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*To*: John Cowan <cowan@LOCKE.CCIL.ORG>*Subject*: quantifiers*From*: "John E. Clifford" <pcliffje@CRL.COM>*Date*: Tue, 25 Jul 1995 10:35:05 -0700*Reply-To*: "John E. Clifford" <pcliffje@CRL.COM>*Sender*: Lojban list <LOJBAN%CUVMB.BITNET@uga.cc.uga.edu>

Quantifiers in Lojban differ markedly from those in standard logic. The quantifiers of logic always occur outside the core sentence and only in association with a variable, never -- like the quantifiers of Lojban -- attached to a designating term nor in the argument place of some predicate. The scope of a logic quantifier is always explicitly given and definite, never extending indefinitely nor requiring either conventions or separate terminators. And, in most standard systems, quantifiers in their own right are not restricted to subsets within the universe of discourse; such restrictions are done sententially after the quantifier is expressed. In most respects, then, the nearest thing in Lojban to a logic quantifier is a prenex quantifier + DA, without POI, and followed by zo'u (eventually, since we could have a sequence of such quantifiers). The chief difference is the matter of scope, sin ce, at at least some stages of Lojban officially -- and almost always in practice, the scope of a Lojban quantifier of this type runs from the first occurrence to a reset marker or the end of occurrences of the bound variable (or requantifying the variable -- more on that in a moment). Assuming that there is no problem about where the quantifier actually occurs --- as there had better not be in a syntactically unambiguous language but often appears to be practice (does the "for all x" in "for all x, x is F only if x is G" go outside the whole conditional or in the antecedent -- I tend to use forethought connectives to be sure), this open scope creates some minor problems of interpretation. In particular, the syntactically specified restriction (antecedent of a conditional for universals, otherwise a conjunct), if there is one, has to be assumed to be satisfied at each occurrence after the initial introduction. This is, of course, not a problem except as a possible source of misunderstanding when, se veral pages after the start, we have to remember what sort of thing da is now referring to. One upshot of this carry-along, however, is the thought that a variable could be requantified to mark out a subset of the original set, three of the original four, for example. As I noted a couple of weeks ago, logic allows a quantified variable to be requantified, but the effect is not a subset of the original set, but rather a new subset of the universe of discourse, just like the same quantification of a different -- as yet unquantified -- variable. The scope of the requantified variable is again de finite and, at the end of it, the variable resumes its original value (if the original quantifier is still in force). With indefinite scopes, the new quantifier would simply supercede the original one, whose scope would thus end at the requantification, even if that were a subset function. On the other hand, the subset notion is difficult to express in standard logic, except the move from all of some sort to a more restricted quantity. Moving from one number to a lesser one requires unabbreviated notat ion and a complex disjunction of conjunctions of new identities, while the move from one indefinite to another can hardly be done adequately at all, without the introduction of a new device, the generalization of a cover symbol for the complex in the nume rical cases -- or an appeal to sets. Even though sets are the individuals that Lojban deals with most naturally (I'll get back to that one), I think that subset quantification should be dealt with by the other standard method, a cover device relating the variable of the original quantifier to a new variable with the new quantifier. The set approach is inherently more complex and less natural, even in Lojban, where the variable, if not the other terms, still refer to individuals. The requantification approach is conceptually confused, trea ting a variable at once as for ordinary individuals and for a set. But, more importantly, requantification with open scopes means that we cannot easily get back to the original quantifier once the new is introduced and such return is often something we do want to do. The only problem for explicit connection is to find the right predicate to use (attached by poi or syntactically to the new quantifier). None of the "obvious" choices works well literally, since they also would result in taking variables as sets and individuals equivocally. So, since the form in logic is an artifical term made up for the purpose (and partially defined in most particular cases), I suggest we use a new artifical term (temporarily xu'u, since almost all the other experiemntal terms are probably already up in the air) for the purpose. Yes, hitting cmavo space yet again to solve a problem! Once prenex bare quantifiers are dealt with, prenex restricted quantifiers present no special problems. Although they are not part of the most widely-used logical notations, they have a fairly respectable place in the acceptable variants. The only thing to recall is that in logic, as usually in Lojban, quantifier DA POI broda requires that there be broda, even when the quantifier attached is "all" or "at most" (unlike the cases where the restriction is in the sentence rather than the phrase, "for all x , if x is broda, then..." etc.). That aside, explaining these quantifiers in logical terms (or conversely) is no more problem than for unrestricted quantifiers -- simpler, in fact, since we do not have to account for the extension of the restricting sentence when there is one. In both cases, the secret is a massive iterated conjunction which encompasses all the occurrences of the bound variable and which begins immediately after the prenex in which the quantifier occurs (or in the consequent of the condi tional in the case of conditionally restricted universals and the like). When we turn to variable-binding quantifiers in Lojban that occur within the sentence, in sumti position, the translation to logic is somewhat more complicated, as the frequent changes in rules in Loglandic history shows. Of necessity, these sumti quantifiers must be equated with prenex forms in logic and so have to be fronted somehow. The changes have been back and forth about whether the quantifiers are fronted as they are, regardless of negations and other potentially affecting marks along the way, or whether they undergo the logical changes as they move -- whether they are to be interpreted where they are are as already prenex regardless of where they are placed, in other words. The usual quantifiers, "all" and "some" tend to pull toward the need to change -- and many objections to that are removed by careful use of the leaper, which does allow a quantifier to be treated as prenex wherever it occurs. On the other hand, the numerical quantifiers tend in natural languages to be read large scope as they are small and so to read prenex as the same as in place, regardless of what intervenes. While a coherent system could be built either way, the system that requires transformations in fronting (together with the leaper) is ultimately simpler. (I cannot remember -- indeed am unsure it is settled -- what the present official line is.) There is also the problem of how far foreward we need to move a given buried quantifier, but that, once the quantifier has been prenexed to its original basic sentence, can be handled by the process that deals with the indefinite scope. It has always been the case that this prenexing is done in order, left-to-right in the sentence gives left-to-right in the prenex (barring leapers always). However, recent examples have tended to show that, at least when quantifiers are around the same predicate, they are forced to interact. Thus, the second quantifier to come out (y, say, following x) comes not simply xFQy but fromAz(z xu'u x => zFQy) (or Az poi z xu'u x), using the device mentioned earlier for subdividing the established bound variab les. That is, a new qantifier is introduced for each instantiation of the first. And so on. (It would be possible to take all the quantifiers here as on a level, all instantiated together, however, this makes the case now covered much harder to do. The altogether instantiation format is covered in the form suggested first by starting with prenex form or using leapers.) This addition complicates the algorithm leading to the final interpretation but does so in a systematic way, so as to be manageable. W e would rarely run the transformations, of course, but the underlying universal quantifiers added here explain why the quantifiers can no longer be considered independent: non-universal quantifiers cannot be rearranged around unversals equivalently and the freedom of order marks the independence of the prenex forms. A similar factor enters with the quantifiers on descriptions in Lojban. These are even more remote from standard logic, since logical descriptions always refer to individuals and so cannot meaningfully be affected by quantifiers, which are all integer in standard logic. The occurrence of quantifiers on Lojban descriptions means that they must refer, somehow, to pluralities, i.e., in logic, to sets. But the most straightforward version of this is incompatible with the factional quantifiers, since sets and most related objects are atomic -- without parts and so not meaningfully fractioned. The easiest satisfactory account is already somewhat complex. To begin with, we can eliminate concern about quantifier expression that may appear between the descriptor and the predicate. That is simply a cardinality marker on the underlying set and appears in an expanded version as the claim that the set is an n-leton (nymei in Lojban). It plays no further role in the quantifier problem, except that the fact that -- barring an explicit 0 in that place -- the set is always non-empty allows us to use restricted quantifiers without qualms. For the rest, Lojban has at least three ways of identifying sets: by what its members are (lo...), by what we describe the members as (le...) and by what we call the members (la..., strictly a subclass of the previous type). These sets may be referred t o in four ways (at least), which give rise to one special form and two further objects in logic. The four ways and the corresponding forms or objects are directly with the set itself, distributively with a complex involving quantification over the member s of the set, collectively with the massification of the set, and statistically with the averaging over the set. The three objects are the set itself, {x : x broda} or just {broda}, the massification of the set, Mas ({broda}), and the average, Avg({broda }). Each of these is a unique individual and so has no use of integral quantifiers. But each is also atomic, a partless point in logic, and so has no use for fractional quantifiers either. Both are legal in Lojban, however, the fractional ones are take n as implicit in all cases (I think that is a needless complication but I will try to explain what it might mean anyhow). So far as I can tell, no one has proposed a use for an integer quantifier and any of the objects here (except the question about two average brodas, which turned out to hang on a conflation of the descriptor version, a statistical look at the set, and the possibly related predicate notion of typical or average). The fractional quantifiers have, however, seen some use that requires a translation. Since 0.n of {broda} makes no sense as it stands we have to take it as an idiom and the most likely referent sought is a subset of {broda} whose cardinality is 0.n that of {broda}. The form is only slightly misleading since we probably assu med that a fraction of a set is itself a set, but spelling this out makes the situation much clearer: Ey c {broda} and k(y) = k({broda}) * 0.n and y.... The corresponding forms for massification and average are the same except that the last y is replace d by Mas(y) or Avg(y) as needed (I am not sure the Avg case has ever occurred). That is, 0.n of a mass of a set is the mass of 0.n of the set, which, in turn, is actually another set. (By the way, properties of masses are properties of the whole mass typically. It is an interesting study to find what prpoerties of masses are also properties of submasses and which are not. We know that "inhabits Africa" is one that covers both and presumably there are others, but "wins the game" does not seem to descend. This problem is one version of the fallacies of composition and division and a more interesting version than the usually discussed one between members of a class and the class itself -- usually meaning the mass). In all of these cases, as with the internal quantifiers, the quantifier aspect -- binding a variable -- disappears or, rather, is relocated. A shift also occurs in the remaining case, the distributive reference to the set, taken (perversely, from the logician's point of view, though not linguistically) as the basic forms: le, lo, la. The set now defines the restriction on the quantifier, Q lo broda cu brode is Q x e {broda}, x brode. As with embedded variable quantifiers, the expansion of quantified descriptions within predications become more complicated when more than one such expression occurs in the same predication (unless all the quantifiers are universal). The format is the same, a universal over the xu'u of the established bound variables. As with variable cases, independent choices can be declared by putting the expressions in the prenex form (though Lojban does not have very consistently worked out anaphora for these, com parable to the repetition of variables). Details of some cases of various expansions still need to be worked out, as do more careful explanations of double descriptions like lo ci lo mu broda and of lua. This is a tentative sketch explanation for comments and development, but it does seem to work, so far as it goes, for both Lojban and logic.

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