species

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This is about using species to give a coherent explanation for a set of changes proposed for the gadri {lo}. I do not believe these changes are either needed or useful for anything in the language (with the possible exception of contrasting {lo} with {su’o} as general and specific “a”) but, if the changes are to be made, I believe they should be based on something reasonably intelligible. Here then is a gleaning from a variety of species theories that will account for generality, non-existent objects, and opaque contexts (I don’t consider that appearing to relieve speakers of the need to learn how to deal with quantifiers is a legitimate goal). As noted, these points are drawn from a number of theories with varying terminologies. The pieces have been put together to do what is wanted, but the terminology is non-standard (because there is no standard).

A species is a node on a net of concepts, possibly more complex than the Jeweled Net of Indra (jijimuge -- by which everything is connected to everything else – but only causally). It may be viewed usefully as having two aspects, being on two subnets at once, a factual and a conceptual (extensional and intensional, to sound more official and be more confusing). The factual side is (represented by) a set – all the real things that fall under (are specimens of) the species. For lo broda (I am using this form to refer to a species in deference to the point of all this), that set is just lo’i broda (or lo’i ro broda, if another proposed change goes through). The conceptual side is a property, le ka ce’u broda, in the above case (the {le} recognizes that there may be several properties going with the word {broda}, but I want the one I have in mind -- it is not meant to be a proposal for the issue of what gadri – if any – is proper with abstracts).

In keeping with the perceived dual character of a species, it enters into what are taken to be two more or less separate groups of relations to other species and to ordinary objects. On the extensional side are relations with species that we can call (without too much misleading) “intersection,” “mingling,” and “inclusion” and one with objects called “specimen.” The intensional side has (more or less correspondingly) “overlap” and “pervasion” for other species and “locus” for objects. All of these relations are fuzzy; that is, whether a relation holds between two appropriate objects is a matter of more or less, leading implicitly to a scale from 0 to 1 – clearly does not hold to clearly does. For the eventual use of these relations in language, each situation divides these continua into two halves – yes and no. Where the break comes may be different in different contexts but is always fixed for the context (one advantage of this is that we can explain {le} by saying it is a context in which the specimen relation breaks at a much lower point than usual -- it would be odd to use {le broda} for something that is not at all like a broda).

The relations of the two groups are related to one another in a variety of ways: if two species mingle, they also overlap, but not necessarily conversely (bnnc). If a species pervades another, that other is included in the first bnnc. If an object is a specimen of a species, it is a locus of that species, bnnc. If one species intersects another, the two also mingle, bnnc. If two species mingle there is an object that is a specimen of both—and, in this case, conversely. One species may overlap both another species and its complement and also mingle with both. However, if one species pervades another, that other does not overlap the complement of the pervader. And similarly for the includer and mingling (and thus intersection). Generally, the fuzzy values of specimen and locus are proportionate, if not the same. The value for mingling is proportionate to those of the specimen in the two; break point is always such that any thing that passes as a specimen of both species will guarantee that the mingle relation holds. Mingling and overlap are symmetric relations. Intersection, on the other hand, is asymmetric, as are the others except mingling. Intersection is the most complicated of these relations: its fuzzy value is based on the values of the specimens of the intersecting species that are also in the intersected, roughly the sum of their discounted values (the discount varying with circumstances, of course). The break point is also most sensitive to circumstances. Note, all this talk about values is purely theory; it allows for the actual patterns but does not permit actually doing any of the calculations.

It is useful to note that every individual (in the outer domain yet) is the sole focus (and of degree 1) of an infima species. Thus, focus value in higher species is a direct function of the overlap of the infima species with that genus (i.e., covaries with: a thing overlapping higher than another as infima species does so also as focus). "Genus” here means just “species higher than another mentioned in association with it. If the object happens also to be in the inner domain, its value as a specimen is related directly to its value as a locus and thus with the overlap of its infima species.

Putting the mechanics all together, we get that a particular broda is a brode just in case its value as specimen for each species is at or above the break point for that species. The asymmetry here is just that an object does not even get questioned about lo brode unless it first has past the test for lo broda. This one case is enough to support the claim that lo broda and lo brode mingle and thus also that lo broda overlaps lo brode. It also gives some small summand to the question whether lo broda intersects lo brode. What that summand is is a direct function of the value of it as a specimen and the details depend upon what lo brode is and in particular what the overlap value of lo broda is to lo brode and on the size of lo’i broda. When this sum reaches the point where mingling become intersection depends upon external circumstances that surround the occasion of the claim being uttered. Note, the theory here has at least three points of wiggle room to accommodate usage.

So far the species involved have been treated as though 1) they were monadic, reaching out to only one other species at a time as a source for mingling and 2) that connection with other species involved only the extensional side of the species, only specimens. For probably the majority of species, the second claim is true, but the exceptions form particularly interesting clusters that need special attention. The first restriction is generally not the case – most predicates (to get back to the language for a moment) are not monadic. In the theory, however, polyadic predicates are reduced to monadic ones when considering the relation between two species, by holding the other places constant for determining the specimen value for the place of concern. With particular fillers for the other places, each object can be given a unique value. If the other places are filled by ranges of objects (either quantification or by assigning them to particular species) the value of specimenity can be calculated (very much in theory) from those particular cases as they are similarly categorized (as a quick rule of thumb, we could take either the maximum or minimum appropriate value and save some time in précising the results; we can compensate at other wiggle points as needed). Thus this restriction can be taken as not being a restriction.

Restriction 2, however, does take us into new territory. All that has been said so far could be done just talking about extensional aspects of a species, just about (fuzzy) sets. The parts that so far depend upon locus and the related higher relations could be dropped and replaced by arbitrary evaluations of specimen and the like directly. The more expanded species, those that relate directly to intensional features, require species (if everything is to be handled by a single conceptual object type). One unfortunate feature of Lojban (and every other language I know of or even have heard of) is that such predicates are not specially marked but most be learned individually along with their various definitions. Although they tend to fall into groups that are generally the same, even within these groups most members have some peculiar features and no set of groups will contain all the anomalous predicates. All they have in common is the need at some point to refer to the property side of a species to determine something that is properly on the set side (whether a predicate applies to an object remains always an extensional matter). And this dependence is not buried in the values of specimen and intersection, where they are even in the best cases, but are overt in the definition of the species itself.

To begin the discussion, consider the ultimate non-problematic predicate, {zasti}. An object is a specimen of lo zasti just in case it is a specimen of any other species (the possibility that the only species it is a specimen of is lo zasti we leave in for theologians to play with). That is, its infima species mingles with at least one species (ahah, there will always be another species, since lo zasti is not ultimate -- or is it theologically?). There are an array of predicates that are defined in terms of not existing, i.e., of infima species not mingling with lo zasti (at least – and one other?): {xanri}, and its derivatives mainly for now. In later times we will be able to distinguish other notions by what sorts of species it does overlap “mentioned in a book,” say, or “acts in Joey’s worst nightmares.” {xanri} also throws the rest of its sentence into intensional mode: what comes after the word is about locus and overlap, not specimen and intersection. So, {lo pavyseljirna cu xanri danlu} says not only that there are no unicorns but that lo pavyseljirna overlaps lo danlu (indeed, lo danlu pervades it, but that is not said here) and {la bellerophon cu xanri listigni gi’e kavbu la pegasus} says not only that Bellerophon is imaginary but that his imfima species is represented in a story and that it overlaps lo kavbu be la pegasus. One large class of words of this sort is that of words about, loosely, mental events: knowing, talking, thinking dreaming, and so on. These officially take abstracts as their objects and abstracts are inherently intensional – events exist even if the participants do not, for example. Up until now, people who would have someone talking, for example, about a nonexistent had either to use an abstraction in which the non-existent is referred to or use a short form with {tu’a} and the direct referent to the non-existent. People who forgot their {tu’a] were strictly speaking either saying false things or at best misleading ones (that a nail is a kind of abstraction, for example). Now we can say that what we talk about is a species (its intensional part directly) and so, whether that species is an abstraction or an infima species or something between is irrelevant. So the {tu’a} can legitimately be dropped in these cases. It an also be dropped in other case where it was required. What one needs, for example, is a focus of some species that also happens to be available (including existent), a relation between oneself and a few species, not between oneself and some specimen. And there are countless other examples, which will emerge as the need arises.

Finally – before applying all this to actual sentences – a word about logical relations, in particular negation, generalization, instantiation, Leibniz’s law and movement.

Negation. Species are virtually a stone wall to negation. The fact that lo broda does not mingle with lo brode does not mean that it mingles with the complement of lo brode, for it may have no specimens at all. Similarly, the fact that lo broda does not intersect with lo brode – even if it does have specimens – does not mean that it intersects with the complement, for neither side may have made it up to the break point for either species (the rules require that zero specimens mingling cannot be intersection and all specimens mingling is intersection, but does not limit the choices in between). In the opposite direction, the fact that lo broda mingles or intersects with the complement of lo brode does not mean that it may not do so with le brode itself; the break point may be below 0.5. So, negation passes through species talk in neither direction.

Nor can species talk be generalized nor instantiated to if quantifiers are to be limited to existents. You can argue for the move in many particular cases, but the intensional cases are not grammatically distinct from the extensional ones, so the move from {lo broda} to {su’o broda} or even {su’o da} does not work. Nor does the move from {ro broda} or even {ro da}, for much the same reason: the species may be functioning as representing a locus, not a specimen at that point. On the other hand, so long as the specification of the species does not refer essentially to the quantified term outside it, references to species can apparently move about quite freely relative to quantifiers. To be sure, this condition is rare – the quantifiers and species reference are where they are for a reason.

Those intensional cases shoot down Leibniz’s law as well. Since they involve the intensional side of a species it may well be that the extensional side of two species are identical but the intensional side different (else why would they be two species rather than one twice?) and so the identity that would justify the replacement does not in fact hold.

And all of this does what? Well, assuming that we use {lo broda} to talk about the species Broda, then strictly speaking {lo broda} ought to occur only in sentences involving specimens, intersections, minglings, inclusions, loci, overlaps and perversions (for none of which concepts does Lojban have an adequate expression at the moment). At best, every interest in the species would have to be replaced by the corresponding comments about the set and the property. But no one wants to talk that way; they want to talk as much as possible about things. Soooo, we devise a convenient way to talk about species that looks and acts (until disaster strikes) like thing talk. First, quantifiers apply only to existents; an unnegated quantifier on a species that has no specimens automatically generates falsehood. Secondly, sentences involving unquantified occurrences of {lo brode} refer to lo broda and, in particular, when the selbri are not species talk, to the intersection of the subject {lo} term and the selbri’s species -- except, of course, in those cases which call for overlap or locus instead. Assuming that intersection can be fiddled with as needed to reproduce the intuitive notion of generality (as it seems that it can), then this trick allows us to say (albeit not literally) all the things that the proposers want for their {lo} using it. And a whole lot more besides (if Lojban gets the vocabulary), including the literal versions of all the other stuff.