C-sets

[John E Clifford]

I was reading up on xorlo today, trying to figure out what it says as opposed to what various people seem to think it says – and possibly what later changed actually make it say, when I was reminded that Lojban still has C-sets in its repertoire. The not said that these are little used. What a pity! C-sets have some advantages over L-sets (mereological hola, pluralities, etc.) which could be exploited in some situations rather handily. The problem is that we tend to take sentences involving them in a fairly literal way, following (Cantorian) set theory rather than Lojban. In set theory of the usual sort, the only properties that sets can have are size and taking things as members or other sets as subsets (and various things derivative from these). But there is no reason, other than habit, why a set can't be said to carry a piano or surround a building. Allowing this, that letting a C-set to have properties that grow out of the properties of its members, has certain advantages in some cases. The simplest is that C-sets have two “among” relations, one exclusively for individuals and another not. This removes – or at least alleviates – a lot of complexity with L-sets, which has to keep shifting back and forth on the issue (quantifiers on L-set expressions – some of them anyhow – are restricted to individuals, even when subsets might be more handy, for example.). Another useful feature of C-sets is that they can be empty, so that, unlike 'lo no broda', 'lo'e no broda' is a meaningful, referential, expression. Exactly what all sets might be used for and just how to exploit them is not clear, though I think all the needed pieces already exist and are grammatical. For now, I just want to place the possibilities into the mix of expanding consciousness for usage.

[Andrew Browne]

Incase anyone else is looking for some more background to understand this:

http://pckipo.blogspot.com.au/2009/09/c-sets-and-l-sets-draft.html

PC: what your previous blog post, and this post, is missing is showing/explaining some examples of these different kinds of sets in Lojban. This would be quite helpful if you want to promote usage.

Andrew / DerSaidin

[MorphemeAddict]

The draft at that link seems to treat C-sets as the more common, standard set theory kind, with L-sets as the less common kind, just the opposite of John's post above, AIUI. 

stevo

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[John E Clifford]

More common in the world at large.  Ordinarily, if you say "set', it is assumed you mean C-sets (even if your hearer knows there are other kinds).  In Lojban, however, C-sets have always occupied a secondary place, behind whatever {lo broda} and the like referred to (and those have been so truly odd things at times).  The reasoning was that, since in set theory C-sets were used in a way that would not do what was needed, they could not be used in Lojban to do what was needed.   It turns out this is not true but does require some added work, so we stick with L-sets (or equivalent) which do things more easily and are finally respectable entities.  My point is just that C-sets can do somethings a bit more tidily that L-sets and should get used more in Lojban.

[.arpis.]

I may be missing something. When you say "used more", do you mean at the language design level or at the speaker level. Could I, as a speaker, distinguish between C-sets and L-sets in my speech as lojban is now?

Also, do you have any references for someone who would like to learn more about L-sets, preferably accessible by a computer science graduate student who has studied neither logic nor linguistics in depth?

mu'o mi'e .arpis.

[John E Clifford]

[John E Clifford]

Both.  C-sets get very short shrift in CLL, a mere mention of lV'e in the gadri section and little else expository, even in MEX.  And no uses them in the language, partly because of the CLL situation and partly because of the belief that they are virtually uselss (i.e., can only be used as in set theory).  As for useful places to look for information about L-sets, the various things Google supplies (under Lesniewski and mereology) tend to be rather technical and wrapped up in the particularities of the system (and related systems) rather than the matters of interest to Lojban.  You can eventually work out the good stuff but it is a bit uphill.  I can't find a popular or directed discussion anywhere -- of L-sets or related notions.   

[Jorge Llambías]

On Sat, Sep 13, 2014 at 10:15 AM, 'John E Clifford' via lojban <loj...@googlegroups.com> wrote:

 In Lojban, however, C-sets have always occupied a secondary place, behind whatever {lo broda} and the like referred to (and those have been so truly odd things at times).  The reasoning was that, since in set theory C-sets were used in a way that would not do what was needed, they could not be used in Lojban to do what was needed.   It turns out this is not true but does require some added work, so we stick with L-sets (or equivalent) which do things more easily and are finally respectable entities.  My point is just that C-sets can do somethings a bit more tidily that L-sets and should get used more in Lojban.

 Let me try to do some of the "added work" you say is required, and see what happens. My conclusion is that C-sets, i.e. "lo'i", are indeed useless, but do tell me where I'm going wrong.

Let C-sets be the natural argument for "bevri1". This is nothing more than a re-definition of "bevri". Instead of meaning "x1 carries/carry x2 to x3 from x4 over path x5" it now means "x1's member(s) carries/carry x2 to x3 from x4 over path x5". Nothing is gained or lost in terms of expressiveness by making this redefinition, we just have to put a reference to the set of carriers instead of to the carriers themselves in x1. (You mentioned that by using C-sets one gains the possibility of having the empty set there, more on that later.)

I don't think you can define "bevri" unambiguously in such a way that you can have either C-sets or non-sets in x1, But that doesn't matter, if we are having sets of three carrying things around, there's no reason not to have sets of one carrying things around either, so we would just say "lo'i ci prenu cu bevri lo pipno" or "la'i djan cu bevri lo cukta". It wouldn't make much sense to say that both John and the set whose only member is John carry the book, because then we would have two different carriers when in reality there is only one. And if we were to say that John and the set whose only member is John are one and the same thing, then we would no longer be dealing with C-sets.

Now, doing that to bevri1 alone would be rather pointless. If we are doing it to bevri1, we should do it to every argument place of every gismu. There doesn't seem to be any reason to have a mish-mash of argument places, some of which take C-sets and others take non-sets. So we would redefine "bevri" as "x1's member(s) carries/carry x2's member(s) to x3's member(s) from x4's member(s) over x5's member(s)". So we would say "la'i djan cu bevri lo'i cukta lo'i vimku'a lo'i tidyku'a" instead of "la djan cu bevri lo cukta lo vimku'a lo tidyku'a".

So far we don't seem to have gained anything with this change, other than making our gadri longer. What about quantification? For the inner quantifier nothing changes either. With "lo" we inform the number of referents, with "lo'i" we inform the cardinality of our single referent. 

For outer quantifiers, we first need to specify what would they mean with "lo'i". Since all our predicates now take sets as arguments, we want to quantify over sets, and the natural quantification is over subsets of the set that provides our domain. (Quantifying over members of the set would be pointless at this stage, since we don't have any predicates left to say anything about the members directly.) But quantifying over all the subsets is usually not what we want either, there are too many of them, so we define "ro" and "su'o" (and therefore "pa", and "re", etc) to quantify over the singleton subsets only. We will also keep "ro'oi" and "su'oi" to quantify over all the subsets for the cases when that is required.

With that definition of quantifiers on C-sets, all our quantified expressions with "lo'i" also have identical meaning to what we had originally with "lo". Still nothing gained.

Now, what about the empty set? What happens when we claim that it's the empty set that carries the book from the library to the toilet? Where is the book after this happens? The only sensible reading I can give to this is one where the book ends up in the toilet, and the empty set plays the role of "zi'o". Calling the empty set "lo'i no smacu", "lo'i no mlatu" or "lo'i no gerku" shouldn't make any difference, unless we are saying that there are different empty sets. There can't be different empty C-sets, can there?

So, do we gain anything by making all our predications about C-sets rather than about non-sets?

Alternatively, if someone prefers to think in terms of C-sets rather than in terms of plural logic, is there any reason why they can't think of "lo broda" as referring to a C-set and all predicates as being about C-sets rather than about non-set things? Is there any benefit in defining predicates in such a way that some arguments take C-sets and other arguments take non-sets, so that the distinction between "lo" and "lo'i" would be relevant? Which argument places should be chosen to take C-sets and which should be chosen to take non-sets?

mu'o mi'e xorxes