quantifiers

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

I've been on deadline for a couple of weeks, not to mention new semester
and the annual campus meeting (University administrators do not have all
their faculties), so, while I have been following along, I have not
commented.  So this is catch-up time, alas.
After several years, my university e-mail line has kicked in.  It is now
subscribed to Think-L and Phil-Logic. I tried to find Logic-L but the local
wizard cannot find it; any suggestions?

iain:
[T]he logical notation I'm familiar with assumes that
consecutive quantifiers are nested.  (Coordinate quantifiers
are not possible in this sort of notation.)
       Ax Ey x broda y
means that for each x there is a y (which is a function of x)
such that (x broda y) holds.
I had naively assumed that this corresponded to an expression
where both quantifiers were in a prenex.
If this is indeed the case, it is not at all obvious why
        3x 3y x broda y
should not also imply that the 'y' quantifier is subordinate
to the 'x' one.
You didn't appear to like Jorge's suggestion that we use multiple
prenexes (prenices?) to denote the nested situation, so I'm not sure
a)      How you represent nested quantifiers
b)      How you decide what's nested and what's absolute if you have
        a mixture of universal, existential and numeric quantifiers
        in a single prenex.
pc:
        Well, if you have a mixture of universal and particular quantifiers,
the rule to remember is that you cannot move a quantifier of one type
across one of another equivalently; fronting a particular across a universal
is invalid and fronting a universal across a particular goes validly to a
much broader situation, from which there is (see above) no return.  But
with strings of quantifiers of the same sort (universal or particulars) the
order makes no difference; the quantifier in the scope of another is not in
fact a function of the larger scoped variable (as a particular is in the scope
of a universal). Numerical quantifiers are abbreviations for complexes of
universals and particulars and the same rules thus apply.  Specifically, all
of these begin with a string of particulars and then may have, buried away
somewhere, some compact universally quantified complexes, which can
be moved independently within the particular scopes but need not be
fronted.  So all of them are also order-independent.
        So, while it is true that all cases of multiple quantifiers are nested
in terms of scope, only AE strings are nested in the sense that the choice
of the second is a function of the choice of the first, what I take to be the
crucial meaning of "subordinate" here.  The other sense of "subordinate",
"in the scope of," is an accidental feature of a mode of speaking, since we
could say the same thing with this subordination reversed.
        I don't approve or disapprove of multiple prenices, I just do not
know what they mean, except, perhaps, as a device in Lojban (but not in
logic) to solve the subordination problem.  I am just trying to find another
way of doing this without introducing anything logically new into Lojban
(not lojbanically new, of course, since the form is legal already).  And, of
course, we may find that we do want to deal with particulars overtly in the
scope of universals where we want neither to change the apparent order
nor require that the particular is a function of the universal.  And then, we
might need double prenices and rules for dealing with them.  I am just not
sure that we are at that point yet.  We do need a sort of double prenex
already, of course, for quantifiers with longer and shorter scopes; a
complex sentence and a single component, for example.
        As for how to represent nested quantifiers that we want to keep
that way:
in Lojban, I agree with xorxes that (though he would not put it this way)
keeping both quantifiers in the matrix of the sentences does that, i.e., not
prenexing them
in reconstructing the logic of the "logical" (more remotely each minute)
language, I have been toying with introducing explicit universal
quantifiers to do the job (which are usually there implicitly in the Lojban).
I have also toyed with a variation of Skolem function throughout in place
of particular quantifiers, but that goes too radically against the structure of
Lojban, which has quantifiers of both sorts and almost no name-like
expressions.  So, I would take the eternal  ci nanmu cu pencu ci gerku as
(skipping reams of essential but here irrelevant details)  3x( x nanmu &
Ay(y =x => 3z (z gerku & y pencu z))).  The idea is that without the the
apparently useless universal, the "3z" can come to the front and change
places with the "3x."  This trick does not quite work with, as I have it here,
"y=x," since that form is provably equivalent to the form without the
universal and the antecedent an with the consequential "y" replaced  by
"x" -- just the case we want to be different from.  Finding the right
antecedent (I think I proposed something like _xu'u_ for what is needed) is
the hard part here.

xorxes:
My question is: supposing that {le ci nanmu} is not universally
quantified, but is rather a "multiple referring expression", i.e. just
like a singular term but with more than a single referent, then how is
any expression containing {le ci nanmu} with this referring
interpretation any different from the same expression as a universally
quantified term?
I don't see any possible difference. {la djan e la bab e la haris
cu batci lo gerku} is just like {ro le ci nanmu cu batci lo gerku}.
I don't see what is the difference that referring is supposed to
make. The same thing with {la djan a la bab a la haris cu batci
lo gerku} and {su'o le ci nanmu cu batci lo gerku}.
pc:
        Since I have said that I seem to be unable to explain the difference
to those who will not see it, I am not sure why I keep trying, but here at
least xorxes offers three possibilities and a couple of them offer some
clear separations.
        First, a general point.  The biggest advantage of  referential
expressions is that they have no scopes or, what amounts to the same
thing, their scopes are always much larger than the context in which they
occur and so never get in the way of one another or of actually present
quantifiers.
        On the other hand, conjunctions and the like do have scopes and so
do muck up present quantifiers.  Now, given the plethora of variations on
the logical connects that Lojban has (requiring which is the strongest
reason I know to doubt that YACC has found the real grammar of a
human Lojban), it should be obvious whether _lo gerku_ in the above
examples lies inside or outside the scope of the _a_ or the _e_.  I suspect
that it lies inside, though I have no idea what the form is to assure that it
lies outside.  And, if it does lie inside, I think that that means that
it it is carried over verbatim into each of the replications in the
expansion.  If it
lay outside, then it need appear only anaphorically in the non-first juncts
of the expansion.  la djan cu batci lo gerku ije al bab cu batci lo gerku ije
la haris cu batci lo gerku (different mutts for different men) vs. la djan cu
batci lo gerku ije la bab cu batci gy ije la haris cu batci gy (same for all).
Insofar as conjunctions of instances are just like universals (and
disjunctions particulars), this is what you would expect  But notice this
still does not say anything about "multiply referring expressions" -- unless
they inevitably give rise to conjunctions and of one kind of scope rather
than another.  And that is not obvious to me.
        Of course, since we can express the same facts in either style of
language, we have to expect that we will probably get matching problems
in the two styles, as here the problem of scope of   "and" matches scope of
"all."  But not really, I think, since the "and" scope is, presumably, local to
the sentence containing tne "and", while the universal's scope continues
(in the Lojban system) indefinitely. (I won't swear to any of this until I get
further along in this reconstruction, but that is the way the likelihoods look
at the moment.)
x:
When using {le},
the quantifiers range over particular referents. If you call the
form "John and Bob and Harry" direct reference to individuals, then
so should be "each of the three men in question".
pc:
        Hey, Ax(x=a v x=b v x=c => Fx) is as much a universal sentences
as Ax(Gx => Fx) and a long way in both form and content from Fa (not
even considering whether Lojban has any way of saying a).

sos:
> Ah yes, the other advantage:
> no prenexes needed, since referring expressions can occur in matrices.
But in Lojban no prenexes are needed for the quantifier case either.
pc:
I meant in the reconstruction, throughout.

And's reference to McCawley clarifies And's position nicely, as does his
example, and validates it, since I tend to think McCawley is more often
right than most linguists.  I like the set notation (though xorxes does not),
mainly because it allows for somewhat easier clauses about just which
broda are involved  ("relevant broda" in some earlier version of this
discussion).  But the whole story is still controversial and xorxes's
immediate question is to that point.

I'll just reiterate my line.  In logic, the several particular quantifiers, E,
in the prenex are coordinate in the interesting sense: no choice of one
depends upon the choice of another.  It seems to me that the natural way
to represent this situation in Lojban is with the directly parallel form,
quantifiers in prenex with variables (or whatever) in the places in the
matrix. The only other reasonably natural form for representing this would
be to have the quantifiers already in the matrix, not the prenex. But that
form, we all seem to agree, requires that the second quantifier be
subordinate to the first (and the third to the second and first and so on,
apparently) and so does not represent the logical situation we are starting
with.  However,  as xorxes points out, there is a Lojban habit of prenexing
simply by fronting, so that the shift from _ci nanmu cu pencu ci gerku_ to
_ci nanmu ki ci gerku zo'u ny pencu gy_ ought to be meaning-preserving,
like the move to _ci namu gu ci gerku cu pencu_ (roughly, at some stage
in the language's development).  I think this move has never been a safe
one (maybe even with the first quantifier, surely not with later ones) and I
am not too sure any more about simple SVO=>SOV changes either, but
that must wait for a fuller analysis.  For now, I am proposing this just as a
solution to a problem: how to express  "(3x man)(3y dog) x pat y" in
reasonably Zipfy fashion, coherent with logic and at least some intuitive
sense.

xorxes
We are much more likely to talk about "some apples" as a mass than about
"all apples" as a mass. The reason why {pisu'o loi plise} is more useful
than {piro loi plise} is the same reason why {su'o lo plise} is more
useful than {ro lo plise}: we very rarely want to talk about all the
apples there are in the universe.
What properties of masses appear best in the whole? If I say that I
carry, buy, take, give, ask for, eat some apples, it is clear that
I'm not talking about the mass of all apples in the universe.
pc:
Well, in saying these things, it is not clear that you are talking about
masses at all, rather than distributively about individuals.  But, IF you are
talking about masses, then what you say is true (if at all) of the mass of all
apples.  It happens that these claims are probably also true about
some"submass" (that expression is just shorter than "the massification of
some subset of the set of all apples"), but that is not always the case and
thus is worth noting when it is.  So "all" is the default and anything else is
marked.

sos:
>the prenex form, by logic, covers three
> men and three dogs, the same for each man).
I suppose that "by logic" means "by the usual convention used in logic",
since there seems to be nothing illogical with the other possible
convention.
pc:
The only convention involved here is the one that takes  _ci nanmu ki ci
gerku zo'u ny pencu gy_ over to _(3x man)(3y dog) x pats y_, which is not
a logical convention.  That the latter, logical, expression is equivalent to
_(3x dog)(3y man) y pats x is not_ conventional but a theorem of logic
(including all the non-standard logics I can think of off hand).

sos:
> On the subject of quantifiers, I take it the standard logic rules
> apply: the universe of discourse -- the range of unrestricted
> quantified variables -- is nonempty, as is any explicitly restricting
> set, the broda of da poi broda.  Thus, ro implies su'o in all
> contexts.
I think that's unfortunate. For the cases when the speaker knows that
there are no broda, it doesn't really matter much, because no one would
want to use {ro broda} in that case unless with intent to mislead.
But the problem appears when the speaker doesn't know. For example,
I couldn't say "I promise to pay an extra dollar to everyone who
finishes their work before noon", unless I am also promising that
at least someone will finish before noon, which is not normally
what I would want to do.

pc:
Then the speaker just is making a conditional claim: "if anyone does then
I'll" and might be well-advised to do so explicitly -- which is how Lojban
already handles cases of possibly unsatisfied predicates.  Or he can sneak
in with some stuff about _no_, the obverse of the original claim about all
(but probably usually too legalese or just plain unintelligibly negative).

sos:
> Xorxes' lu'o may seem to agree pretty much with what I took to be the
> official line above: lu'o <individuals>  is "at least one mass whose
> components are <individuals>."  But, for a given bunch of individuals,
> there is only one such mass, of course,
Unless the individuals are not being referred to, but simply quantified
over. {lu'o ci gerku} is a mass of three dogs, but there are more than
one possible mass of three dogs.
> so the wording leads to the
> question whether this means that  lu'o lo broda (for example) is loi
> broda or rather takes the implicit su'o into account and so covers
> masses that have only some of the brodas in, what is elsewhere
> described as pisu'o loi broda
That's my intent, yes. With the added benefit of being able to
quantify over them, which is not possible for {[pisu'o] loi broda}.
pc:
I first took this last remark to mean that we could specify how many
brodas were massified in one of these masses.  But I now think that xorxes
means (and I hope he will correct me if I have him wrong here) also (?)
that expressions like _ci lu'o lo broda_, "three masses whose components
are are brodas" are legitimate, so that _luo lo broda_  becomes a general
term -- a predicate in effect, if not in grammar -- referring to masses of
this sort.  I am not sure what the consequences of this will be, but it does
seem to be somehting we want to be able to express.  On the other hand, it
also seems to be something we can express directly with predicates and
quantifiers back at the lowest level.  So, I am still ot sure just how the lu'a
series does offically nor does according to xorxes nor should go.
pc>|83