Re: [lojban] Re: C-sets

On Sat, Sep 13, 2014 at 10:15 AM, 'John E Clifford' via lojban <lojban@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