The close relation between
quantification and modality has been noted before. Melvin Fitting & Richard
Mendelsohn, First Order Modal Logic,
1998, p. 108, state "The modal operators are like disguised
quantifiers". Arnold Koslow, A
Structuralist Theory of Logic, 1992, p. 312, states "It seems to be no
accident that the quantifiers look very much like modals". In this
presentation, I propose, following Koslow, that the
first-order universal and existential quantifiers qualify as necessity and
possibility modals respectively, but that the favor is not returned --
the standard modals are not quantifiers.
To
show this, I first work through some simple examples of alethic (only analytic
propositions are necessary) and epistemic (analytic and known synthetic
propositions are necessary) modals. I then take up a simple example involving
universal and existential quantifiers over one-place predicates that
demonstrates why these quantifiers are modal, and how they differ from alethic
and epistemic modals.
Next
I turn to the question whether there are any other quantificational modals, and
show that there are: In a domain with infinitely many entities, such as
positive integers, to quantify over, 'for all but finitely many' is a necessity
modal and 'for infinitely many' is a possibility modal.
However,
other quantifiers, such as 'for all but at most one', 'for at least two', 'for
most', 'for many', and many others, fail to qualify as modals, but come
sufficiently close that they may be called 'quasimodals'; the ones that
"count down" from the universal quantifier (like 'for all but at most
one', 'for most') being quasinecessities and the ones
that "count up" from the existential quantifier (like 'for at least
two', 'for many') being quasipossibilities.
Finally, I ask whether there are any non-quantificational quasimodals, and conclude that there are, including the epistemic quasinecessity 'probably' and quasipossibility 'not improbably'.