(This is my opinionated memory of how things went and where they ended up. I am open to factual corrections and will listen to other (wrongheaded) opinions.)
Logjam (Loglan up to about 1984 and Lojban) has basically three articles: la, le and lo (there are others out there, some of which arose from the concerns that follow, but none of which have much practical significance in either language).
The easiest is 'la'. It turns whatever follows it (assuming solutions to RHE problems) into a name, a purely designating expression, one that refers to one "thing", however that plays out. If what follows 'la' is a name form (ends in a consonant), this is obvious; if what follows is a legitimate Logjam expression (or close to it or whatever), then whatever content it may appear to have is to be disregarded: the referent of the name is not subject to the expression's supposed message applying to it. As a historical example, Afraid-of-horses was not afraid of horses or much else, on the contrary, his enemies were afraid of even just his horses (the English loses some of the point of the original Lakota). But even the latter point is not relevant to the applicability of the name; his son had the same name but was of a very different character. The rules for 'la' has scarcely changed throughout the history of Logjam (though, early on, everything had to be whipped into name form).
The second article, the one often associated with "the", is 'le.' This derives ultimately from the definite description operator in formal logic: ixFx, "the one and only thing that is F". This expression, once introduced into formal logic (Peano, I think, but Frege had something like it , too), caused a problem (Russell to raise and "solve"): what if there is no "one and only thing that is F" and the expression therefore does not designate anything by virtue of its internal structure? What if there are no Fs, or more than one? Russell solved the problem by claiming that the expression wasn't there at all, but was, when appearing as an argument to a predicate, GixFx, part of a short hand for a much longer expression, which asserted that there is exactly one F and it is G. So, if there isn't such a thing, for whatever reason, the simple sentence is, when expanded, simply false, as false as it would be if there were such a thing but it wasn't G. Others took other views, mainly insisting that the expression was a real expression, not a shorthand and really referred to what it appeared to do when that reference was legitimate. But when reference failed? Some said that sentences involving such expressions were simply not truth valued or had truth values other than True and False (logicians will take any excuse to go to a non-standard logic, which are much more fun than the usual one). Others said that the expression always referred to an existing object -- the unique F if there was one, the null object, ix x=/=x if not. Since the identity of the null object is in the semantics and might be anything at all, the syntactic problem was solved. But some went further in positing that the expression was simply referring, without any necessary connection to the property mentioned in it. To be sure, if there are Fs it might be a good idea usually to have it refer to one of them, but even that was not a requirement. That screwed up some of the standard theorems, of course, but had advantages in other areas. Logjam 'le' comes out of this last position, with modifications: 1) the identity of the referent was not put off to the semantics but had to be established (at least in the speaker's mind) at the time of use, and 2) the referent could be plural. The second modification raised problems within logic (that were eventually assuaged by plural quantification or a better understanding of L-sets) and the corresponding problems in Logjam, which were generally solved either by ignoring them (which turned out to be the right way to go) or by generating a number of half-hearted compromises, most of which ended up in new articles (for sets, for example, and masses, and mushes, and Lord knows what all else), and slopped over into 'lo'.
Probably, 'lo' started life closely parallel to 'le', differing in that it was nonspecific (or indefinite or whatever) and that it was "veridical", that is, the referent of 'lo broda' had to be broda. The second feature follows from the first, since, if you don't have a particular thing in mind, the only way to get to what you want is through what it is -- unlike the 'le' case, where, since you have a particular in mind, anything that gets a person to that thing is OK (though calling broda broda is generally a good way of doing that, but not always, as witness then do of The Crying Game or an early Mike Hammer story). Somehow this notion did not catch on, or rather, other notions came into play. Without rehearsing the arguments involved, 'lo broda' was variously explained as:
A.all the broda, taken disjunctively: that is, if even a single lion lives in Africa then lions live in Africa (the stock example). This, of course, raised the question of how to say all -- or some group of -- broda did something conjunctively, working together (the stock case here is boys carrying a piano). So
B. a mass of broda acting conjunctively. But then how say they acted disjunctively? Eventually, this generated a new article. But then
C. the substance of which broda are made (stock example is "There's dog on the windshield, the tires and all down the left side" after hitting a large dog with a car). This arose from the notion of a mass as a product of a blender, which might also act in broda shaped units under certain conditions, which proved hard to define.Whence
D. brodahood, which is indefinitely related to the sense of 'broda' (and comes somehow from Quine's remark about the meaning of 'gavagai'). This solves the problem of there being no broda, for brodahood is presumably always around. On the other hand, it (like the others above) make predicates mean something other than they mean or involve some other way to get from a warehouse of broda to the individual broda that predicates apply to. This was done by hiding the mechanism as much as possible, using default quantifiers and default expressions after the quantifiers, too. So, in 'lo broda cu brode' either 'brode' is not a property of broda but rather of the set/mass/goo/hood of broda (one which the set has, presumably, by virtue of individual broda having some other property not mentioned) or 'lo broda' does not really refer to the presented abstract object, but to some unspecified member/participant/chunk/avatar of that object. Not very transparent for a logical language.
Lojban inherited this muddle, went through most of the same discussion again and finally settled on basically choice A, with bits of the rest creeping in as needed. So, 'lo broda cu brode' probably really means "some member of lo broda is brode" Or "members", of course, but that raises another issue, namely that what you get out of a set this way are not individuals, but a subset, which, being a set, presents the same problem anew. Enter xorxes with the reasonable suggestion that 'lo' be just like 'le' except indefinite (or ...) and hence veridical. We are still dealing with sets though, so the problem above arise still -- and using the empty set (which is always available) when there are not broda has some undesirable consequences. So there emerged (again) Mr. Broda, who combined somehow (hours spent trying to figure out how) the best of the indefinite set with the best of -hood, to cover all eventualities. And then (ta-DA!) in the midst of all this, xorxes found a book about plural quantification and plural reference, where reference and assignment were not a function from a referring expression to an object, but rather a relation between an expression and objects (one or many). Of course, such a change from standard logic was anathema to tired old logicians who had lived with Cantor sets for years; it looked not to be a consistent, coherent logical system at all. But then it turned out to be exactly Lesniewski's mereology (or Quine-Goodman-Leonard's part-whole logic), a known -- but previously unintelligible -- system with all its metalinguistic credentials in place (and now intelligible). So the problem brought on by always needing a single referent and that being something like a set disappeared. The referent could be several things and, because of the way the logic works, these could individually or collectively do various things, without hidden mechanisms (collective action is the norm, independent action is indicated by external quantifiers).
So, 'lo broda' refers broda, one or several or many, not specified beyond what is given in the sentences in which the expression occurs. These broda take on properties individually or collectively as indicated by predicates which can take individuals separately or collectively as arguments. All the various problems which arose around 'lo' (and occasionally 'le', since the same problems occur there) disappear.
Except when there is no broda. Plural reference or Lesniewski sets don't have an empty set to fall back on. Sometimes, bits of Mr. Broda have emerged to deal with this, but the basic solution seems to have been in defining the appropriate universe of discourse (what things there are available to be referred to in a given context). When last we went round on this, xorxes seemed to be holding a view on this that led to some bizarre (to me, with my ingrained habits) consequences. But there is another approach which avoids these consequences but still yields all the right results.