Descriptions

There are nine and ninety ways of writing tribal lays and every single one of them is right.

Kipling’s lines apply to descriptions in Logic as well. I am not sure that there are actually 99 different proposals for descriptions, but I think it would not be hard to generate that many using just factors used in various actual proposals. This is a summary of the main candidates and of the factors involved in their proposals.

The most famous descriptor in Logic is Russell’s definite description, “iota (here i) x Fx,” which referred to the one and only thing that was F. Raises immediately the question what to do if there is not one and only one F, if there are no Fs at all or are more than one. Russell did not answer this question directly because his iota was not a basic symbol of the language but rather an abbreviation: “GixFx” was short for “[Ix]( Gx & [Ay](Fy <=> y=x)).” If there was not a unique F, the particular quantifier or the universal biconditional would be false and so the sentence in which “ixFx” was most immediately inserted. This last point required a system of scope markers to distinguish – in the simplest example – between inserting “ixFx” into \yGy and then negating the whole (giving a “~GixFx” which would be true if there was not a unique F) and inserting the description into \y~Gy (which gives a false “~GixFx,” since the “ixFx” is not covered by the negation): the difference between (when debreviated)
“~[Ix](Gx & [Ay](Fy <=> y=x))” and “[Ix](~Gx & [Ay](Fy <=> y=x)).” Most subsequent – and previous – proposals have taken the description to be a proper term, not an abbreviation, but some have still found equivalent expressions not involving descriptions and used these to decide what happens when the description fails to describe correctly.

The first descriptor in modern Logic antedates Russell’s by a decade or so and appears in Frege’s Ideography. Frege’s notation is altogether different from the usual, requiring two-dimensional arrays, so it is usual to present it in Russellian form using “i” again. For Frege “i” is a proper symbol and, further, always refers to something: the unique F, if there is one, the null entity otherwise. The null entity is an arbitrarily selected object, selection of which is a part of the specification of a model in formal semantics. The sure way to refer to this object is “ix x =x,” which, unless there is only one object – which makes the matter moot, is always improper. This is the second technique for dealing with failed uniqueness conditions. It has been elaborated since Frege in a variety of ways. On the one hand, descriptions that fail because there are no Fs can be treated differently from those that fail because there are too many Fs. All null cases usually get the null entity, which might be anything at all, but an abundance case usually gets an arbitrarily selected member of the set of Fs. This latter choice can be keyed not only to the set involved – according to which choice all Fs that have the same extension have the same selected representative – or each F (sometimes even those which are necessarily equivalent) gets a separate choice. This latest alternative is much more complicated that the former so has not ever really been used in Logic, though it makes sense in natural languages (which Logic is trying to reconstruct – among other things).

A third approach is to decree that descriptions always describe what they seem to describe. Thus, if no such thing exists in any ordinary sense, then it must be in some other sense. That is, descriptions can refer outside the realm of existents (the “inner domain of a model”) to other items (the “outer domain”) among which may be all the things that can be described (coherently or not). This has the effect of making descriptions (and often names and the like as well) behave somewhat differently from what is expected. Bound variables are usually taken to range only over the inner domain, and thus both particular generalization and universal instantiation do not hold for all such terms. A second round of quantifiers might then be introduced to cover the outer domain as well, or, indeed, the usual quantifiers could cover the outer domain with the inner marked off by a special predicate, typically “exists” or “is real” (which are usually defined in ordinary systems simply as “\y [Ix] x=y”). This does correspond to something in natural languages, but it complicates the logic one way or another and so is only used in very special cases. It also meets only the null failures of definite descriptions, not the cases where there is more than one F. When there are too many Fs, this approach is usually combined with a “selected member” treatment of the too-many case.

Another non-standard approach is to tinker with the underlying logic (or semantics) by introducing a third truth-value or allowing some sentences not to have a truth-value at all. In this, every “real” description has a reference and the ones that don’t are meaningless, rendering the sentences in which they occur meaningless as well (with perhaps some few exceptions: sentences which say that there is no such thing as the descriptor purports to describe). This breaks into a range of possibilities: meaninglessness (whether a third “truth” value or the lack of a truth value) may propagate out from the atomic wffs in which the failed descriptors occur or it may be overridden when a compound has an assignable truth value on the basis of other components (a true disjunct might render the whole disjunction true even if the other disjunct were meaningless, for example). And in the truthvalueless case supervaluations may be used (and apparently meaningless sentence gets a truth value if every sentence of the same form with only meaningful components gets the same value; so contradictions and tautologies are preserved, though not some rules of inference: addition or generalization, for example). While each of these approaches does correspond to a natural way of dealing with failed descriptions, the complications they introduce into logic makes them unappealing to anyone who is not using such a deviant logic already for other purposes.

Finally, at the furthest remove from the ideal situation where a definite description refers to the one and only thing that has the property in question, is a combination of the null entity notion and the selected member notion. This, Kaplan’s d-that, assigns to each description an object, potentially a different one for each description. So, each description that fails for lack of an F may get a different null-entity. And so may a description that fails from too many Fs. But there is not requirement that the selected entity be an F. Indeed, even if there is exactly one F, “dxFx” need not refer to that thing. Such descriptions are “purely referential,” their reference is given by a referential function in the model, without being calculated out from other assignments and interpretations. They are, despite their appearance, essentially just names. To be sure, in a natural language, such an expression would be bound by pragmatic considerations, requiring the referent to either be an available F or something that an interlocutor might reasonably be expected to think to be an F or to see as obviously related to Fs. But these considerations do not hold in pure Logic.

This dozen or so approaches to definite descriptions can be matched more or less by approaches to indefinite descriptions, “an F.” Theses tend to fall into two groups, both giving false (or meaningless) when there are no Fs but otherwise having a proper reference, even if there is more than one Fs. The simplest approach is to say that “a F is G” is just a variant way of saying “There is an F which is G,” a quantified variable, not a free term at all. If we devise a symbol for this locution (alpha, eta and epsilon have all been used), it will just be an improper symbol, an abbreviation (though often longer) for a primitive expression. Furthermore, this “description” will not be a referring expression, only a variable. Using descriptionlike expressions in this case will raise the same sort of scope problems that arose with Russell’s description: whether the description lies in a component or in the compound as a whole. Obviously, this approach has little useful to say – except, as we will see, as a lead-in to a later development.

The other major approach is the selector approach, which we have already seen as a way to deal with the “too many” failure of definite descriptions. In the model is a function which assigns to each predicate that refers to a non-empty set in that model a member of that set, whether on the basis of the set itself or directly on the predicate expression. When there are no Fs, either there is no object assigned (the function is appropriately partial) or some null entity is used (one for all or one for each). In the former case, there is the choice whether the unreferential descriptive phrase renders its sentences false (the referent of the term is not in the set of the predicate since it is not anywhere) or meaningless, with all the variations the latter choice allows. Of course, there is also the approach of expanding the domain so that the filed descriptions have referents in the outer domain, but that has all the problems in this case as it did in the definite case.

The whole selection approach seems at first glance to be problematic anyhow — for applied uses at least. Suppose that we have a selected representative of the Fs, exFx, say. Then, given [Ix: Fx] Gx, one could presumably infer GexFx. But, of course, the mere fact that some F is G is no guarantee the selected F is G; it might be another F altogether. The notion of preselecting an F goes against the whole mechanics of particular quantification, where each instantiation has to be to a new term in inferences. To do otherwise would be to legitimate the inference from [Ix: Fx] Gx & [Ix: Fx] Hx to [Ix: Fx](Gx & Hx), for example.

Of course, the presumption here is just wrong. All that is required for the selection notion is that
[Ix] Fx entails FexFx. [IxFx] Gx would really need to entail G [exFx] Gx or (to simplify a bit) Gex(Fx & Gx), a different property and thus possibly a different selected item. And this holds: the particular sentence guarantees that the relevant set is nonempty and so the descriptor has a referent — and that referent must have the property in question. For heuristic purposes, we might call exFx “the most likely F,” attaching it to the strange looking theorem of standard logic, which also can be read in that way: [Ix]([Iy]Fx => Fx) “if anything has F, this thing does,” as it were. Taking this view does have consequences for the semantics, however, for if we allow that a sentence like FexFx is meaningful when exFx does not refer, the sentence must be false and thus entail ~[Ix]Fx, as it should. On the other hand, if it is meaningless and this is a third value, then [Ix]Fx => FexFx is either meaningless or true, depending on the way meaninglessness interacts with compounds (overriding or going with whatever is decisive in two valued systems—here false antecedent gives true conditional). Of course, if meaninglessness is not a truth-value, this would be meaningless, but supervalued to true, since all meaningful cases evaluate to true.

The consequence for applications is more severe. In order to be sure that we continue to be true, the scope of the original quantifier apparently has to extend to all the sentences involved “A man walked into a bar. He ordered a drink. He sat in a corner” cannot be “GexFx. HexFx. JexFx” as we seem to want. The original quantifier has to range over the whole “[Ix: Fx](Gx & Hx & Jx) and keep extending as new items enter the narrative: “He played “Misty” on the jukebox” means conjoining on “&Mx” still the scope. Then, moving to the description must take all of that along: Gex(Fx & Gx & Hx & Jx & Mx) & Hex(Fx & Gx & Hx &Jx & Mx) &…. The whole tale about this fellow is already spelled out in the initial description of him. There is nothing like introducing an item, fixing the reference, and then going on to say new thing about – things not already inherent in the description used for the introduction. The referential function of the description has been totally absorbed by the descriptive part. Anything else, it seems, would open the possibility that something selected for the set so far described (the intersection of the Fs, the Gs, the Hs, the Js and the Ms) might not also be the selection for the next thing, “He danced a few steps,” unless that too were part of the description.

This comes from taking the association between particular quantifier and descriptor a little too close. To be sure, if we have the extended particular sentence above, we can infer the one in terms of the extended description above. But that is not the only way we can get a sentence with the right description. Although we talk about there being a function in the model that selects an item for each nonempty set or nonvacuous predicate, we do not require that the model be complete in order to use the notation. Indeed, in an applied case, a developing narrative, we are constructing the model as we go along. In particular, once we have selected our object and linked it to a description, we can follow that object wherever it goes and talk about it still suing that initial description. The initial object does not even have to be the selected item of the newly arrived at set (the intersection of all the sets so far) as long as it is in that set – which is just what we are building the model to do. Once we have an initial referential usage, “a man,” the rest of the narrative is descriptive, about that man, not specifying further which man it is (although, of course, we can eventually say enough to identify the man in question; in retrospect we could see that the model we constructed could be taken as one for the extended sentence with the extended description). The identity of the item may be no better spelled out in the sentences with the description than it is in the totally nonreferential particular quantification that follows from the narrative up to some point; it is just the item this part of the story is about.

Because the e notation was introduced in such a way as to be tightly connected with quantification, we can use a different notation here, [Yx: Fx], where the Y still recalls the I of the quantifier but is also clearly different. We still have that [Ix] Fx is equivalent to F [Yx: Fx], of course, but this does not even seem to imply any tighter connection. In particular, this equivalence is not the only way to get a description in; indeed, we can start with a description without a previous quantified expression at all: “A man walked into a bar” is (say) G [Yx: Fx] and can begin the narrative.

So far we have talked only about descriptions in the context of singular quantification. Most of this carries over with only minor changes to plural quantification. The definite description now gathers all of the Fs, those F such that any F is among them (no longer identical with them, note, although we could define a plural identity as aAb & bAa): GixFx iff [Ix: Fx] [Ay: Fy] yAx & Gx. Similarly, [Yx: Fx] is now “the most likely Fs,” some selected Fs, picked by a selection relation (rather than function) to represent the Fs (which are ixFx now, for all practical purposes). We can get back to the old singular notions by restricting quantifiers to single items by using the predicate “is exactly one in number,” “\x, 1x,” conjoined F. However, since the insistence on singularity leads to so many problems, we should rarely have need for these versions.

Given that the problem of more than one F has been removed by plural quantification, the only problem that remains for descriptors is non-fulfillment, the case when there are no relevant Fs. This needs to be distinguished when possible from the case where there are relevant Fs but they do not have the further property required; is G [Yx: Fx] false because there are no Fs or because the selected Fs are not G? Obviously, the problem only arises when [Yx: Fx] is first used; subsequent uses in a narrative presuppose that the description is proper and son only the claim that those object have some further property is at issue. But, if the initial occurrence of [Yx: Fx] is in a negation, ~G [Yx: Fx], the issue remains open. We could, of course, introduce scope markers like those Russell used for his iota, but that seems unduly untidy. What is left then is to disallow such an introduction, and that is most easily done by taking what looks to be a forbidden case to be instead an ordinary quantified sentence: not ~G [Yx: Fx] but simply
~[Ix: Fx] Gx. If reference to [Yx: Fx] is taken up later, this resolves the question one way; if it is not then the choice between ~[Ix] Fx and [Ax: Fx] ~Gx is unresolved (but also unimportant for the context). Of course, once we have a way of putting the indeterminate case, we can use the more nearly determinate form, ~G[Yx: Fx], for the case where there are Fs but they are not Gs. (\y ~Gy)[Yx: Fx].

Against this background, what are the Loglan/Lojban descriptors, what Lojban calls gadri? The roles of these items have changed over time and across languages, but the basic patterns emerged fairly early. What became Lojban {le} started as the Russellian iota. Under real-world pressures against uniqueness it expanded to refer to more than one thing (eventually formalized as what was – at least for {lei} – called variously a mass, a group, or a bunch). It was also contextualized: even though there might be several broda, {le broda} meant the one (or several) brodas of interest at the moment. More or less simultaneously, the problem of there being no brodas was dealt with by fuzzying the defining predicate: le broda did not have to actually be a broda; it was enough that the speaker chose to call it a broda in this context. What was important was that the hearer picked out the right things, what the speaker had in mind. In the end, then, {le} became a naturalized version of plural (or bunch) d-that, a purely referential description, in which the description did not play a semantic role at all – although pragmatically it had to work to enable the hearer to pick out the right referents (which meant that they had to be either brodas or related to brodas in some contextually functioning way). {lei} develops in the same way only always referring a bunch (plurality) taken collectively (although there have been times when it has been used for masses in any of several senses, most often go, but sometimes also fuzzy or generalized particulars or mereological sums). {le’i} has always been a set, but was originally the set of all brodas and only later came to be the set of the things referred to by the corresponding {le} expression.

{la} started as a means to introduce names (words ending in a consonant) into sentences. It was first pluralized to cover more than one thing with that name – whole families first. And then it was generalized morphologically to cover predicate expressions that some one might be called, whether or not they actually applied as predicates (it is unclear that So-fierce-that-enemies-are-afraid-of-even-his-horses II was really as fierce as his father, I, but neither actually fit the white-man version, Afraid-of-horses – and don’t get me started on Runs-so-fast-he-breaks-the-wind, called Breaks-wind by his contemptuous opponents). So, in the end it was a plural (bunch) description of things that were actually called by the expression that came after the gadri, whatever its grammatical status. {lai} is just the collective use of these same things and {la’i} is there set.

What became Lojban {lo} started out as a variant form of a particular quantifier. Taken by itself, {lo broda} referred to (the bunch of) all the brodas; in use, it was assumed to have a particular partitive quantifier attached, {su’o lo broda}, and so was equivalent to just {su’o broda} or
{su’o da poi broda}. And similarly for other quantifiers overtly prefixed (infixed quantifiers told how many broda there were altogether). Like particular quantifiers, repetitions of the same expression took on new references. If one wanted to refer again to the same brodas as {su’o lo broda} referred to, one had to use a referential anaphoric pronoun or {le broda} (with some descriptive component still functioning in this case). What has remained constant of this is that the referents of {lo broda} are actually brodas, not just called that or referred to in that way. But the rest has changed – or been proposed to change – in a number of ways, some of them incompatible with others. One trend is to make {lo} more like {le}. In one case the interpretation of basic {lo broda} has shifted from all the broda and then a subseveral of that to an indefinite several broda, but all of them. Standing alone, these two get to the same place pretty much, but the quantifiers are different: the internal Q is now the number of the indicated brodas, not of all the brodas, and the external quantifier is the size of some several drawn from the indicated brodas, not of some brodas drawn directly from the whole. (This version is partially expunded for bunches, plurals and Y-descriptors at Lojban Formulae.) And change reduces the difference between {lo} and {le} to just the difference between the literal application of the predicate (veridical) and “is described as.” Independent of this but occurring in the same proposal (BPFK Section: gadri) is the suggestion – not worked out in practice – that {lo} descritions refer not to in principle identifiable things but rather generically to things of that sort (this is, of course, incompatible with making {lo} merely inspecific {le} and seems to result from the fact that {lo} descriptions would normally be used in generaliztions – a mode parallel to “typically,” for example). The same proposal also contains a different reinterpretation of the quantifiers in {lo} expressions: internal quantifiers become part of the predicate, {lo Q broda} is now some cluster(s) of Q brodas; external quantifiers are multipliers, {Q1 lo (Q2)broda} is some Q1 of whatever lo broda is – clusters of Q2 brodas if the internal quantifier is in place. This creates some problems for fractional quantifiers, which were partitive proportions in the other reinterpretation. The first of these suggestions would make {lo} a d-that with descriptive function as well as referential, roughly between d-that and e. The second suggests the need for another descriptor in Logic, but seems in fact to require a different modal. The third appears to make changes only in the predicate portion, retaining the original interpretation of the descriptors (this is not certain given the ambiguity of much of this proposal). In all these case, {loi} seems to be the collective version of {lo}, covering essentially the same object(s) with a different predication. It has been suggested that {loi} be used for masses, the goo of things, but this proposal has never been worked out in detail (the first proposal here would take the goo to be a collective of pieces of things). All seem to leave {lo’e} as a set, differing only in whether it is the set bemembered by the things in a {lo} description or always being the set of all things satisfying the predicate. {loi} can be handled however {lo} is handled, with differentiation in the predication (though this is known to be inadequate to account for all differences in predication); {lo’e} requires set notation (some variant of an ordinary description of the appropriate sort).