  The concept of Logical Entropy, $S_L = 1- \sum_{i=1}^n p_i^2$, where the
$p_i$ are normalized probabilities, was introduced by David Ellerman in a
series of recent papers. Although the mathematical formula itself is not new,
Ellerman provided a sound probabilistic interpretation of $S_L$ as a measure of
the distinctions of a partition on a given set. The same formula comes across
as a useful definition of entropy in quantum mechanics, where it is linked to
the notion of purity of a quantum state. The quadratic form of the logical
entropy lends itself to a generalization of the probabilities that include
negative values, an idea that goes back to Feynman and Wigner. Here, we analyze
and reinterpret negative probabilities in the light of the concept of logical
entropy. Several intriguing quantum-like properties of the logical entropy are
derived and discussed in finite dimensional spaces. For infinite-dimensional
spaces (continuum), we show that, under the sole hypothesis that the logical
entropy and the total probability are preserved in time, one obtains an
evolution equation for the probability density that is basically identical to
the quantum evolution of the Wigner function in phase space, at least when one
considers only the momentum variable. This result suggest that the logical
entropy plays a profound role in establishing the peculiar rules of quantum
physics.
