# conservative Extension

**A Conservative Extension of SS toward XS**

**What there is, sorta**

Imagine a language that makes every distinction made in any language of the universe, as well as any that affects the behavior of any sentient being or the course of any process. That is, it has a separate predicate for every distinction that makes a difference. Imagine then all these predicates laid out heading columns in an array. Each row then consists of entries in each of the columns, positive, negative or irrelevant, where the “or” is inclusive. Corresponding to each row in this chart is one thing, which has exactly the properties marked in that row, the property if the cell is positive, the denial of the property if the cell is negative, the irrelevance of the property if that is marked. The set of these things is the universal domain.

Things in this domain are very diverse, having from one (Oh, what the Hell, throw in the thing that has “irrelevant” in every column, *materia prima absoluta*) to all of the properties – and their negations and their irrelevancies. Some of these things will be contradictory, either overtly or covertly (overtly, one square containing two or three marks, covertly two squares marked in ways that are incompatible: totally green and totally red, say). Others will be incomplete in an intuitive way that takes volumes to spell out exactly. And some few will be complete (*objects*)– also in an intuitive but indescribable sense.

These things can be combined into *situations*, groups of things that act out their properties together. Some situations are impossible, either because they contain impossible things or because the contain things that are not compossible: a childless man and his oldest son, say. (Logicians to the contrary notwithstanding, an impossible situation does not contain everything, only what is explicitly put into it.) Some situations are incomplete, in a variety of intuitive ways. And some are complete, meaning at least that every thing in them is an object and that all relational properties are reciprocated: if x has -Ry in a complete situation (*world*), there is y that has xR- in that world, and so on even for the modal predicates. The objects that make up a world are said to *exist*, or be *existents*, in that world.

**And so to gadri**

First of all, normal quantifiers in a world quantify over only the existent in that world. That being said, expressions without quantifiers -- or with abnormal ones – can refer to any object in the universal domain, as may be appropriate. In particular, anything in the universal domain may have a name. Here I will take normal quantifiers to be the unmarked ones and abnormal ones to be those with a {ni’u} prefix.

We begin our visible deviations from CLL by saying that a description that begins with a gadri is distinct from one that begins with a quantifier, in the obvious way that the set underlying the bare description may contain appropriate members of the universal domain that are not existents (though they generally have to be complete), while the quantified ones must, of course, contain only existents.

Thus, {le broda} is subset, selected by the speaker, of the universal domain, preferably of at least mainly brodas. {PA le broda} is a PA sized subset of that set, containing only existents. The same holds for {lei broda} and {le’i broda}. The selections for the three kinds of gadri do not have to be the same. They differ in that the members of le broda are treated distributively, those in lei broda collectively and those in le’i broda cumulatively. The quantifierless forms refer to the same set throughout a given context, the quantified forms (except for those with {ro} or {piro}) may change reference at each occurrence (but not with pronoun anaphora). A {PA le broda}, for examples, can also be made a constant by prefixing {le} (and similarly for {lei} on {PA lei} and {le’i} on {PA le’i}. The fixed forms are unaffected by negation (or modal) passage; the quantified forms go through the usual changes.

For a variety of reason, the o forms behave slightly differently. {lo broda} is a designated subset of the brodas in the universal domain, and, like {le broda}, keeps the same referent throughout a context, so is unaffected by boundaries. {PA lo broda} is a PA sized subset of the set of existent brodas, which one quite independent of the choice for quantifierless {lo broda}, and subject to change with each occurrence. A particular subset can again be fixed with {le}.

{loi broda} and {lo’i broda} are the set of all brodas in the universal domain considered collectively and cumulatively, respectively. The quantified forms are what you would expect now: {ro lo’i broda} is the set of all existent brodas; {su’o loi broda} is a nonempty collection of existent brodas, taken collectively. These quantified forms vary with occurrence and are fixed by adding the corresponding e form.

PA quantifiers give the size of the set involved as a cardinal, piPA do the same as a fraction of the original set (of existents).

Unfortunately, in the system so far, we cannot talk about subsets of {lo broda}, since a quantifier will immediately move to a subset of the set of all existent brodas. Yet we want to make this move fairly often. The solution seems to be to use an interposed {lu'a}, the device used to get from a groups to its members distributively. This would appear to be redundant here, since {lo broda} already is a distributive group, but the aim is to keep the initial selection. The same device is not needed for {le broda}, since that does not change referent when quantifiers are added. In a similar way, {lu'i lo'i broda} and {lu'o loi broda} appear to be redundant. They, however, can be used with quantifiers to enumerate submasses or subsets of the original. In particular, if {loi broda} is a Kind, then {PA lu'o loi broda} is PA subkinds of that kind -- not just any old submass/set (for which there is little use). (It should be noted that there is relatively little use for internal quantifiers PA for Kinds, since numerically specified -- rather than predicatively -- submasses are not significant in situations where the Kind is. Thus, as xorxes notes, these numbers could also be used to enumerate submasses. The present suggestions seems to me more systematic, but is also longer.)

tbc

Here's how xorxes think XS works with this ontology:

#### le

The speaker selects a single row, x, and chooses a description to refer to that row. Normally the property chosen will be marked as positive for that row, but this is not necessary. The speaker should choose a property that will help the audience identify row x.

- Well, in the set up of the ontology above, each row is a single individual, a unique collocation of properties, so {le} to a row would be just that one individual (well, actually its singleton, but treated distributively so back to it). Orinarily, we would expect {le broda} to pick from among the things marked positively for {broda}, though not absolutely requiring this.pc
- Are properties like {leka ce'u remei} allowed as one of the columns?

- I don't think so-- no object is a 2-tuple. pc

#### PA le

Given the row x selected with le, the quantification is over all the objects of the world that have a positive mark for the property "is a member of x".

- Some specified sized subset of the existents selected to be le broda. pc
- Yes, that's SS, but I'm describing how things work in XS.

- "Is a member of x" is probably not a property, since it is a nonce choice, but that aside, I think this is about right: the quantification is over the existents that are in the selection. pc
- You talked of properties -Ry and xR- in your exposition. Aren't those just like "is a member of x"? I assumed a property could make reference to a given row based on that.

- "Is a member of x" is probably not a property, since it is a nonce choice, but that aside, I think this is about right: the quantification is over the existents that are in the selection. pc

#### lo

This gadri selects a single row too, but this time we enter the table from a column heading. From all the objects that have a positive mark for that property, we select those that also have a positive mark for the property "is a kind".

If we are still not down to one row, we must appeal to context to identify the single row meant.

- Ahah! You are trying to fit your words into my ontology. I don't think, "is a kind" is going to be on the list, though I may be wrong. Objects aren't kinds but members of kinds. I would say that {lo broda} is a specified subset of the set of all things marked positively for "broda" (and not also negatively or indifferent, that is open to dispute).pc
- You said
*every distinction made in any language of the universe"! Why would you exclude "is a kind"?*

- You said

- No object is a kind. Maybe I should specify first-order predicates. pc

#### PA lo

Given the row x selected with lo, the quantification is over all the objects of the world that have a positive mark for the property "is an avatar of x".

- I'm really sure that being an avatar for x is not going to be a property, though being a broda will be -- and pick out the same things, as far as I can tell. pc
- What is this fascination with single lines? Most Lojban descriptions are of pluralities and only acciddentally of singles -- unlike logic.pc

- Singles are much easier to work with. I take it you don't admit groups as single rows, so either there won't be any row marked positive for "is a group of apples" (which contradicts one of your assumptions, that all marking possibilities are there) or the property "is a group of apples" will not be one of the columns, which means that not every distinction made in any language is there after all.
- I would say that "is a group of apples" does not make a distinction, mainly because nothing is a group of apples per se. They become a group by some action of something, typically a person. Back to basics: groups of every sort are just ways of talking about objects, not objects themselves. pc

- Singles are much easier to work with. I take it you don't admit groups as single rows, so either there won't be any row marked positive for "is a group of apples" (which contradicts one of your assumptions, that all marking possibilities are there) or the property "is a group of apples" will not be one of the columns, which means that not every distinction made in any language is there after all.

- That seems to be the key difference between our points of view. I treat {le ka remei} and {le ka plise} as just two properties. I don't know how to identify any subset of all properties as "first order". Given some broda, how do I know whether {le ka ce'u broda} is first-order or not?

The inner PA is taken to be a part of the property specification.

- I haven't said anything about inner quantifiers yet, but they would not be part of the property specification, I should think: indviduals are not numbered as properties.pc
- Not sure what you mean. I was talking of XS, where the inner pa is part of the property specification.

- I know that; my point is that I don't see how what you are trying to do with this "property specification" fits into this scheme at the moment. pc

I'm not sure how SS deals with {lo plise remei} vs. {re loi plise}. The two expressions would seem to deal with two completely disjoint groups of rows, right? In XS {lo re plise} and {lo pa plise remei} would both zoom in on the same single row.

- Well, they don't deal with rows much at all. {re lo plise} is two (actual)apples, considered distributively, {lo plise remei} is a much more complex notion, since its "members" are already sets. Briefly it turns out to be a selected set of (real or unreal) apples taken pairwise and distributively (with some messy details left out for the moment). pc
- Let me try again. In SS, {lo plise remei} selects from the rows that have "...plise remei" marked as positive, and {re loi plise} selects from the rows that have "...plise" marked as positive. The objects that have "...plise remei" marked (if there are any, which I suspect there might not be in your scheme) are all different objects than those that have "...plise" marked positive.

- Well, there aren't any such rows, of course, as those marked {plise remei}, so they are different from those marked {plise}. {plise remei} is a complex way of talking about apples -- pairwise. That aside, your explanations are gettting more coherent with the scheme I laid out. pc

- I have not worked out what would happen if second-order predicates were allowed. My first guess is that things would get more self contradictory. I am sure they would get more complicated, since it would have to be inherent in each second order predicate what was its relation to things with certain first-order predicates. The two sets of predicates would, of course, form separate object classes. I don't see any reason to do that and one good reason not to-- it short-circuits the project to figure out what expressions of Lojban (or whatever) mean.pc
- How can you tell whether a predicate is first-order or not? Can you use the expression {lo remei} at all? Does {lo} work differently with second-order predicates than with first-order predicates.

- I am not sure what the best thing is to do with internal quantifiers -- other than to make them optional. Right now I would take them as they are in CLL, cardinals of the underlying sets. So, {lo re plise} tells us that only two apples were selected for lo plise. {lo pa plise remei} then apparently means that only one apple 2-tuple was selected. That is close to the same thing, though might not be, depending on exactly how 2-tuple is defined (do the the members have to be distinct?).pc

On the whole, I think you have the scheme off by about 90 degrees. pc

Xorxes asks a good question: how to tell when a predicate is second-order. I suppose the short answer is that a second order predicate always involves the notion of a set in some way, while a first order one can be defined without that notion. But I am not sure that this distinction will always work in practice. Second ordeer predicates, more or less by definition, apply to pluralities directly, not either distributively or collectively or even cumulatively. And so on -- I don't know how to define 'em, but I know one when I see it (most of the time, I think, probably). In any case, Lojban predicates like {PAmei}, {cmima}, {klesi}, and {gunma} are clearly in, and maybe those like {bakfu}. These predicates are used as shortcuts to talk about objects.

- If I say {ro da su'o de zo'u da cmima de}, am I talking nonsense, given that there is no entity
*de*that can fit the bill? Or, are we allowed to quantify over non-entities such as sets?

Given that, Lojban expressions using these predicates are going to be about their members in remoter way than even regular gadri expressions. {klesi} and {gunma} say of a list of things that each of them is to be taken together with the others in a certain way: cumulatively or collectively. {le(')i broda cu mumei} says that there are 5 "brodas" being taken together. And so on. But what exactly a {-mei} or any otehr second order predicate means in a gadri expression takes a bit more work than I have given it. {le broda remei} ought to be a selection things taken pairwise -- but whether as masses or sets is unclear, as is what exactly to do with {le(')i broda remei} in a coherent system. There ar no end of useful things to say here for which these are potential expressions, but a pattern of justification or usage hs yet to emerge. the rpractical results of xorxes' system -- if not the theoretical base -- is one quite satisfactory possibility, but not the only one. pc

- Could you explain what practical problems arise in taking groups themselves as entities?
- This is not about practical problems except finding efficient ways to say certain things. That done, the hard problem is figuring out how this all converts back to logic or applications to the world. It is only in technical logical moves that taking groups as entities gives problems -- or at least I can't think of a practical one. But, as long as we have good systematic ways to deal with everything without doing a lot of violence to the underlying material -- or we can explain how the violence ultimately works -- then nothing seriously bad will happen. So, for example, "quantifying over" groups does not do any damage because we know how to pull that all back to basics: something is a groups just means that some objects are taken to act/be together is in a specified way. And, of course, for efficiency -- and often just because we don't care about the details or don't know them -- we can use all the locutions we always do. I am trying here to build them up as carefully as I can to be able to understand them going back down. I have a scheme laid out for explaining moves one step at a time and I would be delighted if used it -- or explained another one -- to tell me how your ideas work. I have been formulating a translation, but there are still gaps I don't understand well enough to fill. I'll read all XS and related material through again and then try a shot, but I would be delighted if you beat me to it, since yours will be right from the get-go.

You use a different convention for LAhEs than the one xorxes proposed sumti qualifiers (LAhE, NAhE+BO) here. I am trying to understand how this convention works, so what follows is all speculative.

I take it from what you say that:

lu'a le broda = lu'a lei broda = lu'a le'i broda = le broda

lu'o le broda = lu'o lei broda = lu'o le'i broda = lei broda

lu'i le broda = lu'i lei broda = lu'i le'i broda = le'i broda

- No, but only because the selection for {le}, {lei} and {le'i} are separate. It is true that lu'a le broda = le broda, lu'i le'i broda = le'i broda and lu'o lei broda = lei broda

- You and I are extremely bad at communicating with each other. Suppose I want to say something, a collective property, about a certain group of objects that I have in mind. Isn't it equivalent in this scheme to use {lei broda}, {lu'o le broda} or {lu'i le broda}, leaving aside the longwindedness of the latter two?
- Painfully true about the communicationg. As for the rest, if the selected group was referred to already by {le broda}, then the collective property would be assigned to {lu'o le broda}. {lei broda} is a new selection, not necessarily the same.

- You and I are extremely bad at communicating with each other. Suppose I want to say something, a collective property, about a certain group of objects that I have in mind. Isn't it equivalent in this scheme to use {lei broda}, {lu'o le broda} or {lu'i le broda}, leaving aside the longwindedness of the latter two?

And also:

lu'a lo broda = lu'a loi broda = lu'a lo'i broda

lu'o lo broda = lu'o loi broda = lu'o lo'i broda

lu'i lo broda = lu'i loi broda = lu'i lo'i broda

- This is wrong also, since lo broda is a selection, but loi broda and lo'i broda are all possible brodas.

- So the equivalence would be:

lu'a ro lo broda = lu'a loi broda = lu'a lo'i broda

lu'o ro lo broda = lu'o loi broda = lu'o lo'i broda

lu'i ro lo broda = lu'i loi broda = lu'i lo'i broda

- No, because {ro lo broda} is about existent brodas, {loi} and {loi} are about all possible ones.
- So {lu'a lo'i broda} is about existent and non-existent brodas?

- No, because {ro lo broda} is about existent brodas, {loi} and {loi} are about all possible ones.

- I can jump either way on this one, taking either {lu'a} in cleaar cases or treating it like a quantifier. At the moment -- becasue I don't like any more imaginary things that I have to have -- I go with it being only about existents, but I have not thought through the consequences of that choice (or of the other either).

but these differ from lo/loi/lo'i broda in that a quantifier on them quantifies subsets of lo'i broda. So:

- (and loi broda, but members of lo broda)

re lu'i ci lo plise = re lu'i ci loi plise = re lu'i ci lo'i plise

"two sets of three apples"

- I didn't talk about this is just yet: the quantifier over the innermost gadri changes the game. The identities are not meant to be literal, I assume, since there is no reason to think that the three apples picked in one case -- or the two sets used -- is the same as that in another case. You mean they each amount to the same thing, what you say. That said, I expect that the identities do go through, but I haven't checked.

re lu'a ci lo plise = re lu'a ci loi plise = re lu'a ci lo'i plise

"the members of two sets of three apples, distributively"

re lu'o ci lo plise = re lu'o ci loi plise = re lu'o ci lo'i plise

"the members of two sets of three apples, collectively"

It seems a bit wasteful, as only the outer LAhE makes a difference. The choice of inner LE is innocuous.

- It does make a difference without the middle quantifier. That they all collapse together is a feature of LAhE, which works more or less directly on the underlying objects -- wasteful, but coherent.
- And, of course, I am now not sure that they do collapse together. {ci lo plise} is some three apples, so {lu'a ci lo plise} is the distributive group of whatever are members of these three apples (one convention being, of course, just the apples themselves) and {re lu'a ci lo plise} is two of whatever those things are. {ci loi plise} is some three-membered submasses of the mass of all existent apples, {lu'a ci loi plise} is then those three apples taken distributively, and {re lu'a ci loi plise}is two of those apples taken distributively. The same result comes from

{lo'i}. So they are equivalent if the distributive group of an object is that object itself. Otherwise probably not. I am torn now between saying that that holds and saying that the notion of the member of an object is simply incoherent. I hope there is another possibility, since each of these seems to give some trouble.

Presumably the subsets are taken disributively? In other words, when I say something about {re lu'i ci lo plise}, I am claiming that there are two and only two sets of three apples each, such that for each of the sets... something holds. Is that right?

- I think so, yes.
- But now I don't. I think rather than {re lu'i ci lo plise} is a set of two apples: {ci lo plise} is three apples taken distributively, {lu'i ci lo plise} is those same three apples taken cumulatively, i.e., the set of those three apples, and {re lu'i ci lo plise} is a two-membered subset of that set.

What happens with these LAhEs on connected sumti? I would expect (if ko'a and ko'e are assigned to single individuals):

- (assigning anything to a single individual in Lojban is a little hard to do, but OK)
- {ko'a goi le pa plise}, for example. Or {ko'e goi li mu}

lu'i ko'a ce ko'e = lu'i ko'a joi ko'e = lu'i ko'a e ko'e = ko'a ce ko'e

lu'o ko'a ce ko'e = lu'o ko'a joi ko'e = lu'o ko'a e ko'e = ko'a joi ko'e

lu'a ko'a ce ko'e = lu'a ko'a joi ko'e = lu'a ko'a e ko'e = ko'a e ko'e

- The
*e*cases are unclear as always, especially if ko'e and ko'a really are individuals and so have no underlying sets. That aside, the equations look right.

And presumably we can generalise to:

lu'i ko'a (any connective) ko'e = ko'a ce ko'e

lu'o ko'a (any connective) ko'e = ko'a joi ko'e

lu'a ko'a (any connective) ko'e = ko'a e ko'e

Just as lu'i/lu'o/lu'a make the inside LE innocuous, they would also appear to annul any inside connective.

- Surely not with {a} or {enai} or..., even if {e} works as you imply (though I don't see why it should, rather than to give two {lu'V} phrases).

- I don't know. I'm just trying to follow the pattern: lo'i/ce, loi/joi, lo/.e, set, collective, distributive. It would certainly be odd that {lu'i ko'a na.enai ko'e} is {ko'a ce ko'e}, but I don't see what else it could be with this scheme. If {lu'a ko'a .e ko'e} gives {lu'a ko'a lu'u .e lu'a ko'e}, then {lu'a le re broda} should give something like {re lu'a le pa broda}.
- If we have a group, A, say, of two objects, B and C, the {le A} can be replaced with {B e C}
*salve veritate*,, and similarly with your other equations. But that is starting from the group and working downward. Now, {B ce C} names a group and so does {B joi C}, but {B e C} does not, as its expansion shows, so "applying a qualifier to it" (which can't actually be done but can be mimicked grammatically) yields uncertain results, since it is unclear what the members of an individual are. I suppose that the case of {loi} and {lo'i} with individuals yield the singletons. which, in the case of {loi} comes back to the individual, but not in the case of {lo'i}. For convenience, we could decree that {lu'o B} is just B itself again, but that is a convention, not a necessary consequent of what goes before.

- If we have a group, A, say, of two objects, B and C, the {le A} can be replaced with {B e C}

- I don't know. I'm just trying to follow the pattern: lo'i/ce, loi/joi, lo/.e, set, collective, distributive. It would certainly be odd that {lu'i ko'a na.enai ko'e} is {ko'a ce ko'e}, but I don't see what else it could be with this scheme. If {lu'a ko'a .e ko'e} gives {lu'a ko'a lu'u .e lu'a ko'e}, then {lu'a le re broda} should give something like {re lu'a le pa broda}.

- I take the claim that your "pattern" yields that {lu'i ko'a na.enai ko'e} amounts to {ko'a ce ko'e} as good evidence that the pattern is not the right one, though I can see no reason why the earlier examples you give, with {e} and {ce} and {joi} lead you to this in the first place.

I am still not at all clear on what happens with these LAhEs when they act on sumti based on "second order predicates", such as {selcmi}. Does {lu'a le selcmi} give {le selcmi} or {le cmima}?

- On the ground that {lu'a} blanks out on {le}, I assume the former. Why would it be the latter? {le sellcmi} names some sets distributively and {lu'a le selcmi} does the same for the same sets.
- I now (after what is below) see that this is based on two assumptions, one dubious and one clearly false. It is doubtful that {lu'a le broda}, where {le broda} refers to an individual, reduces to just {le broda}. It is false that {lu'a le broda} behaves the same when {le broda} refers to a group as when it refers to an individual.

- That's what I would expect too, but then:

Consider something like:

mi cuxna fi ko'a goi le'i plise

i ko'e goi le te go'i cu selcmi ci da

Are ko'a and ko'e assigned to the same thing? Or is lu'a ko'a = le plise and lu'a ko'e = le te cuxna?

- I don't follow. There are no {lu'a} here.
- Suppose that I use {lu'a ko'a} and/or {lu'a ko'e} in the next line. What do they mean?

- The members of ko'a or ko'e taken distributively: a selection of sets in this case.
- A selection of one set in the case of ko'e, and a selection of three apples in the case of ko'a, right?

- The members of ko'a or ko'e taken distributively: a selection of sets in this case.

- I am not following here. If they are the same, then there members must be the same. The way that the set is referred to should not make a difference at this level. I see that you have misread my previous note (and I therefore miswrote it):{ko'a} refers to a set as a selection of apples, {ko'e} refers to the same set as a selection from among the possible sets to occupy x3 on {cuxna}. But it is the same set and so has the same members. {le'i te cuxna} would refer to a set of sets that might occupy x3 of {cuxna} and might, indeed, be a singleton, whose only member was a set --le'i plise, say. But {le te cuxna} refers to the groups of sets distributively and thus, in this case, to the one set selected, le'i plise. I think I am beginning to see your point. If {lu'a le broda} reduces to {le broda}, doesn't this mean that for each member of le broda (and maybe calling them members is a part of the problem, too), lu'a it reduces to it, even if it is a set. I've said I don't see the necessity of this. If one of the referents of {lo broda} is an object, then lu'a it may be it, but if it is a set, then lu'a it is its members, taken distributively. And I don't see that as necessary either, though it makes pretty good sense. I suppose this means that I need to rethink whether {lu'a le broda} is the same as {le broda}, since that came out of thinking about le broda as both a set and its members distributively, and only the latter is correct. Thanks. So, I retract that bit (happily, I did not say who was suffering from the confusion).

- Does {le te go'i} actually pick what's in the third place of {cuxna} above or is it just saying that to fox me?

- {le te go'i} has a {le}, so I would expect it to behave like any other {le broda}. Use {le te cuxna} if you prefer. {le te go'i} picks le te cuxna, i.e. le'i plise. le te cuxna is le'i plise, but is lu'a le te cuxna lu'a le'i plise? It seems not, because lu'a le te cuxna is a set and lu'a le'i plise are apples.
- My point was just that {le broda} is not veridical, so could mean something totally different from what it says it means. I agree it rarely does, but in these kinds of disputes one gets a bit paranoid. Huh? If le te cuxna is le'i plise, then they must have the same members, else the equation is false. I see no reason to think the equation false, so they have the same members. Why is lu'a le te cuxna a set? le te cuxna is, so one expects its members not to be. I think there is a use-mention confusion here or something akin to one.

- {le te go'i} has a {le}, so I would expect it to behave like any other {le broda}. Use {le te cuxna} if you prefer. {le te go'i} picks le te cuxna, i.e. le'i plise. le te cuxna is le'i plise, but is lu'a le te cuxna lu'a le'i plise? It seems not, because lu'a le te cuxna is a set and lu'a le'i plise are apples.

- You said that {lu'a le broda} always reduces to {le broda}. I asked whether this holds even for second-order broda, and you seemed to confirm it. In this case, we have a second order predicate {te cuxna}. {le te cuxna} is a collection of sets taken distributively (a collection of just one set, in the example). {lu'a le te cuxna} gives the members of the collection, i.e. the set in question. {le'i plise}, on the other hand, is a collection of apples. {lu'a le'i plise} are the members, the apples, taken distributively. So {lu'a le'i plise} takes the apples distributively, {lu'a le te cuxna} takes the collections (the one collection in this case) distributively. {le'i plise cu du le te cuxna}, but {lu'a le'i plise na du le te cuxna}, we are comparing a set with a set in the first case, but apples with sets in the second case.
- Yourt point taken, although I question whether {cuxna} is actually a second order predicate: the reference to sets does not seem to be essential (but that may be because I went on to unpack the concept of choice).

- You said that {lu'a le broda} always reduces to {le broda}. I asked whether this holds even for second-order broda, and you seemed to confirm it. In this case, we have a second order predicate {te cuxna}. {le te cuxna} is a collection of sets taken distributively (a collection of just one set, in the example). {lu'a le te cuxna} gives the members of the collection, i.e. the set in question. {le'i plise}, on the other hand, is a collection of apples. {lu'a le'i plise} are the members, the apples, taken distributively. So {lu'a le'i plise} takes the apples distributively, {lu'a le te cuxna} takes the collections (the one collection in this case) distributively. {le'i plise cu du le te cuxna}, but {lu'a le'i plise na du le te cuxna}, we are comparing a set with a set in the first case, but apples with sets in the second case.

- Suppose it does, then that would be le'i plise, some set of apples, presumably the same throughout the context. In any case, {lu'a le'i plise} and so {lu'a ko'a} does not reduce to {le plise}.
- Is {ko'a du ko'e} true and {lu'a ko'a du lu'a ko'e} false? In other words, is {le'i plise cu du le te cuxna} true, and {lu'a le'i plise cu du lu'a le te cuxna} false?

- No in the way you seem to mean it. The laws of logic have not been repealed. But, of course, the {lu'a} equations suffer from the usual problem with identities and groups, namely that, unless the groups is a singleton (which is odd in its own right), the distributive form will be false. Each member of the group is identical with some member of the group (trivially) but no member of the groups is identical with every member of the group (also trivially). But, of course the sets on the groups and the masses, too, will be identical. Consider the corresponding problem with {le broda cu remei}.
- Sorry, I'm still confused. {le'i plise cu selcmi ci plise} is true, and so is {le te cuxna cu selcmi ci plise}. But whereas {mi citka lu'a le'i plise} makes sense, I eat the apples, {mi citka lu'a le te cuxna} makes no sense, I don't eat mathematical sets.

- No in the way you seem to mean it. The laws of logic have not been repealed. But, of course, the {lu'a} equations suffer from the usual problem with identities and groups, namely that, unless the groups is a singleton (which is odd in its own right), the distributive form will be false. Each member of the group is identical with some member of the group (trivially) but no member of the groups is identical with every member of the group (also trivially). But, of course the sets on the groups and the masses, too, will be identical. Consider the corresponding problem with {le broda cu remei}.

- Right. Nor have you said that you do: since {le te cuxna} means the same as {le'i plise}, {lu'a le te cuxna} means the same as {lu'a le'i plise} and my earlier muddled claim that {lu'a le broda} reduces to {le broda} is just wrong; it reduces to the lu'a of the selected "brodas" taken distributively. I think. Are there hideous results from this? And thanks again.
- I think there are hideous results. Sometimes we do want to quantify over sets of sets: Suppose there are {le ci te cuxna}, three sets of choices, each to be used under different circumstances. Why shouldn't we be able to access each of these sets, the same way that we access each apple in {le ci plise}. I don't see why lu'a/lu'o/lu'i should care what type of objects brodas are.

- Right. Nor have you said that you do: since {le te cuxna} means the same as {le'i plise}, {lu'a le te cuxna} means the same as {lu'a le'i plise} and my earlier muddled claim that {lu'a le broda} reduces to {le broda} is just wrong; it reduces to the lu'a of the selected "brodas" taken distributively. I think. Are there hideous results from this? And thanks again.

- (with as many *s as needed) I agree we want to do this and that we should be able to say it. I just don't (now at least) think that {lu'a} is going to be what we use to do it -- at least in this simplest form. We could make it that, of course, but then we would have to take longer forms to get from sets back to the members (from cumulative or collective to distributive) which seems to me a more common shift. But also, I don't quite see the problem, for {le ci plise} and {pa le ci plise} access the apples in any sense that seem relevant here. So why doesn't {le ci te cuxna} or {pa le ci te cuxna} get down to the sets? {lu'a} goes a step farther down, to the members of the sets, which is not, I gather what you mean by "access." {le ci te cuxna cu selcmi} means that this one is and that one is and the other one is something with members, which seems right. What more access can we ask? The example we have been talking about is odd only because the referent of {le te cuxna} is taken to be a single set, rather than several, as is usually the case.