quantifiers

["John E. Clifford" <pcliffje@CRL.COM>]

        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.