This paper studies a family of
monotonic extensions
of first-order logic which we call modulated logics,
constructed by
extending classical logic through generalized quantifiers called modulated
quantifiers. We give an uniform treatment of modulated logics,
obtaining
some general results in model theory. Besides carefully reviewing the
“Logic
of Ultrafilters” and the “Logic of Most”, two new monotonic logical
systems
are introduced here: the “Logic of Many” and the “Logic of Ubiquity”,
which
formalize inductive assertions of the kind “many” and “almost
everywhere”
through new modulated quantifiers and, respectively. Although the
notion
of “most” can be captured by means of a modulated quantifier
semantically
interpreted by cardinal measure on sets of evidences, it is proven that
this
system, although sound, cannot be complete if checked against the
intended
model. This justifies the interest on a purely qualitative approach to
this
kind of quantification, what is guaranteed by interpreting the
modulated
quantifiers, respectively, as families of upper closed sets and
pseudo-topologies. Modulated logics can be used to provide
alternative foundations for fuzzy concepts and fuzzy reasoning, for
reasoning on social choice theory, and for
gaining a new regard on certain problems in philosophy of science.
