, Dept of French & Italian Studies Is a knife University of Melbourne, Australia Whose wife nickn@hidden.email Is a scythe http://www.opoudjis.net --- Zoe Velonis, Aged 14 1/2.