  The notion of a partition on a set is mathematically dual to the notion of a
subset of a set, so there is a logic of partitions dual to Boole's logic of
subsets (Boolean subset logic is usually mis-specified as the special case of
"propositional" logic). The notion of an element of a subset has as its dual
the notion of a distinction of a partition (a pair of elements in different
blocks). Boole developed finite logical probability as the normalized counting
measure on elements of subsets so there is a dual concept of logical entropy
which is the normalized counting measure on distinctions of partitions. Thus
the logical notion of information is a measure of distinctions. Classical
logical entropy also extends naturally to the notion of quantum logical entropy
which provides a more natural and informative alternative to the usual Von
Neumann entropy in quantum information theory. The quantum logical entropy of a
post-measurement density matrix has the simple interpretation as the
probability that two independent measurements of the same state using the same
observable will have different results. The main result of the paper is that
this increase in quantum logical entropy due to a projective measurement of a
pure state is the sum of the absolute squares of the off-diagonal entries
(`coherences') of the pure state density matrix that are zeroed (decohered) by
the measurement, i.e., the measure of the distinctions (`decoherences') created
by the measurement. The Von Neumann entropy provides no such analysis of
measurement. That result is also classically modelled using ordinary partitions
and density matrices for such partitions in the pedagogical model of QM/Sets
with point probabilities--which for a fixed basis is just sampling a real r.v.
in classical finite probability theory.
