Re: semantics of "most"

[David Elworthy <cbmvax!uunet!computer-lab.cambridge.ac.uk!David.Elworthy>]

I wrote:

>> Do you know how
>> Lojban handles the semantics of "most"? If it is done by a quantifier similar
>> to "exists" or "all", be warned that it can be shown this will not work! I'll
>> forward details if you're interested.

John Cowan replied:

> Please do!  Currently, "most" is a number, like "two".


I'll try to give an informal description of the problem. If you want to have a
formal proof of why you can't express "most" in this way, see Barwise &
Cooper: reference below (theorem C12).


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

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

Secondly, I should make it clear what I mean by "most". A paraphrase is "more
than half of", i.e. "Most men walk" is equivalent to
(1) "more than half of the individuals who are men are individuals who walk."


Now, look at how we translate sentences into first order logic and interpret
the resulting formulae:
English	A man walks.
FOL	Ex.(man(x) & walk(x))
Interp	Examine the universe U, and see if you can find at least one
	individual i from U for which man(i) and walk(i) both hold.	

English	Every man walk.
FOL	Ax.(man(x) => walk(x))
Interp	Examine the universe U, and check that for every individual i from U,
	if man(i) holds, then walk(i) holds. (And if man(i) does not hold, we
	don't care if walk(i) does or not.)

English	Two men walk.
FOL	2x.(man(x) & walk(x))
Interp	Examine the universe U, and see if you can find at least two different
	individuals i from U for which man(i) and walk(i) both hold.	

And if we try it for "most":
English	Most men walk.
??FOL	Most x.(man(x) c walk(x)), where c is a logical connective.
Interp	Examine the universe U, and see if more than half the individuals i in
	U are such that the fact of i being a man is related by c to the fact
	of walking.
The question is, then, can we find a connective c, which means that this
interpretation is equivalent to (1)? Try the most likely candidates: "&" and
"=>".
&	Examine the universe U, and see if more than half the individuals i in
	U are such that both man(i) and walk(i) hold. Paraphrase: "more than
	half the individuals in the universe are men who walk", which is not
	the same as (1).
=>	Examine the universe U, and see if more than half the individuals i in
	U are such that if man(i) holds, then walk(i) holds. Paraphrase: "more
	than half the individuals in the universe are either men who walk, or
	are not men at all", which is not the same as (1).
... and similarly for all other possible connectives.


One solution which is widely used is to adopt what are called "generalized
quantifiers". Now, instead of translating a sentence into a quantifier over a
variable, and a formula which uses that variable, we express the quantifier as
a relation between two sets. Thus
English	A man walks.
GQ	The set of men has a non-empty intersection with the set of walkers.

English	Every man walks.
GQ	The set of men is a subset of the set of walkers.

English	Two men walk.
GQ	The intersection of the set of men and the set of walkers contains at
	least two individuals.

English	Most men walk.
GQ	The intersection of the set of men and the set of walkers contains at
	least half as many members as the set of men.


I don't know if I've explained this terribly clearly. I haven't found any
reference which states the problem with "most" (and some other quantifiers) in
an intuitive way, but if you want to follow up the formal details, and see how
GQ theory works, have a look at:

Jon Barwise and Robin Cooper. Generalized Quantifiers and Natural Language.
Linguistics and Philosophy Vol.4 (1981), pp. 159-219.

Jan van Eijck. Quantification. In A. von Stechow and D. Wunderlich (eds.)
Handbook of Semantics. (The copy of the paper I have says "to appear", so I
can't tell you the date or publisher, but Reidel or Kluwer would be a fair
bet.)

-- david elworthy