  Square roots of probabilities appear in several contexts, which suggests that
they are somehow more fundamental than probabilities. Square roots of
probabilities appear in expressions of the Fisher-Rao Metric and the
Hellinger-Bhattacharyya distance. They also come into play in Quantum Mechanics
via the Born rule where probabilities are found by taking the squared modulus
of the quantum amplitude. Why should this be the case and why do these square
roots not arise in the various formulations of probability theory?
  In this short, inconclusive exploration, I consider quantifying a logical
statement with a vector defined by a set of components each quantifying one of
the atomic statements defining the hypothesis space. I show that conditional
probabilities (bi-valuations), such as $P(x|y)$, can be written as the dot
product of the two vectors quantifying the logical statements $x$ and $y$ each
normalized with respect to the vector quantifying the conditional $y$. The
components of the vectors are proportional to the square root of the
probability. As a result, this formulation is shown to be consistent with a
concept of orthogonality applied to the set of mutually exclusive atomic
statements such that the sum rule is represented as the sum of the squares of
the square roots of probability.
