Quantifier
The general name for a logical operation that constructs from a predicate a statement characterizing the domain of validity of
. In mathematical logic, the most widely used quantifiers are the universal quantifier
and the existential quantifier
. The statement
means that the domain of validity of
is the same as the domain of values of the variable
. The statement
means that the domain of validity of
is non-empty. If one is interested in the behaviour of the predicate
not on the whole domain of values of
, but only on the part singled out by a predicate
, then one often uses the restricted quantifiers
and
. In this case, the statement
means the same as
, while
has the same meaning as
, where
is the conjunction sign and
is the implication sign.
Comments
The subject of quantifiers nowadays is far more involved than suggested above, since there are many more quantifiers (e.g. game quantifiers, probability quantifiers) than just the two (or four) discussed above.
More generally, the model-theoretic interpretation of an arbitrary "quantifier" (with the same syntactic behaviour as
and
) can (according to A. Mostowski) be given by a mapping associating with each model
a class
of subsets of
. Then one stipulates as a truth-definition for
that, e.g., a sentence
holds in
if and only if the set
is in
. Thus, with the existential quantifier
is associated the class of non-empty subsets of
and with the universal quantifier
is associated the class
. However, there are many more possible quantifiers, e.g. given by
,
(the Chang quantifier),
, etc. This set-up can be generalized to "quantifiers" binding more than one variable occurring in more than one formula (example: the equi-cardinality quantifier
binding two variables
and
in two formulas
and
, yielding the formula
, which is interpreted by
). Even more general is the Lindström quantifier. And each quantifier has its own logic.
References
[a1] | J. Barwise (ed.) S. Feferman (ed.) , Model-theoretic logics , Springer (1985) |
Quantifier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantifier&oldid=14021