Re: quantifiers

[Iain Alexander <ia@stryx.demon.co.uk>]

In a separate post I've made the point that we are approaching
this subject from completely different directions, and that
what we probably need is to thrash out some way of allowing
our different cultural preconceptions to become explicit and
thus coexist.  Here I focus on details.

In message  <9509072350.aa18293@punt3.demon.co.uk> pcliffje@crl.com writes:

> 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.

Of course, from my point of view, the alternative you suggest is pretty
implausible, since a proposition like (3x) (3y) F(x,y) cannot be
understood by breaking it down into its component parts, e.g. as (3x) G(x),
where G(x) is (3y) F(x,y), but can only be interpreted as a whole.

When I see something like X Y Z, I expect it to be either
left-associative, equivalent to (X Y) Z as would be the case
in combinatory logic, or right-associative, equivalent to
X (Y Z), which is probably the more common of the two.
(I'm assuming here that X and Y are each some sort of
function or operator, ignoring e.g. the case where Y is an
infix operator between two arguments)

> 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."

The point is that we were discussing multiple quantifiers,
where the F in Ax(Fx => x=... ) is itself defined as a numerically
quantified expression, whose leading particular quantifier thus
comes within the scope of that universal.  You obviously don't
consider this situation to arise in the case of _consecutive_
numerics, and it is not clear whether you have some alternative
means of expressing that situation.

>         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.

I'm not convinced by this.  I can't help feeling that the
overzealous application of this rule-of-thumb results in
the unnecessary replication of natlang confusions of
different concepts.  Just because "lions" in English can
mean a number of different things (e.g. {lo'e cinfo}:  the typical
lion, {[su'o] loi cinfo}:  some of the mass of lions, {ro cinfo}:
all lions), doesn't mean that we have to define a concept of
"mass" which magically covers all the possibilities.  We can
and should make many of these distinctions explicit.  A rule-of-thumb
is a heuristic, to be considered as perhaps our first choice,
but examined critically and discounted if it does not make sense.
"Lions live in America" means simply that some of the mass of
lions live there, not that the whole mass does so.

>
> 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.

Well, we have different opinions about what is clearer, but then
we have different preconceptions about what a restricted universal
quantifier means, so I guess that's not surprising.

>
> 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.

Well, I'm reasonably confident that what we have here is a
compound predicate applied to the initial term, so that the
nagi'e is inside the scope of the quantifier.
(At first I assumed we had selbri connection (rather than the
actual bridi-tail connection), which would have been even more
clearly thus.)

> xorxes:
> 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.

That may be the case, but think that should be a matter for
the pragmatics rather than the logic.

> 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.

At first glance, it may not be obvious what such a value should be.
But it turns out that the practical way to define the minimum of
an empty set is the maximum possible value, and conversely for
the maximum.  This is a fairly well-known trick where I come from.
It appears counter-intuitive when you first come across it, but
it works, and you soon get used to it.

> 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,

This only holds if the set is non-empty, in which case it is
trivial.  Otherwise, see above.

> 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.

In your culture, perhaps, but not in mine.

>         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.

I think this is confusing pragmatics and logic.

> (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.)

The fact that we tend to misread certain constructions such as
this emphasises the need to have clear, simple, consistent rules
for interpreting them, in this case by converting into prenex
form.  Then, while you are learning the language you can do
occasional spot checks on the meaning of various things, by
converting them to other equivalent forms, until it becomes
second nature.

Unfortunately, we appear to have different preconceptions
of what the prenex forms mean.

> 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.

As I say above, in the case of interest, F is itself a quantified
expression.

> 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.

I've obviously lost the thread completely on this one.
What's a "multiply referring expression", that you don't
think we've covered.  You've covered {la bab. e la djan. e la haris.}
and {le ci nanmu} yourself above, so it's neither of those.

> 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.

The common wisdom arises (to the best of my knowledge) from exporting
the quantifiers to the prenex, where they are nested, and the later
is therefore presumed a function of the earlier.  But the fact that
consecutive universals _do_ commute is a _theorem_, and in this
case I cannot think of any interpretation which makes the
apparent dependence of the later on the earlier into a real one.

> 1) it does not deal with restricted quantifiers and so takes
> the quantifier to be defined by its entire scope

I was trying to keep it as simple as possible.  I don't see any
problem in principle with adding in the restrictions.

> 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).

If your question is how to say it in Lojban, my preferred solution
at the moment would be an explicit {ro}

        ro ci nanmu cu rapypencu ro ci gerku

which would be equivalent to

        ro lo ci lo nanmu cu rapypencu ro lo ci lo gerku

> 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,

Unfortunately, nobody told us that. :-)

> 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.

I'm sorry, this appears to have been a red herring, since it's
talking about unrestricted universals - I can't find a definition
of restricted quantifiers in H&C.  The problematic case is of
course that when S is empty, (Ax e S) F(x) is false if the
universal has existential import, whereas ! (Ex e S) ! F(x)
is true.  (Yes, ! is negation.)

> 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.

There is certainly at least one way to get a universal quantifier
with explicit existential import - {ro lo su'o broda} (restricted)
or {ro lo su'o da} (unrestricted).  It's not perhaps quite as snappy
as we might wish, but I think we need {ro PA broda} to mean
{ro lo PA lo broda} as above.
--
Iain Alexander                    ia@stryx.demon.co.uk
                    I.Alexander@bra0125.wins.icl.co.uk