Quantifier: Qualitative term that states the proportion of
something in some domain. In
logic the standard quantifiers are
"for all" or "for every" or "for each" and "for some" or "there is" or
"there are", and in standard logic the two are interdefinable, in that
"for all x Ax" is true precisely if "it is not true that for some x not
Ax" and "for some x Ax" is true precisely if "it is not true that for
all x not Ax".
There are many subtle issues involved in the logic of quantifiers,
which was first thought of and created by Frege and, independently,
Peirce and Mitchell.
Five problems relating to quantifiers are as follows.
First, there is the problem that "there is an x such that Fx" and
"for some x, Fx" are not intuitively the same, as shown by "Some Greek
Gods dallied with human damsels, though  of course  there are no Greek Gods".
This can be rectified in various ways, e.g. by a predicate of
existence, or by laying down that a thing exists iff some thing is
identical to it, as in "(Ex)(x=Napoleon) & ~(Ex)(x=Jupiter)". Both remedies have their own problems.
Second, there is the problem of quantifying
subjects and
predicates. Standard logic and
Set Theory has only quantified
subjects and not quantified predicates, which is to say it has a
First Order Logic.
The problem is that it is quite intuitive in
Natural Language to
quantify predicates ("You don't have all properties Einstein had") and
that quantified predicates make it easier to deal with quite a few
logical and mathematical concepts, but that, on the other hand, there
are logical problems involved in quantified predicates.
One basic reason for this is that quantified predicates and relations
assert far more than constant terms with quantified subjects only, and
that this easily leads to paradoxes and problems. Logics with quantified
predicates are known as Higher
Order Logic a.k.a. HOL.
There are now good introductions to
HOL, but it also has
unresolved problems.
Third, Henkin showed that not all things can be said precisely and
uniquely using formulas with sequences of quantifier prefixes. This is
known as the problem of Branching Quantifiers, which is essentially due
to dependencies between universally and particularly quantified terms.
To accomodate such quantifiers, some things have to be adjusted or
assumed.
Fourth, there are other quantifiers than "for all x", "for some x"
and "there is an x such that", such as "for most x" and "for a few x",
and also quantifying expressions in temporal logic, as "for all times",
"for some times" etc.
Fifth, there are conceptions of mathematics and logic (intuitionism
and constructivism) that insist that "there is an x such that Fx" is
true and sensible only if one can show how to construe something that is
F, and that one cannot merely hypostasize this or conclude if from a
reductio ad absurdum.
The problem with this is that most mathematicians have liked
constructive approaches, but found them insufficient for some things
they wanted to express, and that at least sometimes such
nonconstructive statements seem both sensible and true, as in "there
are particular numbers no one has ever thought of, specifically".
Even so, it has been shown, e.g. by Errett Bishop, that far more can
be said and proved constructively than most mathematicians would have
guessed  and the advantage of constructive proofs is that involve fewer
assumptions than nonconstructive proofs.
In general terms:
The standard quantifiers have been most researched and studied,
mostly by mathematicians and logicians, but more recently also by
linguists, computer scientists and even philosophers, and there are,
next to the standard approach to quantification, systems for alternative
quantifiers, and temporal logics with temporal quantifiers, and there
also are different approaches to quantifiers, notably in intuitionist
and constructive mathematics.
And in any case, the logical notion of a quantifier is an example of a
very neat and useful abstraction thought up by logical mathematicians in
the 19th Century,
of which the clarification of their logic  the rules and assumptions
involved in their logical use  was most useful for both mathematics and
logic.
