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RE: [jboske] Kludgesome Solution #1

And, thank you for hanging in there. My immodest prediction (and Bob actually adumbrated this on the board) is that if we (as fundie vs. progressive formalists) get our story straight, this will be accepted at large. So let's see.

A first reply to Nick:
> Mad Propz to And and all, but I simply could not understand XS4
That is unfortunate, because it is probably substantially different
in SL-compatibility. The main differences in expressive power
between 3XS and 4XS is that 4XS allows you to quantify over pairs
(and other n-somes) of broda and over halves (and other fractions)
of broda. I will post another message giving the gist of 4XS.

I'll wait for that before posting Kludgesome Solution #2 (and I'll be putting in more and more illustrative examples.)

> 2. The Nicolaic properties aren't my idea, they're Johannine.
You pinned them down to prototype & made a ruling that solved the
problem of uninheritable properties. ("The zoologist studies the lion"
!= lo'e cinfo)

Sure, but both of these were ultimately at John's behest.

> 4. I'm not clear on the difference between extensional and intensional
> sets, but the solutions And proposes are compatible with a more
> fundamentalist Lojban
Two extensional sets are identical if they have the same membership.
Two intensional sets are identical if they have the same defining
feature (the same membership criterion).

Oh. Got it. As in, the former is the denotation of lx.P(x); the latter is, more or less, the expression lx.P(x). OK, I'll try and incorporate this.

> 7. Inner quantification does not properly include tu'o, since the set
> of all possible portions of substance does have a (transfinite)
> cardinality --- which is assuredly not mo'ezi'o.
Okay, but I can't promise that we won't find differences between
"is water" and "is a portion of water".

Well, there are three things one can speak of:

1. substance(x): x contains at least one bit that is P, but x contains no atom of P

2. bit of substance(x): the transinfinitely many bits of x. Transinfinitely means uncountable because, if you think you have counted all the possible portions of the substance (top 16/th cube, middle 16/th cube, central 19/th sphere), you can always come up with a portion you haven't accounted for. Just as with Real numbers and the diagonalisation proof.

In fact, since bits of substance are delimited by 3D space, which are delimited by 3D points, which are specified by real numbers, that cardinality is necessarily the same as real numbers.

You objected that real numbers are surely countable, and substances have no individual bits. But you see, they do. Take the 1m * 1m * 1m cube. You can divide that into 1/2 cubes (there are 2**3 = 8), 1/3 cubes (27), 1/4 cubes (81), and so on. You can also combine any two adjacent bits and get a bit (2/3, 3/4, 2/5...) And the cardinality of such fractional subcubes is Q, the number of rational numbers, which is aleph-0: fractions are countably many. But you can chop it into non-cubes, too, and for any fraction, you can always find a real number not covered.

So no, there are countable bits of the substance. It's just that they are not all the possible bits of the substance. Just as natural and rational numbers are countable subsets of the uncountable set of real numbers. So I believe the analogy holds. A substance is something with aleph-1 bits, a non-system has at most aleph-0 bits.

3. physically distinct bit of substance. I defined this at the end of Ontology #3 as spisa. This is a portion of substance of P, wholly surrounded in 3D space by non-P. The glassful of water, physically separated by the glass from the pitcherful of water. These spisa are countably many.

By portion of water, you probably mean (3), which is countable. But we also need to be able to reason about (2) (bits of water), which is uncountable.

> These are prolix inner quantifiers, and I will not shed a tear if we
> revert to ro and tu'o. But ro clearly applies to transinfinites as
> well, so I believe this is kind of cheating
It is cheating under the 'bit of substance' analysis.

I believe it is. When the founders said a substance was pisu'o loi ro (and they did), the substance was ro of something. This can't just be ro of spisa (because you can say half the glass is irradiated, and so the spisa is not atomic.) So I contend it is quantificaation over bits of substance.

> pisu'o loi djacu = some water
> re lo pisu'o loi djacu = two pieces of some water
> = re lo djacu

Why not abbreviate thus:

pa lo su'eci'ino [spisa be piro loi ci'ipa] broda

instead of thus:

pa lo [su'eci'ino spisa be piro loi] ci'ipa broda

? That seems more SL-conformant.

Because I am going to use pa lo su'eci'ino broda, *always*, to refer to individuals. Individuals always have at most aleph-0 cardinality, they are countable. Substances always have su'opi'ica cardinality: their bits are uncountable. Am I conflating substance and bit of substance here? Yes, and that's the point of substances: you can, there is no intrinsic difference between a substance and a bit of substance.

> {tu'o lo broda se pamei} = {tu'o lu'i loi pa lo broda} = Mr One Broda
> {tu'o lo broda se remei} = {tu'o lu'i loi re lo broda} = Mr Two Broda
> (the members of Mr Pair of Broda)
> {tu'o lo broda se romei} = {tu'o lu'i loi ro lo broda} = Mr All Broda
> (the members of Mr Collective of All Broda)

We need to quantify over members of Mr Broda Pair. How do we do that?
{PA lu'i tu'o lo(i)}?

Hm. Hm. I have only introduced lu'i as a last minute thing, and so I didn't really think this through, but I suspect the answer is yes. I'll need to revisit the INDIVIDUALS IN COLLECTION definition.

I haven't worked out how to quantify over subkinds of Mr Broda, either.
Ah: here it is:
> PA Kinds of the Kind expressed by {tu'o lo broda}... would need to be
> expressed by {PA lo tu'o lo broda}. But since this introduces ambiguity
> (I've been using non outermost tu'o to mean ci'ipa = ci'ipa loi
> su'osi'e be), and it is messy anyway, I would prefer it to be expressed
> by bridi
"by brivla", you mean? If you set aside your abbreviations, and use
tu'o only where it isn't an abbreviation, then {PA lo tu'o lo broda}
makes perfect sense.

Perhaps more pedantically, PA lo ro lo tu'o lo broda. I would rather use ci'ipa than tu'o, but we do at any rate have tu'o doing two different things (uncountably many vs. kind), and that's not quite right anyway.

> lo broda remains an individual rather than a kind. Or rather, lo broda
> expresses both an individual and a kind, but the latter is marked as
> tu'o lo broda
4XS is compatible with lo broda remaining an individual, if loi broda
becomes a kind. Of course, then {loi} would not mean a jbomass.

Yup. But KS1 keeps loi as jbomasses, lo as individuals, and has unquantified jbomasses and individuals, which both end up as kinds (though the kind of jbomass is an individal, and the kind of individual is a jbomass!)

> I clean up a logical confusion between tu'o = uncountably many and tu'o
> = uncounted
That's not a logical confusion. It's a confusion about the meaning
of substance selbri ('substance' vs 'bit of substance').

Well, it's a confusion, anyway.

I'm not sure how kludgesome yours really is. A lot of the verbosity
is due to conversions between types that in practise would be
left to glorking. I don't think we often want to convert between
su'omei and substance.

Oh absolutely. And the whole {loi vo lo remna} is something that is in real life done by {lo vomei} (though it does roll off the tongue.) But I keep both loi as a jbomass and lo as an individual, and I have space for default quantifiers. I really want these kept for me to call it SL-compatible. Be ruthless with it, but understand the spirit it is crufted in. (Well, I know you do, of course.)


--
Life Dr Nick Nicholas, Dept of French & Italian Studies
Is a knife University of Melbourne, Australia
Whose wife nickn@unimelb.edu.au
Is a scythe http://www.opoudjis.net
--- Zoe Velonis, Aged 14 1/2.