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"Most" again

A number of people have replied to my comments on "most". I want to clear up a
few points here and throw in a few things just for interest's sake, and then
leave it at that for a while (before the Lab bans me for using too much
bandwidth!). Besides, as lojbab says:

> I'll be running this topic by pc (John Parks-Clifford) for a more authoritative
> opinion on the logical aspects.

and it would be nice to have a "real logician" look at this. After all, this
work is no more than the one of the cornerstones of my PhD research, and so
can scarcely be considered authoritative (:-).

(DE = david elworthy, DM = Dave Matuszek, JC = John Cowan, RC = Robert
Chassell)



DE> First, to restate the claim in a more formal way:
DE>  no quantifier with the same meaning as the English word "most" can be
DE> defined in first order logic, i.e. a logic in which there is
DE> quantification over individuals drawn from some (finite or infinite)
DE> universe.

DM>The first comment/question is, does this really cause any problems for
DM>Lojban?  In other words, though Lojban certainly tries to be logical,
DM>I don't believe it tries to mirror first order logic, does it?  FOL is
DM>certainly powerful, but equally certainly it is provably inadequate
DM>for many tasks.

Certainly: my comments arose from John Cowan saying:
JC> Currently, "most" is a number, like "two".
All of my reply is really an attempt to make sure that in giving a semantics
to Lojban, we avoid some of the pitfalls of adopting a logic which is not
expressive enough. I'm not so much concerned with the content of the language
itself. I say this in response to

RC> Anyhow, Lojban does have numbers and counting predicates.  What this
RC> tells me is that if you want a lojban computer to handle most, you
RC> will need to program the computer to compute (or at least, give it the
RC> ability to pair and compare).
Which is exactly what I was meaning to get at.


To continue,

DE> As an aside, it is a standard result that if a quantifier can be
DE> defined in first order logic, then it is reducible to some combination
DE> of "All" and "Exists" (which I will write here as A and E). For
DE> example, the derived quantifier "two x" can be reduced to "Ex.Ey.x /=
DE> y" (where /= is "not equal").  So we can translate "Two men walk" as
DE> "two x.(man(x) & walk(x))", i.e.  "Ex.Ey.(x /= y & man(x) & man(y) &
DE> walk(x) & walk(y))".

DM> This is a little puzzling; it sounds more like this is a definition of
DM> what FOL is.  You seem to be saying that if a quantifier can be
DM> defined in FOL, then it can be defined in terms of the concepts
DM> already in FOL.  This seems more of a truism than a "result" (which
DM> implies to me that it is somehow provable).

I was being a bit vague when I wrote this paragraph: the point I was really
trying to get across was that you can't just arbitrarily invent new
quantifiers and glue them on top of first order logic - you have also to show
that you can give them the semantics you need. In papers on quantification,
you occasionally find people just throwing in things like "Most x.F(x)"
without specifying how they will interpret this. The point of the mail was
really to give an informal sketch of why you can't give a semantics to most in
the same way that you can for "all" and "every". The formal proof is in
Barwise & Cooper.

DM> My problem came in attempting to show that two sets were equal in size
DM> by forming a 1-1 mapping between them  ... 
DM> Is this basically what the proof is about?

Yes: this is very similar to Barwise & Cooper's method.

DE> One solution which is widely used is to adopt what are called
DE> "generalized quantifiers". Now, instead of translating a sentence into
DE> a quantifier over a variable, and a formula which uses that variable,
DE> we express the quantifier as a relation between two sets. Thus
DE> ...
DE> English	Most men walk.
DE> GQ	The intersection of the set of men and the set of walkers contains at
DE> 	least half as many members as the set of men.

DM> While I like the notion of generalized quantifiers, I still don't see
DM> how things like "at least half" can be expressed without bringing in
DM> the same sorts of problems.

There are several ways of stating GQs. The one which in some sense is most
fundamental (though not necessarily the clearest) is as a set of sets. To
interpret a sentence of the form NP VP, we translate the NP into a GQ, and the
VP into a set of individuals who do what the VP says. If the VP set is a
member of the GQ from the NP, then the sentence is true. The GQs for "a man",
"every man" and "most men" are:
a man		{X <= U | X ^^ man' != 00}
every man	{X <= U | man' <= X}
most men	{X <= U | card(X ^^ man') >= card(man')/2}
(where U is the universe of individuals, <= is subset, card is set
cardinality, ^^ is intersection, man' is the set of men, != is not equals, and
00 is the empty set). So "most men walk" has the truth condition
 walk' ee {X <= U | card(X ^^ man') >= card(man')/2}

This in turn means that the translation of a determiner is a function from a
set (namely the N in the NP) to a GQ. Compare the FOL approach, where a
determiner is translated into a function from a predicate (the N) to another
set (the VP) to a quantified formula. For example:  (with \ being a lambda,
other symbols as above)

"every"
GQ	\P.{X <= U | P <= X}
FOL	\P.\Q.Ax.[P(x) => Q(x)]

DM> In other words, if FOL quits being FOL when we introduce quantifiers
DM> like "most" or "same number of," then neither can we arbitrarily throw
DM> in new generalized quantifiers.  So:  there must be a "standard set"
DM> of generalized quantifiers.  What is it?

If you mean by this, is there a fixed list of GQs, the answer is that there
isn't one. In fact, this is one of the advantages of GQ theory: that you can
always invent a new one to cope with complex determiners (such as "at least
three but no more than seventeen or alternatively exactly twenty-nine").
However, it is claimed that *all* GQs encountered in natural language satisfy
certain properties, namely (if D is a determiner, and A and B sets)

Extension: if DAB holds (i.e. B ee DA) in universe U, then it also holds in
universe U', where U <= U'.

Conservativity: if DAB holds in a universe U, then DA(A^^B) also holds in U.

Isomorphy: if DAB holds in U, and f is a bijection from U to another universe
U', then Df(A)f(B) holds in U'.

A consequence of these properties is that the truth of DAB depends purely on
card(A) and card(A^^B).

Note that there are constructs in natural languages which at first sight
appear not to be expressible in GQs with these constraints. So far, every one
has been explained away somehow (:-). Examples are "only" and "mainly", as in
"only/mainly donkeys neigh", which are explained by claiming that they are not
really determiners. I think a number of the non-exact quantifiers mentioned by
Lojbab will come into this category.


-- david elworthy