quantifiers

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

Some general background:  Lojban, like earlier Loglan, claims to
be built upon the language of formal logic.  This language of logic is
remarkably uniform over many practitioners, with only minor surface
differences by and large.  I take the standard language as a given in all of
this, so that its structure is the target toward which Lojban structures are to
be transformed for explanatory purposes.  In the logic language,
everything is to be taken as explicit, not as involving further interpretation
-- except as this is also a part of that language.  Thus, two basic adjacent
quantifiers in prenex in the logic language are subject to no further
analysis, other than theoremic transformation.  Two non-basic quantifiers,
numericals, for example, are subject only to the expansions defined for
them in the language: into a mass of particular quantifiers and whatever
further clauses are needed (non-identities and closed "at most" phrases as
needed) and them the theorems that apply to the results.
[The one large exception to the uniformity of the structure of the language
of logic  is Peirce, whose system does allow quantifiers (or what
correspond to them) in matrix sentences.  His system is also at least two
dimensional (and works better at three or four), though it (or the part we
are concerned with) is easily projected onto a linear scheme (in fact is
better linear than two dimensional).  But Loglan (hence, Lojban) follows
his system in several respects -- the embedded quantifiers and their
indefinite scopes are relevant here.   The beginning of my project to bring
out the logic of the "logical language" was, in fact, the standard translation
of Peirce's graphs into the usual notation.  One thing that is emerging is
that the basic translation scheme does not work very well when the
statements involved are at all complicated.]
        The strategy of converting into the language of logic for
explanation has some serious shortcomings eventually, for that language
does not do a lot of things that Lojban -- and virtually any natural language
-- does.   When we get to those places, we have to look at non-standard
logics first that operate more or less in the logic tradition.  Unfortunately,
in most of the problem areas to come there either is no such logic or there
are several incompatible ones or there is only one but it is a mere sketch.
We use what we can and what seems to work and, in the process, may
help the construction of these later logics.  Masses are a case in point,
where there are several largely incompatible sketches, which do agree on
a few things, enough to give some rules of thumb.  But some practical
cases suggest that the rules of thumb are occasionally wrong or -- more
likely -- are for a restricted set of cases which are not the most usual ones.
So, I present the official line(s) and then get on to looking at the problems,
if there are any.

sos:
>xorxes:
> 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).
When I asked for an example of a multiple referring expression, you
gave "John and Bob and Harry". If you didn't mean that to be {la djan
e la bab e la haris}, then you must have meant it as {la djan joi
la bab joi la haris}. In that case, I understand it as a singular
term, with a single referent, namely the mass of the three men.
Is that what you mean by a term with multiple referents?
pc:
Assuming for the moment that _la djan_ etc. are direct referring
expressions in Lojban, rather than quantified expressions (as the official
line has it), then, when I said "John  and Bob and Harry,"  I meant "la djan
e la bab e la haris" and not "la djan joi la bab joi la haris" and certainly not
 _le du la djan a la bab a la haris_ , i.e., each of the three men in question
(also _(ro) le ci nanmu_), which is presumably still (as officially) a
quantified expression.  We could easily suspend that presumption as well
but have not done so, although xorxes occasionally talks as though the
quantifier were not there even implicitly and at other times insists upon it
(to contrast with _lo_ expressions, for example).

sos:
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).
It may look that way to you, because you are used to thinking that
_(3x man)(3y dog) x pats y_ already has a meaning, but for someone who
has never seen this notation before, it is equally reasonable to expect
it to expand as (in the spirit of And & McCawley):
        (Ex a set of men) cardinality(x,3)
        & Ay ( belongs(y,x) -> ( (Ez a set of dogs) cardinality(z,3)
                                 & Aw (belongs(w,z) -> pats(y,w)) ) )
In other words, it could expand as: (3x man) F(x), where
F(x) = (3y dog) x pats y.  There is nothing illogical about that, at worse
it may be unconventional.
I suppose the conventional way would be:
        (Ex a set of men) (Ez a set of dogs)
        cardinality(x,3) & cardinality(z,3)
        & Ay Aw (belongs(y,x) & (belongs(w,z)) -> pats(y,w))
which is also logical, and it even has a simpler structure. But I don't
see where coherence comes into it. Each is coherent once things
have been defined in its way.
pc:
Well, we are doing logic, so, if someone wants to play, they need to learn
how the game goes.  Yes, the rules could have been set up otherwise, but
they weren't, so we will have to learn to live with this set.  Of course, the
alternative you suggest is pretty implausible, since it buries a pile of just
the things that logic wants to make explicit, quantifiers and conditionals in
an interlaced -- rather than detached -- order.

sos:
> So "all" is the default [for masses and sets] and anything else is
> marked.
This is not what the documentation says
pc:
Yes, at this point I am recommending a change (as I noted earlier in the
original) or, as I would say, a correction.

sos:
> 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.
That is clear. There is no problem when only Es are involved. The
question arises with more complex quantifiers like the numerical ones. Do
they simply consist of a string of existentials, or do they also have a
universal that can take a following particular under its scope?
pc:
Numericals all start with a string of particular quantifiers, Es.  What
happens after that depends upon the type of numerical.  For "at least,"
each of the newly introduced bound variables has to be differentiated from
all the others and each has to be assigned the defining property: "at least
two Fs" is  ExEy: x=/=y & Fx & Fy.  Larger numbers just repeat the three
parts: n particular quantifiers, (n!/2!*(n-1)!) different non-identities
conjoined, then n conjoined F's.  No subordinate universals at all and so
none that affect more subordinate particulars.  For "at most n," the
quantifier string is followed immediately by Ax(Fx => x=... ) where the
"..." is filled by a string of  n identities, each of the particularly bound
variables being identified with the universal x.  Now we have a universal
in the scope but no subordinate particulars and the universal is a closed
item, capable of being placed anywhere in the immediate scope of the
particulars.  "Exactly n" simply conjoins "at most n" and "at least n" and
then does a variety of theoremic reductions.  But in no case do we get a
particular in the scope of the universal.  So, in short, the numericals can be
move around as though they were only particulars, as, for the most part,
they are.  The underlying propositional structure is all conjunctions, thus
associative and commutative and expansive, and so allows for all the
various rearrangements that one might like, particularly the relevant "(3 x
men)(3 y dogs) x pats y" to "(3 y dogs)(3 x men) x pats y."

sos:
xorxes:
> 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.
It can be made clear by adding an appropriate context, e.g. "I buy the
apples for three dollars", which doesn't mean that each individual apple
cost me three dollars.
> But, IF you are
> talking about masses, then what you say is true (if at all) of the mass of
> all apples.
Did I buy the mass of all apples for three dollars? It doesn't sound right.
> 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.
I think these claims are usually true _only_ of a "submass", very rarely
> about the whole mass.
pc
The rule of thumb about masses is that a mass has the sum of the
properties of its components, where "sum" is taken in a rather broad sense,
which includes "logical sum"., i.e., disjunction.  So, by that rule, if I buy
an apple, the mass of all apples (and a large number of its submasses) is
also bought by me (and so, presumably, I buy it too).  As xorxes says, this
just seems wrong and the more so when we throw in the price of the
purchase.  On the other hand, the rule does a good job of distinguishing
masses from other things in important ways, where confusion is rife in
ordinary English. for example, so we do not want to get rid of it entirely
either.  What seems to be needed is a specification of the scope of
application of the rule -- both when it applies (or, better, when it doesn't)
and to what extent it moves from submass to supermass (or conversely).
This is probably best done with case-by-case examinations for a while,
until we have some hypotheses to test.
        Take the present case: "I buy the apples for three dollars."  This is
pretty clearly about a mass, the three dollars is the sum of the prices of the
individual apples, which prices are either uniform or weight-dependent (if
the latter, the individual prices are not calculated but rather we get the
sum of the weights and price the mass accordingly). But what mass is it?
Not the mass of all apples, clearly, but just the mass of these apples, the
ones I bought (offered to purchase, originally).  Buying has a specifying
effect on what is bought, moving it, in effect, from _lo_ to _le_ (or at least
_lovi_).  But then, it IS the whole of the mass that one buys for $3.  So, the
first maxim of this investigation is to get the right mass to talk about to
begin with.  Now, loivi plise  is a submass of loi plise and we know that
the property "bought by me for $3" does not go from submass to mass.
Given the specifying effect, even "bought by me" may not (the deed --
implicit though it be --does not cover any apples but these few).  But, note,
"bought" surely does, even though some apples are not bought (cf. lions
living in America). So, the second observation is to be careful about the
property being projected.  The logic literature (what little there is of it) is
not very good on this, but I think it is probably time to check McCawley
(and other linguists), if they have anything to say on the subject.

sos:
xorxes:
> 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.
Yes, it can be done like that, at the cost of losing clarity and conciseness:
        ro karce poi se stagau fi ti ba se lebna
        All cars parked here will be towed.
        ro da poi karce zo'u da se stagau fi ti nagi'e ba se lebna
        Every car is either not parked here, or it will be towed.
pc:
Well, the latter is less concise, but it is clearer -- the deliberately chosen
muddling translation notwithstanding -- or at least more unequivocal, even
with the a more generous quantifier.

xorxes
I am not sure whether it is even possible to avoid using the
prenex. What would this mean:
        ro karce cu se stagau fi ti nagi'e ba se lebna
Is it: "If every car is parked here, then every car will be towed."?
Or is it: "For every car, if it is parked here, then it will be towed."?
It is hard to tell, since we don't really have well established rules
for the relative scopes of quantifiers and logical connectives.
pc:
As I have said, you'd think that with the proliferation of  "and"s this
situation would not arise.  And it probably does not.  But I -- nor xorxes,
apparently -- knows which of the two readings this one gets: is the
quantifier inside the scope of _nagi'e_ or not.  I *think* it is and thus the
reading is the first and so not what is wanted, but...  Here, at least, we do
know how to disambiguate, using prenex forms.
xorxes:
Anyway, the simple {ro karce poi se stagau fi ti ba se lebna} can't
be used, because it claims additionally that at least one car will
be parked there and towed, while the purpose of the sign is precisely
to stop that from happening.
> 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).
Let's see:
        no karce poi se stagau fi ti naku ba se lebna
        No car parked here will not-be-towed.
As you say, legalese and not very easy to understand. But why go to all
that trouble? Under what possible circumstances is the universal entailing
the particular a useful thing to have?
pc:
Most ordinary possible circumstances: we regularly do it.
xorxes:
If it's a matter of avoiding odd-sounding sentences being true, then that
is not really accomplished, because with the existential interpretation,
things like:
        ro pavyseljirna na blanu
        It is not the case that every unicorn is blue.
are true. And they are just as confusing as {ro pavyseljirna cu blanu}
being true with the other interpretation. Nobody who knows that there are
no unicorns would want to say either of those anyway. But when one
doesn't know whether there are or aren't satisfiers of some broda, then it
is often the case that one wishes to make universal claims without being
committed to their existence. It would really be much easier if {ro da}
was simply {naku su'oda naku}, i.e. if {ro} and {su'o} were truly dual.
pc:
Even if there are no unicorns, what compels us to claim that _ro
pavyseljirna cu blanu_ is true?  It is a universal claim, so the minimum
truth value of its instances.  It has no instances, so, presumably, it has no
truth value.  Or, since I suppose _su'o pavyslejirna cu blanu_ is false and it
is the maximum value of its instances (since it is a particular claim) and
the minimum is never greater than the maximum, it follows that the
universal claim is false to.  The only way to make universals plausibly true
when they are about unthings is have -- overtly or covertly -- a conditional
form for the universals.  And, if we have it, we should, for our logical
language, have it overtly. Just because we allow that  classes might be
empty does not mean we have to allow that all their members are
therefore endowed with some property.  In fact, we regularly object to that
inference.
        I agree that no one who knows there are no unicorns -- or, as I
would say, is in a universe of discourse that does not include unicorns --
would say any sentence with _ro pavyseljirna_ in it.  It follows that
someone who does say such a thing is in a universe which does include
unicorns, so the subject is non-empty.  That is why -- almost exactly -- it is
most useful to take all affirmative quantifiers as having existential import.
(Part of the unplausibility of _ro pavyseljirna cu na blanu_ is also that we,
following English and most languages I can think of, tend to read that
internal negation as predicate rather than sentential, a particularly
common problem with universals.)

iain:
(I presume your "particular" quantifiers are what I would call
"existential".)
pc:
Yes; a theological distinction having to do with whether the quantifier
refers to what exists in reality or merely to what is in the universe of
discourse.

nss (new setta suspects - typically pc in quotes and iain commenting):
> 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.
By "validly", do you mean that the new proposition is implied by
the original?  (I didn't understand what you meant by this at all
at first.)
pc:
Yes, and "equivalently" means each implies the other.  Jargon!

nss:
> 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.
I'll come back to this later.  (I don't understand the bit about the
compact universally quantified complexes.)
pc:
The "at most n F" condition takes the form  Ax(Fx => x=...) where the
dots are a list of all the particularly bound variables involved with F.
There are no further quantifiers relevantly in this block and the rest of the
material involved is conjunctive, so this little piece can be moved
anywhere that suits.

nss:
>         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 see how to do this.  Could you give an example?  E.g. how do you
express Ax Ey F(x,y) with the subordination (scope nesting) reversed?
pc:
I just said that you can't change the order of AE , only AA and EE (and,
with a loss of specificity, EA).

nss:
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.
Put it first.  The general rule we've been working to is that order
determines scope determines subordination determines what is
(potentially) a function of what - irrespective of whether the terms
occur in an (explicit) prenex or in the "matrix" of the bridi (sentence).
pc:
I am not sure that will work.  The sentence is _la djan e al bab e la xaris
cu pencu lo gerku_.   We come up against the question whether
conversion changes meaning (like xorxes question about whether fronting
or SVO=>SOV order does) as well as the question about scopes.  But it
may be that _lo gerku cu se pencu la djan e la bab e la xaris_ does solve
that problem; it sure would be convenient (for now.  I can just hear
someone sharpening up a plausible counterexample somewhere -- as well
as citing the texts on this.)  For the general point (order determines...) I
keep having to come back to the point that this is a logical language and a
certain point, when we get down to the logic, what logic says goes. So,
scope maybe, but that need not always cover subordination at least in the
functional sense, and so on.  At least I hope not, or Lojban is going to be
very hard to be accurate in, much worse than English.

nss:
> And, if it does lie inside, I think that that means that
> 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.
Are you saying that if {le ci nanmu} is a singular term, then
{le ci nanmu cu batci lo gerku} (or even {li'o... da poi gerku})
there is only one dog, because there is no universal quantification?
pc:
My turn not to understand.  _le ci nanmu_ can hardly be a singular term,
since it says on its face it is about three men, taken distributively (and
conjunctively).  Further, there is universal quantification (presumptively,
since we haven't bracketted that rule yet in this discussion, that I can
remember), so its absence can't be the cause of anything  here.  And I
would be inclined to say (did say, indeed) that, because of the universal
quantification, there may be several dogs, a different one for each man.
What I take to be the standard view on this construction.

nss:
> 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.
Hang on a minute, that's not the way I view the situation at all.
To me, consecutive quantifiers are nested (as above), irrespective
of whether they occur in a prenex or in the "matrix".  The second
is in principle subordinate to the first, etc.  It just so happens
that in certain circumstances (consecutive particulars, consecutive
universals, but I don't see it at the moment for consecutive numerics)
you can prove that the meaning is unchanged by reversing two quantifiers
- there's a commutative law in effect - and thus the subordination
is irrelevant.
pc.
As far as the prenex is concerned, we agree, however much our
terminology may differ.  In the matrix, however, the common wisdom --
which I am going along with (willingly in this case) -- holds that the order
changes things and that even with consecutive quantifiers of the same
type, nesting makes the later a function of the earlier, i.e., they are not
commutative.  The consecutive numerics are just a special case of
consecutive particulars.

nss:
> sos:
(One of these days I'll figure out what you mean by
pc
"Same old suspects," pc quoted and xorxes commenting unless otherwise
noted.

iain:
If (3x) f(x) means 3=k{x: f(x)}
then (3x) (3y) f(x,y) means 3=k{x: 3=k{y: f(x,y)}}
If we rewrite this as (3x) f(x) means (E S) (3=k(S) & (Ax) x e S <=> f(x))
then (3x) (3y) f(x,y) means
(E S) (3=k(S) & (Ax) x e S <=> (E T) (3=k(T) & (Ay) y e T <=> f(x,y)))
["e" represents set membership, k(S) represents the cardinality of set S.]
(If you don't like using numbers, you can expand (3=k(S)) to
Ex Ey Ez: S={x,y,z} & x != y & y != z & z != x)
With this formulation, I don't see any way of avoiding the interpretation
that set T is a function of x.  You obviously therefore mean something
different somewhere.
pc:
While "Three is the cardinality of the set of brodas" would not be my
preferred translation for _ci da broda_, it is equivalent under the usual
assumptions.  In like manner, while I do not necessarily care for all the
moves through sets and back, I think that your formula is a good
representation for _ci da brode ci de_.  I see two problems with it wrt the
problem at hand: 1) it does not deal with restricted quantifiers and so takes
the quantifier to be defined by its entire scope and 2) it does not deal with
question of commutativity of prenex forms.  So the question is how to say
what might be symbolized briefly as ExEy(3=k(x) & 3=K(y) & x c {men}
& y c {dogs} & AzAw (z e x & w e y => z pats w))  (The universals could
be replaced by restricted quantifiers here, since x, y are guaranteed non-
empty; "c" means "is included in")  The prenex form in Lojban is
apparently precluded because prenexation is taken as meaning preserving
from the embedded form (the same problem as will conversion).
> pc>|83
(Is this just an elaborate smiley?)
pc:
Well, maybe.  It is a pictoglyph: turned a quarter clockwise gives a geek
with glasses (8), a fuzzy beard (3) and hair going in various directions (>|)
and those are recognizable features of mine, perhaps definitive in the
present context.

nss:
> 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.
This may be the case in logic qua logic.  I do not believe it to be
the case in logic as applied in other fields.  All my training leads
me to believe that any such inbuilt assumption that a set is
non-empty is fraught with peril.  I cannot emphasise this too strongly.
If you train someone for three years in pure logic, they may be able
to cope with this sort of complication - for the majority of people
who pick up a smattering of logic as a tool for use in the study of
another subject, it's a disaster waiting to happen.
Formal systems rely on simple equations between different
representations of the same thing.
        (Ax) f(x) === !(Ex) !f(x)
        (Ax e S) f(x) === (Ax) x e S => f(x)
are axioms or even definitions.
(I've even managed to dig up a citation.  An Introduction to
Modal Logic, G. E. Hughes & M. J. Cresswell, University Paperbacks
UP431, 1972, Methuen & Co. Ltd., first published and (c) 1968,
S B N 416 29460 X:  p. 134 defines the existential quantifier
in terms of the universal quantifier.)
The empty set is a normal set, just as zero is a normal number.
If I forget to allow for the case where the range of a variable
turns out to be empty in some particular run of my computer
program, nine times out of ten it will lead to a bug.
I don't care whether even the Universe of Discourse is empty -
it's none of my business.
pc:
Well, it is arguable that an empty universe of discourse is impossible,
since all discourse is about something.  In a similar way, the empty set is a
perfectly normal set in some sense, but its member(s?) are (almost?) never
the topic of conversation nor the sort of thing one wants to make claims
about.  So, in language construction (including by the natural process) we
tend to make the classes we talk about non-empty.  We are sometimes
forced to talk about classes where we are not sure and then we do the
proper thing: be tentative and, in particular, hypothetical: "every broda -- if
there are any --"  It is to that end that Hughes and Cresswell (following a
tradition of at least a century even back then) introduce the definition for
restricted universal quantifiers (the one fore restricted particulars is, of
course , (Ex e S) Gx =df Ex(x e S & Gx)).  We cannot do this in Lojban,
however, because, through a series of decisions, each taken for its own
good reasons but without (some would argue) adequate attention to long
range effects, Lojban has identified three originally very distinct notions,
_ro da poi broda_, _ro broda_ and _ro lo broda_.  Since the first of these
was created exactly to have a universal quantifier with existential import,
the rest fell into that pattern as well, leaving nothing for a snappy version
of _ro da zo'u [if, some version of _ganai_] da broda [then, some version
_gi_] da brode_.  So, the logical language just has to be fully logical,
without shortcuts, this time. (Or we could try to break the cluster, but that
never seems to get anywhere, since the equivalences are written down in
the prototextbook, cf. the rules about various transformations mentioned
above.)
As for duality, H&C's equation ( I assume ! is a negation sign) is, of
course, based on a universe of discourse guaranteed non-empty, the
standard model, in fact.  And, under that guarantee, duality holds
everywhere.  Not that duality is actually that interesting in itself; what
people really want is a good way to get rid of those initial negations and I
have suggested (with little noticeable effect) the easy way to do that:
restore the full set of  traditional quantifiers with the Carrollian
interpretation -- a good idea even if we could get the desired results in
other ways.
        As I have mentioned often enough, I want to get back to working
this stuff out and to stay out of polemics, which use up my hour a day.
this looks like a good time to try again at that.  The discussion of three
dogs and of existential import seem to be in tis-taint cycles and the
questions about _lu'a_ and about masses are at the point where study is
clearly required and nicely sketchedout as well.  But more examples of
both of those latter two would be helpful.
pc>|83