Re: quantifiers

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

In message  <9509032158.aa14688@punt3.demon.co.uk> pcliffje@crl.com writes:
...
> pc:
>         Well, if you have a mixture of universal and particular quantifiers,

(I presume your "particular" quantifiers are what I would call "existential".)

> 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.)

> 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.

I'll come back to this later.  (I don't understand the bit about the
compact universally quantified complexes.)

>         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?

...
>         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.

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).

> 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.

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?

...
> 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.

> 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.

This reads like nonsense to me, so I must be misunderstanding something.
Just to be sure we're talking mass not individuals, let's talk about
milk.  If I drink a quantity of milk, then I don't want to claim that
I drink all the milk in the world.  That would be a different claim.
But Jorge's already said this, so we must be talking at cross-purposes.

...
> sos:

(One of these days I'll figure out what you mean by "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).

(Back to numeric quantifiers, at last.)
If this is a _theorem_, then something about

        (3x man)(3y dog) x pats y       (1)

means something very different to you from what it means to me,
either the (3x broda) construction, or the juxtaposition of
quantifiers, or something else.

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.

>
> 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>|83

(Is this just an elaborate smiley?)

--
Iain Alexander                    ia@stryx.demon.co.uk
                    I.Alexander@bra0125.wins.icl.co.uk