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(From Lognet 95/2. Used with the permission of The Loglan Institute, Inc.)

Sets and Multiples

By James Cooke Brown

This is the first of two essays on the several ways of handling "collectivities" in Loglan. The second, which appeared in Lognet 96/1, will examine the difference between designating sets and designating masses.

Learning Loglan--especially learning how to use its logical machinery--is partly a matter of becoming aware of what Loglan gives us, in the way of communication tools, that the natural languages we learned at our mothers' knees do not. Regularly distinguishing between sets, multiples, and masses is part of that alien logical machinery. Therefore it is part of what most of us have still to learn about our demanding language. Natural languages do not make these distinctions very clearly. In them sets, multiples, and masses are sometimes all treated as "plurals"; sometimes sets and masses are treated as "singulars"; and almost always these three types of collective entities are used in indistinct and overlapping ways.

All of us, including myself, need to make a determined effort to refine our uses of L's precisely differentiated apparatus for designating sets, multiples, and masses. In this first essay we'll confine ourselves to distinguishing sets from multiples. Probably this pair is the most difficult of the three to learn how to distinguish. Doing so comfortably and automatically will not be easy. Our own natural languages positively mislead us at nearly every point in handling these "collectivities". Indeed, Loglan is felt to be "odd" precisely because it invites us to draw logical distinctions--such as that between e and ze--that our souls, soi crano, feel more comfortable without.

First, let's define our terms. By a "set" I mean a collection of two or more entities about which a speaker has something "collective" to say. (Warning: This is a linguistic definition, not a logical one. A logical one will come later.) Numerous "collective nouns" in E are used to designate sets: words like family, group, band, troop, flock, herd, collection, army, platoon, aggregation, class, category, and the word set itself. They allow us to designate collections of entities that are in some sense "acting together" or have some property "as a whole" that we wish to point out. (Even the pieces of a chess set "act together". If any piece is missing, then that "defective set" cannot be used for playing chess. Notice also that the property that complete chess sets have is not a property possessed by any of their members.)

Suppose, for example, we want to talk about a group of farmers who are raising a barn together. It is not true that any one of them is "raising a barn"; no one alone could do that. So we use the word group/grupa, a set-designating collective noun/predicate, to talk about what they're doing together. Suppose we're interested in a troop of monkeys moving through the rainforest canopy; we use the word troop/goigru ("go-group") to talk about their collective action. Suppose we are police and have found an odd assortment of objects in a suspect's pocket; we use the E words collection/assortment and, in L, spasei ("spatial-set") to talk about the oddness of the members of this set being found together. (Notice that no single item need be odd.) Suppose a company of soldiers attack a fort together; we use company in E, tergru ("third-group": third military group, that is, after squad and platoon) in L. Suppose a group of shareholders own a company together; we might use shareholder group or simply shareholders in E, parpongru ("part-owner-group") in L. No member of this set owns the company; but collectively the whole set of them do. Suppose the set we're interested in is a widely distributed class of animals who have some common feature or features--say horns, hair, live birth, split hooves, and double stomachs--that allow scientists to think about them "as a single entity", i.e., as a "species", and so to consider how, as the current manifestation of some "lineage", they and their ancestors have evolved. In this case we use E-words like species/genus/taxon/class/lineage or the related L-words speci/gensu/klesi to deal with such spatially and temporally scattered entities that are thus only thinkable as "acting together", but may nevertheless be treated as "collectivities" of organisms that are in some way "acting together", or "being acted on together"...for example, by a sequence of habitats.

Considered as a linguistic phenomenon, then--that is, as part of the machinery of human languages--the things we say are "members of sets" may sometimes be "concrete", as books or coins or antelopes are concrete, or "abstract", as prime numbers and regular polygons are abstract. (What we apparently mean by "abstract" here is "exists only in our minds". This is a strange sort of "existence"; but it is a very important kind to the members of our species. Dodecahedrons do not exist in nature; but they exist for us--or at least for some of us--"inside our heads". For we realize that even the physical models we make of dodecahedrons are only rough approximations of the "ideal" objects we can precisely define in every detail geometrically.) But whatever the members of some set may be--whether they are observable together in the reality of nature (like that troop of moneys) or only "thinkable" together in that strangely useful "unreality" that we call "our minds" (in which prime numbers, dodecahedrons, and the taxa of biology evidently reside)--those members must be at least "denumerable". That is, we must be able in principle to "count" them, which means being able to put them--either actually or by what Einstein called a "thought experiment"--into one-to-one relations with some first part of the natural number series, that is, with a part that starts with 1.

Now a "multiple" is quite a different sort of linguistic creature. But before we consider the ways in which multiples differ from sets, I must first confess that I have just invented this use of the E word multiple in order to talk about them with you. In fact, the set-multiple distinction has become clear in my own mind only in recent years. There are places in 1989's LI, for example, where I used the word set in ways that make it obvious to me now that I must have meant multiple; see p.262, second para, lines 3 and 13, of L1 for two instances of this confusing usage. But I did not have this use of the word multiple in my own idiolect then, and so could not distinguish clearly between the two ways in which I was then using the word set. Working with L itself in the meantime--and if working with L does nothing else, it is always a speech-clarifying experience--as well as with our two lodtua in recent months, has taught me the virtue of making the set-multiple distinction in E.

As for the set-multiple distinction itself, it has been implicit in L ever since the adoption of ze, for the ze/e distinction presupposes it. A quick look through our old documentation shows me that ze was adopted sometime between 1963 (the date of the first L dictionary, from which it was absent) and 1966 (the date of the 1st Edition of L1, in which ze was present in full panoply...although apparently not yet fully understood). The set-multiple distinction is, of course, strongly implied by the difference in the two designations A ze B (A and B jointly) and A, e B (A and B independently). So if the set-multiple distinction has not been implicit in L from the very beginning, it has at least been deeply buried in it for a very long time. That nobody has talked about this distinction--until this essay and the few memos to the Keugru that preceded it--is perhaps the result of nobody's having invented an E way of talking about it until very recently. (The shadow of Whorf is still long over our Loglandian lives!) Using set and multiple for these two former uses of the E word set is, I believe, one way of doing so now. So let's begin.

What is this old, implicit, but for a long time deeply buried distinction? Well, to begin with, multiples, like sets, are collectivities. That is, designating them in any language usually requires either a plural form (two men) or a connected one (Joe and Pete). Indeed, both sets and multiples come to our attention only when they have more than one member. (The sets that logicians talk about may of course have zero or one member; but that is apparently a technical feature of the abstract objects called "sets" with which mathematicians and logicians deal.) But our linguistic notions of sets and multiples are both related to the linguistic phenomenon of "plurality".

We have just seen that speakers designate collectivities as sets when they have something collective to say about their members. Speakers use multiples to talk about what are sometime the same collections of individuals when they do not have anything collective to say about them. In fact, when speakers use the designative machinery of their languages to designate multiples rather than sets, it is because they do not wish to treat their members as acting or being "together" in any particular way. What they do wish to say is the same thing about each member of that collectivity, speaking of each one of them as if it were a separate entity. It is to do this that they designate it as a multiple. Saying the same thing about each member of some collectivity is therefore the speech-occasion for designating multiples. As far as I know, it is the only reason we ever do it.

What is the point of such linguistic behavior? Well, by designating multiples, we can make numerous claims in short, single utterances that would otherwise be very long. How do the occasions for making multiple claims arise? Well, some collection of things or persons may come along--in our minds or on the ground--and the speaker may notice that all of its members have some common property: say they are all red-haired or that each of them is doing something fairly odd such as listening to his or her own private radio. The speaker may decide to point this out. So he says, for example, All ten of those fellows are red-haired! This is tantamount to saying That fellow is red-haired! ten times. If the speaker does not wish to say some identical thing about each member of a passing collectivity, then s will have no (current) use for the apparatus of s's language that allows s to designate multiples.

Let's consider a richer instance of this linguistic mechanism. (For that is exactly what it is: a linguistic mechanism. Like sets, multiples are not part of non-human nature. They are part of the mental apparatus we humans have developed as a species to talk, not only about nature but also about the things--like dodecahedrons and what we would do if we were kings--with which we populate our minds.) Suppose I see a large group of men strolling together along a street, and I notice that each member of the group is more than seven feet tall. I can of course treat them as a set and say they're walking along together. That's what the set machinery of language is for. But my noticing their extraordinarily uniform great height also gives me an occasion for using the multiple-designating apparatus of my language. So I say to my friend (who may be looking the other way), Those men over there are more than 7 feet tall! What do I mean by this? Do I mean that the set of men is more than 7 feet tall? Suppose there are 25 of these giants. Then if they were to do something truly collective, like stand on one another's shoulders, this set of men would be far more than "7 feet tall". Their collective tallness would, in that case, equal or exceed 175 feet! So when I say "they" are red-haired or 7 feet tall, I am clearly not talking about them as a set; I am talking about each of them separately, but in a "multiple" way. Therefore I must be thinking of them as a multiple. For what I have observed about "them" applies to each one of them separately and not to the set as a whole. The redness of their separate heads or the tallness of their separate bodies implies nothing at all about what they may be doing or being together (like playing basketball, or standing on one another's shoulders, or raising a barn). So when I say Those men over there, what I evidently mean is Each of those men over there. Saying All of those men is another way of making sure that the listener understands that I am talking about multiples and not about sets.

But very often we E-speakers don't bother to do that. We leave off both Each of and All of and just say Those men over there; and we apparently expect our auditors to understand that our use of the plural means that we have something to report about each of them. But the social context in which the remark occurs counts very heavily. If we said of the 25 farmers: Those men over there are raising a barn! our listeners would know that we were not talking about a multiple but a set. Just so, in the case of the 25 very tall men, my friend will be able to tell that I mean to designate a multiple and not a set because he will see that each man in the group I am looking at is more than 7 feet tall. That would be a pretty amazing spectacle, I think you would agree. Perhaps a convention of American basketball players is in town. But in relying on my auditor to complete my sentence for me--in a sense, mentally to insert Each of' into my remark--I am relieving myself of the necessity (to be sure, it is never really a necessity in E) to say what I mean.

So this is what we use multiples for, in speech. We use them to say multiple things about several or many different individuals simultaneously. We do this whenever they have some common property or are engaged in some common behavior which we want to "sum up". We say 55 million Americans smoke cigarettes. In this case the multiple-designating machinery--rough as it is in English--provides us with a very compact way of making a very large number of claims briefly. This is the basic reason why we designate multiples. It is a hyper-efficient speech-tool, a way of saying a multitude of things fast.

Think of what I would have to say to my friend--let's say he has poor eyesight--to convey to him the astonishing fact that twenty-five 7-footers have just walked past if I could not designate them as a multiple...if English did not have the facility I'm now going to call "multiple designation". Using singular designation I would have to say That man over there is at least 7 feet tall while pointing at one of them, and then repeat that claim--while making sure that I was pointing at a different man each time--twenty-five different times! Saying it once with a multiple designation is not quite as economical as not having to say A certain American smokes cigarettes 55 million times, but it's quite an advantage even so. One can easily see how, in the evolution of language, the machinery of multiple designation arose.

Let's examine the difference between designating sets and designating multiples a little more closely. (Notice that I didn't say "the difference between sets and multiples"; that is a much more elusive matter that I'm not talking about here. What I am talking about are designations, not designata...not about the things designations designate, but the designations themselves. The set-multiple distinction can be drawn very clearly between designations; it is not clear how it can be drawn at all between designata!) As I indicated above, we only designate a set when we have something collective to say about its members...something that might not be true, but could be, of its members taken separately. Suppose Jack and Jill own a house together. It would then be true that the pair of them (a 2-set) own that house; that is, they, as a single 2-person entity, own it. If that is true, then it is interestingly not true that each of them owns it separately; for under any reasonable law of ownership, there can't be two separate owners of the same property! What is true of them separately is that each of them is "a part-owner" of some house, namely their house. But notice if we say this latter thing about Jack and Jill, we are treating them as a multiple. Earlier, we had been treating the same two people as a set. So which we do, on any given speech occasion, depends on our purposes as speakers as well as on how the world is arranged. (Have joint-stock companies been invented yet? Has community property been invented? The second question is, of course, likely to be unnecessary, as "community property"--at least in the sense of something belonging to a whole village or tribe--is probably the oldest form of human property.) If the human world of property is arranged in one way, then part-owners will exist. If in another, then only sole-owners will exist. In the former case, we will have many occasions to designate sets to talk about ownership. In the latter, we would have to designate multiples if we wanted to talk about more than one sole-owner at a time.

Let's look at an example of the use of multiples in a world in which all owners are sole-owners. In this world, Jack is the owner of a house, and Jill is the owner of another house, and their friend Sally is the owner of still a third house. Saying Jack, Jill, and Sally own their own houses says this complex thing compactly. It does so by designating one multiple with the connected phrase Jack, Jill, and Sally and another multiple with the plural noun phrase their own houses. Notice that the connected phrase does designate a multiple and not a set. How do we know that? Because it allows us to say the same thing about each member of the designated triple; namely that each is the sole owner of a house.

Interestingly enough, we could say the same thing with what our Lodtua believes are ordered sets. (See Randall Holmes on "The Logic of Respectively" in LN 94/3:24.) We could say Jack, Jill, and Sally, in that order, own the 1st, 2nd, and 3rd houses, respectively, on this street. According to Randall, we are now talking about what had been a multiple as if it were a set, in fact as an "ordered set". For we are saying of the members of this set "collectively" that they can be put in a certain order, and then in a one-to-one relation with the members of a second ordered set, namely the first three houses on this street. Notice that it is not true that each member of either set can be put in this one-to-one relation.

So we can handle the same facts with either sets or multiples, the most important distinction between them being that, when we treat them as multiples, it will always be to say exactly the same thing about each of their members; and that when we treat them as sets, it will always be because there is some sense in which their members are participating together in some joint act or condition. (Randall's treatment of this matter last December made me just a bit uneasy because, in fact, 3 separate claims are generable from A,B,C are married to X,Y,Z, respectively, namely (1) A is married to X; (2) B is married to Y; and (3) C is married to Z. This makes these two groupings "feel" suspiciously like multiples to me and not sets. Perhaps there are two "ordered multiples" here instead of two "ordered sets"! These usages are very subtle, and it may take some time before we have these matters absolutely clear in even our loglaform heads.)

Features may be shared by the members of sets, of course. When the sets are "classes", this is always true of them: for classes are always classes of something, and this gives their members a shared property, namely whatever distinguishes that something. But what we then say about sets is never distributed over their entire memberships, as is always true of our talk about multiples. Whatever is true of a set--for example, its size, its inclusion in other sets, or whatever its members are doing jointly (such as raising barns or standing on each other's shoulders) or being as a whole (such as a useful set of chessmen)--must always be true of a single entity, namely that set "taken as a whole". In fact, logicians have for years used the properties "distributive" and "nondistributive" to distinguish the operations we perform on multiples from those we perform on sets. (This distinction is, in another guise, the set-multiple distinction.)

We are now ready to consider the designative apparatus with which Loglan provides its speakers and auditors for making the set-multiple distinction. Here is a list of examples of L's "multiple designators" at work: A, e B, e C; Le 25 mrenu; Neni le 25 mrenu; Neni mrenu; Ro mrenu; Ra mrenu. Each of these expressions will "distribute" some claim over the entire membership of each of the multiples so designated. Thus A, e B, e C will make 3 separate claims out of any predicate that follows it (for example, A, e B, e C preda "unwinds"--may be interpreted as claiming--A preda, ice B preda, ice C preda); Le 25 mrenu will generate 25 similar claims; Neni le 25 mrenu will generate 10 claims; Ro mrenu, many claims; and finally Ra mrenu stands for that inconceivably large number of claims that we could match one-to-one with the adult human males that exist now, have ever existed, or will ever exist...if we could ever count them. An impossible enterprise, to be sure; but obviously if we could do it, it would yield a very large number of claims indeed. All these expressions designate, and do so plainly in L, multiples. There is no way that an attentive user of L--whether a speaker or an auditor--could reasonably confuse the multiple designata of these designations with the designata of sets, which are always single entities and known to be so.

Here is a corresponding list of "set designators" at work: A ze B ze C; Lau A, B, C (lua); Lou A, B, C (luo); Leu 25 mrenu; Leu 10 leu 25 mrenu; Lea mrenu. Each of these arguments will generate exactly one "undistributed" claim about some "plural" group of objects taken together, that is, a claim that will not apply (except as it may do so accidentally) to the members of the designated sets taken individually. The designata of A ze B ze C may be carrying a heavy log together that no pair of them could manage alone; they are a log-carrying team or set. The set {A,B,C} (which is what Lau A, B, C designates) may be performing a musical trio, a composition that no pair of them could perform; and the ordered set (A,B,C) (which is what Lou A, B, C designates) may be a way of identifying the route along which a planned trip from Omaha to Louisville will pass. Let's say that it does so by designating, in order, the 3 major cities along the way. Let's say that A is Des Moines, that B is Peoria, and that C is Indianapolis. What is important to note about the claim that could be made about this ordered set by designating it with Lou A, B, C is not true of either Des Moines, Peoria, or Indianapolis taken separately. That is, Peoria is not "the planned route from Omaha to Louisville"; nor is Des Moines or Indianapolis. Peoria, like each of the other two cities, is simply on that route. Of course, we could if we chose, turn A, B, C into a multiple--that is, distribute a common claim over all of three of its members--simply by saying that each of them is "on the route that someone has planned to take from Omaha to Louisville." And it is here that an "ordered multiple"--a designative feature of the language we do not (yet) have--would, if we had it, come into play.

We come now to one of the most important points about sets and multiples as designated in L. Happily, it is a very simple point. For we may now note that ze is the "set-analog" of the e we sometimes use to form designations of multiples; that leu is the set-analog of le (leu preda means the set I have in mind whose members are apparently all predas); lea is the set-analog of ra (for lea preda means the set of all the predas there are); and that lau...lua and lou...luo have no analogs among the multiple-designating devices of Loglan, and probably should have.

I admit that this is tricky linguistic machinery. Our native languages--English, for most of us--give us very few clues about how to make the set-multiple distinction correctly. In E we say John and Jane own their own home; but we also say John and Jane are Republicans. In the first sentence, John and Jane are treated as a set; in the second, they are treated as a multiple. But the same E expression suffices for both. In the one case, John and Jane designates the pair of persons who jointly own some house, and in the second, the same phrase designates a multiple of persons, each of whom is--independently, I'll assume--a Republican. In L we not only can make the distinction that E does not make here, we are somehow obliged to do so. We must use ze in translating the first case, and e in translating the second, if we wish to be understood. If we fail to do this, we may end up saying what we do not mean. For, unlike E, whatever we do say in L will mean one thing or the other, but not both. Similarly, if I wish to translate Those 25 men over there (are doing something) into L, I must first decide whether what they are doing is a collective action, like raising a barn, or 25 independent actions, like eating 25 separate hamburgers. In the first case we would say that "Leu 25 mrenu" raised the barn; in the second. that "Le 25 mrenu" ate the hamburgers. It is (unfortunately?) impossible to say one thing in L and expect to be understood as having meant the other...unless, of course, your auditor is as deaf as you are to the distinction you have failed to make, soi crano. Then, of course, by luck or charity--as in E!--you may expect to occasionally get your message across!

Let me end this essay with an acknowledgement that logicians apparently do not use the word set in the way that I have been using it here...indeed, in the way I have been using it for nearly 40 years to talk about one kind of "plurality" in L, namely the collective kind. To logicians, a "set" is an abstract object "attached"--rather, I gather from Randall, as a label is attached to a jar--to its members when these are considered "timelessly" (this I gather from Emerson). From this definition, two rather startling consequences follow: first, sets can be "empty"; that is, this label-like entity can be attached to no elements at all. When this happens, the set--that entity itself--is called "the empty set". Like zero, the empty set is a very useful entity to mathematicians; it makes certain manipulations possible that would not be possible without it. Second, because they are label-like abstract entities, sets cannot do anything. Certainly, as Randall says, "Sets cannot carry logs." Indeed they cannot do or be anything together except include or be included by other sets, and suffer other formal operations involving identity relations among their members.

I understand this. Set theory was, after all, first developed by mathematicians and philosophers of mathematics. Frege and Russell come to mind. And they built this formal theory largely to serve as a tool for exploring the foundations of mathematics, principally arithmetic. But apparently in arriving at this formal conception of sets, the builders of set theory left the uses of the set words and expressions of ordinary language--where, at least for humans, it all began--off to one side. For, as I have tried to show in this essay, it is the very essence of a set when used as a linguistic tool that its members be seen as "being or doing something together". Indeed, that is the occasion for creating sets in the first place. We wish to talk about a couple who own a house together, or about a string of lakes that together establish a canoe route, or about a set of chessmen which, when complete, allows a game of chess to be played. All these are linguistically-created sets, and of course some of them do carry logs. If logical sets cannot do any of these things, then logical sets have moved very far away from the linguistic phenomena by which a kind of "mental net" (not a "label") is thrown over a group of individuals so that they may be considered "as a whole". (But not uselessly far away, note, for numbers, too, have moved similarly far from their observable prototypes in the natural world: the numerosities of things like herds, litters, and eggs.) But note that the linguistic set is ethologically more fundamental than the logical one. For it is the linguistic maneuver that not only enables us to talk about what the members of sets are doing or being together, but it is probably what created our human sense of collectivity in the first place.

Thus, in my own idiolect of E, the word set expresses an evolutionary notion. I believe that as the planet's first speakers, we humans created--and still create--sets by designating the groups of individuals we come upon that were acting, or being acted on, together. We reified these collectivities as single entities precisely by those speech-acts. They did not exist as "things" before we speakers spoke of them. And we do this now whenever we wish to talk about the joint states or actions of similar collections of individuals. Sets, in short, are our inventions, not features of the natural world. We invent them over and over again in order to speak of the collective, wholistic, or affiliative features of the phenomena we observe: fish schooling, troops of monkeys scurrying through treetops, pairs of men carrying logs, species evolving. Therefore of course sets scurry, carry logs, and evolve. For if a pair can carry a log, and a pair is a kind of set, namely a 2-member set, then some sets carry logs.

Next time I'll deal with masses, and with the way masses may be used in L to designate another kind of abstract object, namely numbers. --JCB

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